Matrix and Basic Operations on Matrices
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- What is a Matrix
- What are the Special Matrices
- Times Scalar, Add, Subtract
- Multiplication I
- Multiplication II
- Multiplication III
- Transpose
- Trace
- Exercise 1
- Exercise 2
- Exercise 3 Parts 1-4
- Exercise 3 Part 5
- Exercise 3 Parts 6-7
- Exercise 3 Part 8
- Exercise 3 Part 9
- Exercise 3 Part 10
- Exercise 4
- Exercise 5 Part 1
- Exercise 5 Part 2
- Exercise 5 Part 3
- Exercise 5 Part 4
- Exercise 5 Part 5
- Exercise 6
- Exercise 6 Part a
- Exercise 6 Part b
- Exercise 6 Part c
- Exercise 6 Part d
- Exercise 6 Part e

Matrix Inverse and its Applications
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Properties of the Matrix Inverse
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Elementary Matrices and LU Decomposition
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{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Matrix and Basic Operations on Matrices","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"What is a Matrix","Duration":"4m 29s","ChapterTopicVideoID":9545,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9545.jpeg","UploadDate":"2017-07-26T08:24:20.7470000","DurationForVideoObject":"PT4M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.915","Text":"In this clip, I\u0027ll be talking about matrices."},{"Start":"00:03.915 ","End":"00:06.795","Text":"You\u0027ve probably seen them before."},{"Start":"00:06.795 ","End":"00:10.980","Text":"We briefly mentioned them in systems of linear equations,"},{"Start":"00:10.980 ","End":"00:17.170","Text":"but I\u0027m going to start from scratch assuming that you don\u0027t know what they are."},{"Start":"00:17.330 ","End":"00:22.020","Text":"I\u0027ll give you now a preview of the sections."},{"Start":"00:22.020 ","End":"00:24.360","Text":"The first clip we\u0027ll discuss,"},{"Start":"00:24.360 ","End":"00:25.665","Text":"what is a matrix?"},{"Start":"00:25.665 ","End":"00:30.240","Text":"Definition. Note that the plural of matrix is matrices,"},{"Start":"00:30.240 ","End":"00:32.595","Text":"that because it comes from Latin."},{"Start":"00:32.595 ","End":"00:39.240","Text":"It actually means a womb in Latin and has nothing to do with the movie, The Matrix."},{"Start":"00:39.240 ","End":"00:47.290","Text":"Now, the second clip discusses certain special matrices."},{"Start":"00:47.290 ","End":"00:52.250","Text":"The third discusses arithmetic operations on matrices such as addition,"},{"Start":"00:52.250 ","End":"00:56.870","Text":"subtraction, multiplication by a constant, and so on."},{"Start":"00:56.870 ","End":"01:00.520","Text":"Then there\u0027s something called the transpose of a matrix,"},{"Start":"01:00.520 ","End":"01:04.230","Text":"and there\u0027s a trace of a matrix."},{"Start":"01:04.230 ","End":"01:09.305","Text":"After that, the next series will be"},{"Start":"01:09.305 ","End":"01:15.965","Text":"about something called the inverse of a matrix or this the inverse matrix."},{"Start":"01:15.965 ","End":"01:19.130","Text":"Anyway, I don\u0027t want to get into that."},{"Start":"01:19.130 ","End":"01:21.865","Text":"Let\u0027s just get started."},{"Start":"01:21.865 ","End":"01:23.660","Text":"What is a matrix?"},{"Start":"01:23.660 ","End":"01:26.870","Text":"Let me write down a definition,"},{"Start":"01:26.870 ","End":"01:30.575","Text":"which will seem a little bit abstract until we see an example."},{"Start":"01:30.575 ","End":"01:32.855","Text":"Anyway, a matrix is a rectangular,"},{"Start":"01:32.855 ","End":"01:34.010","Text":"a 2D array,"},{"Start":"01:34.010 ","End":"01:38.380","Text":"a table of elements sometimes called entries."},{"Start":"01:38.380 ","End":"01:42.600","Text":"I\u0027ll use the terms both; the element or entry."},{"Start":"01:43.090 ","End":"01:47.850","Text":"Each of them can be a number, a symbol,"},{"Start":"01:47.850 ","End":"01:52.355","Text":"or an expression, and they\u0027re arranged in rows and columns,"},{"Start":"01:52.355 ","End":"01:57.365","Text":"and it\u0027s enclosed in square brackets, for example."},{"Start":"01:57.365 ","End":"02:00.070","Text":"Here\u0027s a matrix of numbers,"},{"Start":"02:00.070 ","End":"02:01.305","Text":"there are the brackets,"},{"Start":"02:01.305 ","End":"02:04.890","Text":"there\u0027s 3 rows, 4 columns,"},{"Start":"02:04.890 ","End":"02:07.410","Text":"and that\u0027s the array."},{"Start":"02:07.410 ","End":"02:11.270","Text":"Now, the matrix is an entity,"},{"Start":"02:11.270 ","End":"02:13.040","Text":"can be viewed as a single hole,"},{"Start":"02:13.040 ","End":"02:14.930","Text":"and we might sometimes want to give it a name,"},{"Start":"02:14.930 ","End":"02:16.130","Text":"but we often do."},{"Start":"02:16.130 ","End":"02:19.960","Text":"It\u0027s customary to use capital letters,"},{"Start":"02:19.960 ","End":"02:24.874","Text":"such as capital letter A equals this matrix."},{"Start":"02:24.874 ","End":"02:27.650","Text":"This course will often use A, B,"},{"Start":"02:27.650 ","End":"02:33.165","Text":"C as the names of matrices."},{"Start":"02:33.165 ","End":"02:37.235","Text":"Now, sometimes you want to emphasize"},{"Start":"02:37.235 ","End":"02:40.630","Text":"the number of rows and columns that this has 3 rows, 4 columns."},{"Start":"02:40.630 ","End":"02:44.580","Text":"So we would write this matrix as follows."},{"Start":"02:44.580 ","End":"02:47.675","Text":"Here we put 3 times 4,"},{"Start":"02:47.675 ","End":"02:52.500","Text":"and that\u0027s not an x, that\u0027s a cross."},{"Start":"02:52.940 ","End":"02:58.295","Text":"Now, sometimes you want to refer to a specific element or entry."},{"Start":"02:58.295 ","End":"03:03.020","Text":"You might want the third column,"},{"Start":"03:03.020 ","End":"03:06.015","Text":"second row element, this 1."},{"Start":"03:06.015 ","End":"03:08.505","Text":"We want a notation for that."},{"Start":"03:08.505 ","End":"03:12.095","Text":"We use a lowercase,"},{"Start":"03:12.095 ","End":"03:15.485","Text":"usually the same lowercase letter a as this."},{"Start":"03:15.485 ","End":"03:18.515","Text":"This is big A, this is little a, and 2 subscripts."},{"Start":"03:18.515 ","End":"03:23.545","Text":"I is the row and j is the column."},{"Start":"03:23.545 ","End":"03:28.260","Text":"In this example, if you said little a 1,"},{"Start":"03:28.260 ","End":"03:31.605","Text":"2, it means first row, second column."},{"Start":"03:31.605 ","End":"03:33.155","Text":"That\u0027s the first row."},{"Start":"03:33.155 ","End":"03:35.410","Text":"That\u0027s the second column is 24."},{"Start":"03:35.410 ","End":"03:37.865","Text":"Second row, first column,"},{"Start":"03:37.865 ","End":"03:39.700","Text":"1.4, second row,"},{"Start":"03:39.700 ","End":"03:42.015","Text":"third column is that 4,"},{"Start":"03:42.015 ","End":"03:44.670","Text":"third row is here,"},{"Start":"03:44.670 ","End":"03:47.325","Text":"the first column here."},{"Start":"03:47.325 ","End":"03:50.430","Text":"Third row, fourth column, a 100."},{"Start":"03:50.430 ","End":"03:52.990","Text":"I think you get the idea."},{"Start":"03:55.430 ","End":"03:58.710","Text":"Let me give another example."},{"Start":"03:58.710 ","End":"04:00.930","Text":"We\u0027ll use the letter C."},{"Start":"04:00.930 ","End":"04:06.480","Text":"C will be this matrix which has 2 rows and 5 columns."},{"Start":"04:06.480 ","End":"04:09.195","Text":"For example, C 1, 4,"},{"Start":"04:09.195 ","End":"04:11.430","Text":"first row, 1, 2, 3,"},{"Start":"04:11.430 ","End":"04:13.690","Text":"4th column, cosine x."},{"Start":"04:13.690 ","End":"04:18.875","Text":"This is a matrix with expressions not numbers or mixed."},{"Start":"04:18.875 ","End":"04:23.760","Text":"Second row, third column is that,"},{"Start":"04:23.760 ","End":"04:27.525","Text":"and second row, fifth column is that."},{"Start":"04:27.525 ","End":"04:30.430","Text":"I think you\u0027ve got the hang of that."}],"ID":10492},{"Watched":false,"Name":"What are the Special Matrices","Duration":"7m 24s","ChapterTopicVideoID":9546,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9546.jpeg","UploadDate":"2017-07-26T08:24:57.0330000","DurationForVideoObject":"PT7M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"We\u0027ve introduced the concept of a matrix and now I\u0027d like to"},{"Start":"00:03.510 ","End":"00:07.410","Text":"discuss some special kinds of matrices."},{"Start":"00:07.410 ","End":"00:13.605","Text":"A vector is a matrix which has just 1 row or just 1 column."},{"Start":"00:13.605 ","End":"00:18.120","Text":"For example, here\u0027s a matrix with just 1 row,"},{"Start":"00:18.120 ","End":"00:19.365","Text":"so it\u0027s a vector,"},{"Start":"00:19.365 ","End":"00:22.455","Text":"and here is matrix with just 1 column, so that\u0027s the vector."},{"Start":"00:22.455 ","End":"00:26.025","Text":"Sometimes we say this is a row vector or column vector."},{"Start":"00:26.025 ","End":"00:29.000","Text":"As for notation, we normally,"},{"Start":"00:29.000 ","End":"00:32.300","Text":"for the matrices use a capital letter,"},{"Start":"00:32.300 ","End":"00:34.025","Text":"but for vectors,"},{"Start":"00:34.025 ","End":"00:39.860","Text":"we use a lower case and an underline."},{"Start":"00:39.860 ","End":"00:45.980","Text":"For example, this 1 might be called u underline and this 1 might be called v underline."},{"Start":"00:45.980 ","End":"00:47.795","Text":"There are other notations,"},{"Start":"00:47.795 ","End":"00:50.110","Text":"but this is what we\u0027ll use."},{"Start":"00:50.110 ","End":"00:52.100","Text":"Next, the square matrix,"},{"Start":"00:52.100 ","End":"00:54.305","Text":"you can probably guess what it means."},{"Start":"00:54.305 ","End":"00:59.270","Text":"Same number of rows and columns like this matrix B,"},{"Start":"00:59.270 ","End":"01:04.925","Text":"which has 3 rows and 3 columns, same number."},{"Start":"01:04.925 ","End":"01:08.345","Text":"Next is the 0 matrix,"},{"Start":"01:08.345 ","End":"01:11.690","Text":"which is a matrix where every entry,"},{"Start":"01:11.690 ","End":"01:14.120","Text":"every element is 0."},{"Start":"01:14.120 ","End":"01:17.460","Text":"They come in different sizes, these 0 matrices."},{"Start":"01:17.460 ","End":"01:20.825","Text":"This for example, is a 3 by 3,"},{"Start":"01:20.825 ","End":"01:26.965","Text":"0 matrix, whereas this is a 3 by 4, 0 matrix."},{"Start":"01:26.965 ","End":"01:34.415","Text":"The next special kind of matrix will be the identity matrix,"},{"Start":"01:34.415 ","End":"01:36.065","Text":"which is a square matrix."},{"Start":"01:36.065 ","End":"01:39.230","Text":"Must be square because I\u0027m going to"},{"Start":"01:39.230 ","End":"01:43.010","Text":"talk about the diagonal and there won\u0027t be a diagonal unless it\u0027s squared."},{"Start":"01:43.010 ","End":"01:49.735","Text":"Where the entries on the main diagonal are all 1 and the rest are 0."},{"Start":"01:49.735 ","End":"01:54.720","Text":"The notation is capital I, I for identity."},{"Start":"01:54.720 ","End":"01:58.610","Text":"But since it comes in different sizes, it\u0027s a square matrix."},{"Start":"01:58.610 ","End":"01:59.900","Text":"It could be 3 by 3,"},{"Start":"01:59.900 ","End":"02:01.250","Text":"it could be 2 by 2,"},{"Start":"02:01.250 ","End":"02:03.125","Text":"it could be 4 by 4."},{"Start":"02:03.125 ","End":"02:05.195","Text":"Sometimes instead of just I,"},{"Start":"02:05.195 ","End":"02:08.240","Text":"we might say I with a subscript n,"},{"Start":"02:08.240 ","End":"02:10.500","Text":"which in this case is 3,"},{"Start":"02:10.500 ","End":"02:14.670","Text":"here it\u0027s 2, here it\u0027s 4, and so on."},{"Start":"02:14.670 ","End":"02:19.415","Text":"Just want to point out that it\u0027s important that it\u0027s the main diagonal."},{"Start":"02:19.415 ","End":"02:22.535","Text":"This is the main diagonals, so is this,"},{"Start":"02:22.535 ","End":"02:26.690","Text":"and so is this because you notice that in"},{"Start":"02:26.690 ","End":"02:32.230","Text":"square matrices there is another diagonal, this 1 here."},{"Start":"02:33.590 ","End":"02:36.290","Text":"But that\u0027s not the main diagonal,"},{"Start":"02:36.290 ","End":"02:38.900","Text":"that\u0027s the secondary diagonal,"},{"Start":"02:38.900 ","End":"02:42.200","Text":"anti-diagonal, has many names."},{"Start":"02:42.200 ","End":"02:46.280","Text":"We start from the top left and end in the bottom right."},{"Start":"02:46.280 ","End":"02:49.370","Text":"This concept, main diagonals,"},{"Start":"02:49.370 ","End":"02:55.785","Text":"also appears in our next definition an upper triangular matrix."},{"Start":"02:55.785 ","End":"02:57.875","Text":"It\u0027s also a square matrix."},{"Start":"02:57.875 ","End":"03:03.685","Text":"What we want is all the entries below the main diagonal are 0."},{"Start":"03:03.685 ","End":"03:06.540","Text":"This is the main diagonal,"},{"Start":"03:06.540 ","End":"03:08.565","Text":"all zeros below,"},{"Start":"03:08.565 ","End":"03:12.865","Text":"we don\u0027t care what\u0027s on the diagonal and above."},{"Start":"03:12.865 ","End":"03:17.870","Text":"I don\u0027t have to tell you why it\u0027s called an upper triangular matrix,"},{"Start":"03:17.870 ","End":"03:21.140","Text":"I think picture explains itself."},{"Start":"03:21.140 ","End":"03:24.475","Text":"Typical letter will be U for upper."},{"Start":"03:24.475 ","End":"03:26.840","Text":"The next 1, as you might guess,"},{"Start":"03:26.840 ","End":"03:30.690","Text":"will be a lower triangular matrix."},{"Start":"03:32.560 ","End":"03:36.980","Text":"You could probably guess the definition from the upper triangular,"},{"Start":"03:36.980 ","End":"03:38.465","Text":"but I\u0027ll give it to you."},{"Start":"03:38.465 ","End":"03:43.940","Text":"Lower triangular because above the main diagonal it\u0027s 0,"},{"Start":"03:43.940 ","End":"03:47.750","Text":"which means that the shape is also a triangle,"},{"Start":"03:47.750 ","End":"03:52.410","Text":"but it\u0027s the lower part, that\u0027s the triangle."},{"Start":"03:52.870 ","End":"03:55.250","Text":"Let\u0027s see what\u0027s next."},{"Start":"03:55.250 ","End":"04:03.895","Text":"A diagonal matrix where the all the entries are on the diagonal."},{"Start":"04:03.895 ","End":"04:05.650","Text":"I didn\u0027t mentioned, but of course,"},{"Start":"04:05.650 ","End":"04:07.430","Text":"when I put asterisks can mean anything,"},{"Start":"04:07.430 ","End":"04:10.755","Text":"means I don\u0027t care, just some entry."},{"Start":"04:10.755 ","End":"04:12.890","Text":"This is the main diagonal."},{"Start":"04:12.890 ","End":"04:15.200","Text":"Again, it\u0027s the main diagonal,"},{"Start":"04:15.200 ","End":"04:19.600","Text":"don\u0027t get confused because there was another diagonal for square matrices."},{"Start":"04:19.600 ","End":"04:22.775","Text":"Everything outside the diagonal is 0."},{"Start":"04:22.775 ","End":"04:24.650","Text":"Notice just by the way that"},{"Start":"04:24.650 ","End":"04:28.735","Text":"a diagonal matrix is an upper triangular and a lower triangular."},{"Start":"04:28.735 ","End":"04:37.965","Text":"Anyway, continuing the next 1 is a symmetric matrix."},{"Start":"04:37.965 ","End":"04:39.720","Text":"Now the definition,"},{"Start":"04:39.720 ","End":"04:46.490","Text":"square matrix where each ij element is equal to the ji element,"},{"Start":"04:46.490 ","End":"04:52.475","Text":"and I\u0027ll explain that best to give an example."},{"Start":"04:52.475 ","End":"04:55.040","Text":"Here\u0027s an example."},{"Start":"04:55.040 ","End":"04:58.250","Text":"Here\u0027s the main diagonal,"},{"Start":"04:58.250 ","End":"05:01.340","Text":"doesn\u0027t matter what\u0027s on the main diagonal,"},{"Start":"05:01.340 ","End":"05:06.665","Text":"but the elements are mirror images."},{"Start":"05:06.665 ","End":"05:09.245","Text":"If this main diagonal was a mirror,"},{"Start":"05:09.245 ","End":"05:11.870","Text":"it would just cause a reflection."},{"Start":"05:11.870 ","End":"05:15.710","Text":"That\u0027s color-coded like this 2 is a reflection of this 2,"},{"Start":"05:15.710 ","End":"05:17.945","Text":"this 7 reflection of this 7."},{"Start":"05:17.945 ","End":"05:20.090","Text":"Now, the mathematical part,"},{"Start":"05:20.090 ","End":"05:23.555","Text":"the ij element means, for example,"},{"Start":"05:23.555 ","End":"05:29.760","Text":"that if I take i equals 1,"},{"Start":"05:29.760 ","End":"05:31.560","Text":"j equals 3, the 1,"},{"Start":"05:31.560 ","End":"05:32.870","Text":"3 element first row,"},{"Start":"05:32.870 ","End":"05:34.700","Text":"third column is here."},{"Start":"05:34.700 ","End":"05:37.805","Text":"If I then reverse it and take 3, 1,"},{"Start":"05:37.805 ","End":"05:38.990","Text":"it means third row,"},{"Start":"05:38.990 ","End":"05:41.915","Text":"first column, it\u0027s going to be the same."},{"Start":"05:41.915 ","End":"05:45.455","Text":"Another example, if I take 2, 4,"},{"Start":"05:45.455 ","End":"05:49.005","Text":"the 2, 4 element is row 2,"},{"Start":"05:49.005 ","End":"05:51.660","Text":"column 4 is the 7 and the 4,"},{"Start":"05:51.660 ","End":"05:53.580","Text":"2 element is row 4,"},{"Start":"05:53.580 ","End":"05:57.105","Text":"column 2 is the same thing, and so on."},{"Start":"05:57.105 ","End":"06:01.485","Text":"Symbolically, we can say that a_ ij is a_ ji."},{"Start":"06:01.485 ","End":"06:05.910","Text":"For reverse the row on the column, it comes out the same,"},{"Start":"06:05.910 ","End":"06:12.560","Text":"and as I demonstrated this means symmetry about the main diagonal reflection."},{"Start":"06:12.560 ","End":"06:15.215","Text":"Now if we have a symmetric,"},{"Start":"06:15.215 ","End":"06:21.580","Text":"there\u0027s also something called an anti-symmetric matrix."},{"Start":"06:21.580 ","End":"06:26.460","Text":"This time the ij element is not equal to the ji element,"},{"Start":"06:26.460 ","End":"06:29.505","Text":"but to minus the ji element."},{"Start":"06:29.505 ","End":"06:34.570","Text":"This time when we reflect in the diagonal,"},{"Start":"06:34.570 ","End":"06:38.225","Text":"then it becomes negative, the opposite."},{"Start":"06:38.225 ","End":"06:40.850","Text":"Like you see this 2 and the minus 2."},{"Start":"06:40.850 ","End":"06:42.830","Text":"The color-coding should help."},{"Start":"06:42.830 ","End":"06:44.450","Text":"The minus 3 and the 3,"},{"Start":"06:44.450 ","End":"06:49.330","Text":"the minus 7 and the 7 opposite side of the mirror is opposite signs."},{"Start":"06:49.330 ","End":"06:52.185","Text":"The formal definition is this."},{"Start":"06:52.185 ","End":"06:55.235","Text":"Let\u0027s just take another couple of examples."},{"Start":"06:55.235 ","End":"06:58.190","Text":"Let\u0027s say I have the 1,"},{"Start":"06:58.190 ","End":"06:59.390","Text":"3, i is 1,"},{"Start":"06:59.390 ","End":"07:04.200","Text":"j is 3 would mean first row, third column."},{"Start":"07:04.200 ","End":"07:05.760","Text":"If I reverse that and take 3,"},{"Start":"07:05.760 ","End":"07:07.050","Text":"1, it\u0027s third row,"},{"Start":"07:07.050 ","End":"07:10.845","Text":"first column, and there 1 is the minus of the other."},{"Start":"07:10.845 ","End":"07:13.155","Text":"Or second row, fourth,"},{"Start":"07:13.155 ","End":"07:16.305","Text":"column would be this fourth row,"},{"Start":"07:16.305 ","End":"07:20.380","Text":"second column is these minuses of each other."},{"Start":"07:20.510 ","End":"07:25.750","Text":"I\u0027m done with the clip, on special matrices."}],"ID":10493},{"Watched":false,"Name":"Times Scalar, Add, Subtract","Duration":"2m 41s","ChapterTopicVideoID":9547,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9547.jpeg","UploadDate":"2017-07-26T08:25:17.4330000","DurationForVideoObject":"PT2M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this clip, we\u0027re going to discuss arithmetic operations on matrices."},{"Start":"00:04.560 ","End":"00:06.030","Text":"Just like with numbers,"},{"Start":"00:06.030 ","End":"00:07.170","Text":"we can do things with them,"},{"Start":"00:07.170 ","End":"00:09.090","Text":"we can add them, we can subtract them,"},{"Start":"00:09.090 ","End":"00:10.890","Text":"multiply them, and so on."},{"Start":"00:10.890 ","End":"00:14.670","Text":"We want to do certain operations on matrices."},{"Start":"00:14.670 ","End":"00:18.870","Text":"The first of these will be multiplication of a matrix by a scalar."},{"Start":"00:18.870 ","End":"00:20.850","Text":"Scalar means a number,"},{"Start":"00:20.850 ","End":"00:25.665","Text":"as opposed to say a matrix or an algebraic expression, scalar."},{"Start":"00:25.665 ","End":"00:32.070","Text":"What it means is just multiplying each element of the matrix by the scalar."},{"Start":"00:32.070 ","End":"00:34.500","Text":"For example, if this is the matrix,"},{"Start":"00:34.500 ","End":"00:35.710","Text":"it\u0027s a 3 by 3,"},{"Start":"00:35.710 ","End":"00:39.780","Text":"and I multiply it by a scalar 4,"},{"Start":"00:39.780 ","End":"00:43.400","Text":"the 4 is written on the left of the matrix."},{"Start":"00:43.400 ","End":"00:46.580","Text":"Then we take this 4 and multiply each of the elements,"},{"Start":"00:46.580 ","End":"00:48.720","Text":"I guess there\u0027s 9 of them by 4,"},{"Start":"00:48.720 ","End":"00:51.215","Text":"so 4 times minus 1 is minus 4,"},{"Start":"00:51.215 ","End":"00:55.050","Text":"4 times 20 is 80, and so on."},{"Start":"00:55.360 ","End":"00:59.450","Text":"Now, in contrast to future operations that we\u0027ll see,"},{"Start":"00:59.450 ","End":"01:00.890","Text":"this has no restrictions,"},{"Start":"01:00.890 ","End":"01:07.110","Text":"meaning you can multiply any matrix by any scalar, no problem."},{"Start":"01:16.580 ","End":"01:23.670","Text":"In contrast to the addition and subtraction of matrices,"},{"Start":"01:23.920 ","End":"01:27.710","Text":"which will have some restrictions."},{"Start":"01:27.710 ","End":"01:34.320","Text":"Now, the actual operation is performed depending on whether it\u0027s addition or subtraction."},{"Start":"01:34.320 ","End":"01:38.180","Text":"By adding in subtraction each pair of corresponding entries,"},{"Start":"01:38.180 ","End":"01:41.893","Text":"corresponding means in the same positions, row and column"},{"Start":"01:41.893 ","End":"01:44.405","Text":"but in order for this to make sense,"},{"Start":"01:44.405 ","End":"01:50.165","Text":"both matrices that we want to add or subtract have to have the same order or same size,"},{"Start":"01:50.165 ","End":"01:53.094","Text":"meaning same number of rows and columns."},{"Start":"01:53.094 ","End":"01:56.090","Text":"That same size might be 3 by 3,"},{"Start":"01:56.090 ","End":"01:58.310","Text":"in which case we would have this example."},{"Start":"01:58.310 ","End":"02:01.760","Text":"This is 1 matrix, there\u0027s the plus"},{"Start":"02:01.760 ","End":"02:04.250","Text":"but it could be a minus also,"},{"Start":"02:04.250 ","End":"02:05.615","Text":"but in the case of a plus,"},{"Start":"02:05.615 ","End":"02:06.920","Text":"we just take each pair."},{"Start":"02:06.920 ","End":"02:09.440","Text":"For example, suppose I take the second row,"},{"Start":"02:09.440 ","End":"02:12.290","Text":"second column 5, here it\u0027s 11,"},{"Start":"02:12.290 ","End":"02:14.120","Text":"5 plus 11 is 16,"},{"Start":"02:14.120 ","End":"02:15.290","Text":"and so on for the rest of them."},{"Start":"02:15.290 ","End":"02:17.484","Text":"1 and minus 1 is 0."},{"Start":"02:17.484 ","End":"02:22.095","Text":"Or here, this 6 corresponds to this minus 4 gives us 2."},{"Start":"02:22.095 ","End":"02:27.335","Text":"That\u0027s what we mean by pair of corresponding entries,"},{"Start":"02:27.335 ","End":"02:30.165","Text":"just add and if it\u0027s subtraction, then we subtract."},{"Start":"02:30.165 ","End":"02:31.700","Text":"I want to say again,"},{"Start":"02:31.700 ","End":"02:35.075","Text":"you cannot add or subtract matrices with different sizes."},{"Start":"02:35.075 ","End":"02:39.335","Text":"If I have a 2 by 2 and I want to add 2 with 2 by 3,"},{"Start":"02:39.335 ","End":"02:42.480","Text":"it is not defined."}],"ID":10494},{"Watched":false,"Name":"Multiplication I","Duration":"5m 15s","ChapterTopicVideoID":9548,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9548.jpeg","UploadDate":"2017-07-26T08:26:12.9370000","DurationForVideoObject":"PT5M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"Continuing with arithmetic operations on matrices,"},{"Start":"00:03.840 ","End":"00:06.660","Text":"the next 1 is matrix multiplication."},{"Start":"00:06.660 ","End":"00:12.025","Text":"This is much more delicate, it\u0027s complicated."},{"Start":"00:12.025 ","End":"00:14.355","Text":"Don\u0027t worry, we\u0027ll get you through it,"},{"Start":"00:14.355 ","End":"00:19.905","Text":"but it\u0027s not straightforward and not what your intuition would tell you."},{"Start":"00:19.905 ","End":"00:24.510","Text":"Your intuition might tell you that it\u0027s like addition and subtraction that we take"},{"Start":"00:24.510 ","End":"00:30.000","Text":"2 matrices the same size that say both 2 by 2 matrices."},{"Start":"00:30.000 ","End":"00:33.480","Text":"Let\u0027s just fill them up with consecutive numbers."},{"Start":"00:33.480 ","End":"00:39.030","Text":"You might think 1 times 5 is 5, 2 times 6 is 12,"},{"Start":"00:39.030 ","End":"00:42.590","Text":"3 times 7, 21, 4 times 8 is 32,"},{"Start":"00:42.590 ","End":"00:44.540","Text":"turns out this is totally wrong."},{"Start":"00:44.540 ","End":"00:46.340","Text":"This is not useful at all,"},{"Start":"00:46.340 ","End":"00:50.285","Text":"and we define multiplication completely differently."},{"Start":"00:50.285 ","End":"00:53.525","Text":"There is a restriction on the size,"},{"Start":"00:53.525 ","End":"00:57.350","Text":"the order of each of the matrices we want to multiply."},{"Start":"00:57.350 ","End":"00:59.510","Text":"When I multiply A by B,"},{"Start":"00:59.510 ","End":"01:01.520","Text":"then this is the condition."},{"Start":"01:01.520 ","End":"01:04.850","Text":"The number of columns of A has to be the"},{"Start":"01:04.850 ","End":"01:08.640","Text":"same as the number of rows of B. I\u0027ll say that again."},{"Start":"01:08.640 ","End":"01:12.920","Text":"The number of columns in the first 1 on the left has"},{"Start":"01:12.920 ","End":"01:17.720","Text":"to be the same as the number of rows in the 1 on the right."},{"Start":"01:17.720 ","End":"01:21.245","Text":"So if A has size m by n,"},{"Start":"01:21.245 ","End":"01:23.300","Text":"m rows and n columns,"},{"Start":"01:23.300 ","End":"01:26.765","Text":"and B has size n by q,"},{"Start":"01:26.765 ","End":"01:28.965","Text":"n rows and q columns,"},{"Start":"01:28.965 ","End":"01:31.025","Text":"then this has to be the same n,"},{"Start":"01:31.025 ","End":"01:32.930","Text":"number of columns in A,"},{"Start":"01:32.930 ","End":"01:36.050","Text":"number of rows in B."},{"Start":"01:36.050 ","End":"01:39.080","Text":"Here are a couple of examples."},{"Start":"01:39.080 ","End":"01:43.340","Text":"Well, not so easy to see because everything is 3,"},{"Start":"01:43.340 ","End":"01:46.250","Text":"but it\u0027s the number of columns here,"},{"Start":"01:46.250 ","End":"01:49.950","Text":"that\u0027s 3, like 3 columns 1, 2,"},{"Start":"01:49.950 ","End":"01:54.375","Text":"3, and the number of rows here is 1, 2, 3."},{"Start":"01:54.375 ","End":"01:59.850","Text":"That would be this 3 here, that\u0027s in a blue color if you can read it."},{"Start":"02:00.320 ","End":"02:02.595","Text":"That\u0027s 1 example."},{"Start":"02:02.595 ","End":"02:06.950","Text":"In this example, here we have also 4 columns,"},{"Start":"02:06.950 ","End":"02:08.885","Text":"1, 2, 3, 4,"},{"Start":"02:08.885 ","End":"02:12.690","Text":"and over here, the other 1 has 4 rows."},{"Start":"02:15.050 ","End":"02:17.790","Text":"We can multiply them."},{"Start":"02:17.790 ","End":"02:21.760","Text":"This 4 as the same as this 4."},{"Start":"02:25.280 ","End":"02:37.800","Text":"Another example, 4 columns, 4 rows."},{"Start":"02:37.800 ","End":"02:42.460","Text":"That\u0027s this 4, and that\u0027s this 4."},{"Start":"02:42.740 ","End":"02:45.830","Text":"Let\u0027s see what else."},{"Start":"02:45.830 ","End":"02:47.800","Text":"Well, I think you got the idea,"},{"Start":"02:47.800 ","End":"02:52.645","Text":"but I wanted to bring 1 bad example I think you cannot do."},{"Start":"02:52.645 ","End":"02:56.360","Text":"You can\u0027t multiply this 1 by this 1,"},{"Start":"02:56.360 ","End":"03:03.120","Text":"because this has 2 columns and this has 3 rows."},{"Start":"03:03.120 ","End":"03:06.330","Text":"This 1, 2 is not the same as 1, 2,"},{"Start":"03:06.330 ","End":"03:10.840","Text":"3, so we cannot multiply them."},{"Start":"03:11.300 ","End":"03:14.595","Text":"Another bad example."},{"Start":"03:14.595 ","End":"03:17.445","Text":"Here we have 2 columns,"},{"Start":"03:17.445 ","End":"03:22.530","Text":"but here we have 3 rows."},{"Start":"03:22.530 ","End":"03:27.015","Text":"Again, we cannot multiply them."},{"Start":"03:27.015 ","End":"03:29.120","Text":"Now here\u0027s the thing."},{"Start":"03:29.120 ","End":"03:32.780","Text":"The question is, what will be the size of the product?"},{"Start":"03:32.780 ","End":"03:34.070","Text":"When I say product,"},{"Start":"03:34.070 ","End":"03:35.960","Text":"I mean the multiplication."},{"Start":"03:35.960 ","End":"03:38.823","Text":"Let\u0027s suppose we have m by n like we said"},{"Start":"03:38.823 ","End":"03:41.180","Text":"and then multiply by n by q."},{"Start":"03:41.180 ","End":"03:47.780","Text":"So this n is the same as this n. It turns out the result will"},{"Start":"03:47.780 ","End":"03:54.840","Text":"be the size of this number and this number."},{"Start":"03:54.840 ","End":"03:56.830","Text":"Cut out the middle man,"},{"Start":"03:56.830 ","End":"04:02.450","Text":"eliminate the middle n and just take the 2 outer sizes,"},{"Start":"04:02.450 ","End":"04:06.460","Text":"and that will be the size of the product."},{"Start":"04:06.460 ","End":"04:09.320","Text":"Let\u0027s just look at 1 of our examples."},{"Start":"04:09.320 ","End":"04:12.980","Text":"Suppose I look at this example."},{"Start":"04:12.980 ","End":"04:20.445","Text":"Let\u0027s say this 1 is A_2 by 4,"},{"Start":"04:20.445 ","End":"04:23.190","Text":"and this 1 is B,"},{"Start":"04:23.190 ","End":"04:27.450","Text":"which is 4 by 3."},{"Start":"04:27.450 ","End":"04:31.920","Text":"We already said that this 4 is the same as this 4,"},{"Start":"04:31.920 ","End":"04:40.855","Text":"so the size of the output will be A_2 by 3 matrix."},{"Start":"04:40.855 ","End":"04:42.710","Text":"In the future clip,"},{"Start":"04:42.710 ","End":"04:43.970","Text":"I\u0027ll show you how to multiply."},{"Start":"04:43.970 ","End":"04:52.230","Text":"But meanwhile, we can say that the result will be something of this size,"},{"Start":"04:52.230 ","End":"04:57.040","Text":"with 2 rows and 3 columns,"},{"Start":"04:57.770 ","End":"05:03.495","Text":"because here I had 2 rows and here I had 3 columns."},{"Start":"05:03.495 ","End":"05:07.530","Text":"I\u0027m also going to get a 2 by 3."},{"Start":"05:07.530 ","End":"05:12.390","Text":"The 4 is eliminated."},{"Start":"05:12.390 ","End":"05:14.270","Text":"We\u0027ll see this. I\u0027m going to take a break now."},{"Start":"05:14.270 ","End":"05:16.420","Text":"We\u0027ll continue in the next clip."}],"ID":10495},{"Watched":false,"Name":"Multiplication II","Duration":"8m 30s","ChapterTopicVideoID":9549,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9549.jpeg","UploadDate":"2017-07-26T08:27:01.3430000","DurationForVideoObject":"PT8M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.735","Text":"In this clip, we\u0027ll learn the technique of how we actually do multiply matrices."},{"Start":"00:06.735 ","End":"00:11.715","Text":"Let me just remind you how we do not multiply matrices."},{"Start":"00:11.715 ","End":"00:15.900","Text":"We don\u0027t take 2 matrices of the same size and multiply"},{"Start":"00:15.900 ","End":"00:19.965","Text":"the pairs of corresponding elements like so."},{"Start":"00:19.965 ","End":"00:22.680","Text":"This is the wrong way."},{"Start":"00:22.680 ","End":"00:24.750","Text":"Let\u0027s do it in the right way."},{"Start":"00:24.750 ","End":"00:28.780","Text":"Remember we said that not every 2 matrices can be multiplied."},{"Start":"00:28.780 ","End":"00:36.080","Text":"The number of columns in the first has to be equal to the number of rows in the second."},{"Start":"00:36.080 ","End":"00:39.950","Text":"Let\u0027s start off slowly by multiplying vectors."},{"Start":"00:39.950 ","End":"00:44.015","Text":"That is we\u0027re going to multiply a 1 row matrix by a 1 column matrix."},{"Start":"00:44.015 ","End":"00:47.045","Text":"Because the number of columns here has to be the number of rows here,"},{"Start":"00:47.045 ","End":"00:52.310","Text":"the first 1 will be a 1 by n matrix and the second 1 will be an n"},{"Start":"00:52.310 ","End":"00:58.400","Text":"by 1 matrix on these are vectors and I\u0027ll illustrate with an example."},{"Start":"00:58.400 ","End":"01:02.610","Text":"Let\u0027s suppose we have 1 row matrix,"},{"Start":"01:02.610 ","End":"01:05.340","Text":"1, 2, 3 and 1 column matrix,"},{"Start":"01:05.340 ","End":"01:06.610","Text":"4, 5, 6."},{"Start":"01:06.610 ","End":"01:11.755","Text":"What we do is we take the first element here with the first here and multiply,"},{"Start":"01:11.755 ","End":"01:16.400","Text":"then add this times this and then this times this as shown here."},{"Start":"01:16.400 ","End":"01:19.745","Text":"We multiply them pair-wise and then we add."},{"Start":"01:19.745 ","End":"01:22.975","Text":"The answer comes out to be 32."},{"Start":"01:22.975 ","End":"01:26.670","Text":"Another example of a 1 row times a 1 column."},{"Start":"01:26.670 ","End":"01:29.990","Text":"Notice there are 4 entries here and 4 here, that\u0027s important."},{"Start":"01:29.990 ","End":"01:31.940","Text":"We take the 1 with the 0,"},{"Start":"01:31.940 ","End":"01:33.710","Text":"the 2 with the 0.5,"},{"Start":"01:33.710 ","End":"01:40.970","Text":"multiply each pair and then add and we get 38."},{"Start":"01:40.970 ","End":"01:48.010","Text":"By the way, if you\u0027ve studied vectors and you know what the dot product is,"},{"Start":"01:48.010 ","End":"01:50.785","Text":"this is virtually the same thing."},{"Start":"01:50.785 ","End":"01:54.205","Text":"We take a vector dot product with another vector,"},{"Start":"01:54.205 ","End":"01:59.460","Text":"only there we don\u0027t have 1 in a row form or 1 in a column form."},{"Start":"01:59.460 ","End":"02:05.950","Text":"I\u0027m just saying that this is very similar to the dot product of 2 vectors,"},{"Start":"02:05.950 ","End":"02:08.995","Text":"sometimes called the scalar product."},{"Start":"02:08.995 ","End":"02:13.520","Text":"Anyway, let\u0027s continue with another example."},{"Start":"02:13.910 ","End":"02:16.870","Text":"This is length 2,"},{"Start":"02:16.870 ","End":"02:18.430","Text":"this is of length 2."},{"Start":"02:18.430 ","End":"02:20.590","Text":"We take 1 times 3,"},{"Start":"02:20.590 ","End":"02:23.710","Text":"2 times 4, and that comes out 11."},{"Start":"02:23.710 ","End":"02:27.785","Text":"I think it\u0027s time to generalize a bit."},{"Start":"02:27.785 ","End":"02:34.110","Text":"When we have a row vector of length n and a column vector of length n,"},{"Start":"02:34.110 ","End":"02:40.730","Text":"then what we do is we take each element here with its corresponding element here."},{"Start":"02:40.730 ","End":"02:46.180","Text":"Multiply, and then add all the products together like so."},{"Start":"02:46.180 ","End":"02:51.650","Text":"Now, make sure you\u0027ve understood what we just did about multiplying"},{"Start":"02:51.650 ","End":"02:57.575","Text":"a row vector by a column vector because that\u0027s the building block for the general case."},{"Start":"02:57.575 ","End":"03:01.170","Text":"We\u0027re going to multiply 2 general matrices when defined."},{"Start":"03:01.170 ","End":"03:06.430","Text":"As we said, not all 2 matrices are compatible for multiplication."},{"Start":"03:06.430 ","End":"03:09.710","Text":"Don\u0027t be alarmed, it\u0027s not as bad as it looks."},{"Start":"03:09.710 ","End":"03:13.639","Text":"We\u0027re going to learn how to multiply 2 general matrices,"},{"Start":"03:13.639 ","End":"03:16.325","Text":"but we\u0027re going to take a specific example."},{"Start":"03:16.325 ","End":"03:24.390","Text":"I\u0027m going to take this matrix which is a 3 by 3 and this matrix which is a 3 by 3."},{"Start":"03:24.390 ","End":"03:27.170","Text":"Remember, the middle bit has to be"},{"Start":"03:27.170 ","End":"03:32.790","Text":"the same number and the answer is going to be also a 3 by 3,"},{"Start":"03:32.790 ","End":"03:34.635","Text":"which it will be."},{"Start":"03:34.635 ","End":"03:39.095","Text":"Now, each element here can be obtained"},{"Start":"03:39.095 ","End":"03:45.395","Text":"by multiplying a row from here with a column from here as we learned earlier."},{"Start":"03:45.395 ","End":"03:48.395","Text":"The way we get the first element,"},{"Start":"03:48.395 ","End":"03:51.125","Text":"which is first row, first column,"},{"Start":"03:51.125 ","End":"03:59.749","Text":"is to take the first row in the first matrix and the first column in the second."},{"Start":"03:59.749 ","End":"04:01.985","Text":"If I do this by this,"},{"Start":"04:01.985 ","End":"04:04.520","Text":"I get row 1, column 1."},{"Start":"04:04.520 ","End":"04:08.490","Text":"Next, still row 1, but column 2."},{"Start":"04:08.490 ","End":"04:10.785","Text":"Row 1 here, column 2 here,"},{"Start":"04:10.785 ","End":"04:12.110","Text":"and this with this,"},{"Start":"04:12.110 ","End":"04:13.550","Text":"we know how to do that already."},{"Start":"04:13.550 ","End":"04:17.600","Text":"We\u0027ll do the actual computations in a minute and get the numerical value,"},{"Start":"04:17.600 ","End":"04:19.735","Text":"but I wanted you to see how it goes."},{"Start":"04:19.735 ","End":"04:24.780","Text":"Then this 1 is still in place 1, 3,"},{"Start":"04:24.780 ","End":"04:26.630","Text":"so it\u0027s the first row from here,"},{"Start":"04:26.630 ","End":"04:28.535","Text":"the third column from here."},{"Start":"04:28.535 ","End":"04:31.950","Text":"Now this 1. The result,"},{"Start":"04:31.950 ","End":"04:33.855","Text":"there\u0027s a second row, first column."},{"Start":"04:33.855 ","End":"04:35.650","Text":"Here we take second row,"},{"Start":"04:35.650 ","End":"04:40.765","Text":"here we take first column, and multiply those just like we know how,"},{"Start":"04:40.765 ","End":"04:48.200","Text":"and so on and so on and so on until we get to third row, third column here."},{"Start":"04:48.200 ","End":"04:54.180","Text":"We take third row with third column and this is the product we get."},{"Start":"04:54.180 ","End":"04:59.500","Text":"Now, we actually have to do the arithmetic, and in each case,"},{"Start":"04:59.500 ","End":"05:01.195","Text":"I don\u0027t know, let\u0027s pick this 1."},{"Start":"05:01.195 ","End":"05:02.410","Text":"If we choose this 1,"},{"Start":"05:02.410 ","End":"05:06.325","Text":"it\u0027s 1 times 2 plus 2 times 4 plus 3 times 1,"},{"Start":"05:06.325 ","End":"05:12.125","Text":"that would be this 1 that corresponds to this product."},{"Start":"05:12.125 ","End":"05:16.600","Text":"Then to actually multiply out,"},{"Start":"05:16.700 ","End":"05:19.275","Text":"I\u0027ll scroll ahead,"},{"Start":"05:19.275 ","End":"05:24.045","Text":"2 plus 8 plus 3 is 13,"},{"Start":"05:24.045 ","End":"05:26.980","Text":"and so on for all the rest."},{"Start":"05:26.990 ","End":"05:32.360","Text":"That\u0027s how we multiply 2 matrices."},{"Start":"05:32.360 ","End":"05:35.090","Text":"But that was an example of a 3 by 3."},{"Start":"05:35.090 ","End":"05:38.195","Text":"Let\u0027s see if we can generalize a bit more."},{"Start":"05:38.195 ","End":"05:42.340","Text":"Suppose we have an m by n,"},{"Start":"05:42.340 ","End":"05:44.970","Text":"meaning m rows, n columns,"},{"Start":"05:44.970 ","End":"05:51.475","Text":"and multiply it by n rows, q columns, the n here is the same."},{"Start":"05:51.475 ","End":"05:56.870","Text":"The result has to be an m by q matrix."},{"Start":"05:56.870 ","End":"05:58.980","Text":"This is the m, q."},{"Start":"05:59.510 ","End":"06:04.295","Text":"In each place, let\u0027s say this is the location"},{"Start":"06:04.295 ","End":"06:08.600","Text":"of the ith row and the jth column of the result,"},{"Start":"06:08.600 ","End":"06:13.235","Text":"then we get it from multiplying the ith row of the first"},{"Start":"06:13.235 ","End":"06:18.170","Text":"with the jth column of the second and we get this times this plus this times this,"},{"Start":"06:18.170 ","End":"06:19.865","Text":"what we call the dot product,"},{"Start":"06:19.865 ","End":"06:24.270","Text":"and this would be this element."},{"Start":"06:25.070 ","End":"06:30.330","Text":"Well, let\u0027s just give them names."},{"Start":"06:30.330 ","End":"06:32.760","Text":"Let\u0027s say this is A, this is B,"},{"Start":"06:32.760 ","End":"06:34.035","Text":"and this is C,"},{"Start":"06:34.035 ","End":"06:38.250","Text":"so this is made up of little a, i, j,"},{"Start":"06:38.250 ","End":"06:45.000","Text":"are the typical elements, and little b and little c."},{"Start":"06:45.000 ","End":"06:51.200","Text":"We get the i, j element, C_ij, of the matrix C"},{"Start":"06:51.200 ","End":"06:57.010","Text":"which is the product from taking the ith row of matrix A."},{"Start":"06:57.010 ","End":"07:01.320","Text":"Well, in general, it will be ith row, first column,"},{"Start":"07:01.320 ","End":"07:05.511","Text":"ith row, second column, ith row nth column."},{"Start":"07:05.511 ","End":"07:08.250","Text":"The j column of B,"},{"Start":"07:08.250 ","End":"07:12.960","Text":"which is b, anything j, 1j, 2j."},{"Start":"07:12.960 ","End":"07:15.885","Text":"J is the column and the rows go from 1 to n."},{"Start":"07:15.885 ","End":"07:22.345","Text":"That\u0027s the formula for the product of compatible matrices."},{"Start":"07:22.345 ","End":"07:28.700","Text":"This is the same n as this because we make sure that this n and this n are the same."},{"Start":"07:28.700 ","End":"07:32.680","Text":"Now another example of matrix multiplication."},{"Start":"07:32.680 ","End":"07:34.210","Text":"We have this times this."},{"Start":"07:34.210 ","End":"07:38.380","Text":"Notice that this is a matrix which is 2 rows, 3 columns."},{"Start":"07:38.380 ","End":"07:41.340","Text":"This is 3 rows, 4 columns."},{"Start":"07:41.340 ","End":"07:43.815","Text":"This 3 is the same, it has to be."},{"Start":"07:43.815 ","End":"07:49.120","Text":"Then the result you get from the 2 outer numbers is going to be a 2 by 4."},{"Start":"07:49.120 ","End":"07:50.860","Text":"Each of these is 1 number."},{"Start":"07:50.860 ","End":"07:57.360","Text":"There\u0027s 1, 2 rows and 4 columns and let\u0027s just take 1 of the examples."},{"Start":"07:57.360 ","End":"08:00.005","Text":"Let\u0027s say we want this element,"},{"Start":"08:00.005 ","End":"08:02.615","Text":"second row, third column."},{"Start":"08:02.615 ","End":"08:05.380","Text":"We need second row from here,"},{"Start":"08:05.380 ","End":"08:08.925","Text":"third column from there,"},{"Start":"08:08.925 ","End":"08:12.800","Text":"and then we just put them together and multiply out."},{"Start":"08:12.800 ","End":"08:18.560","Text":"4 times 2 is 8, 5 times 0 is 0, 6 times 1 is 6,"},{"Start":"08:18.560 ","End":"08:20.335","Text":"8 plus 6 is 14,"},{"Start":"08:20.335 ","End":"08:23.160","Text":"and that\u0027s the 14 we just computed,"},{"Start":"08:23.160 ","End":"08:25.720","Text":"and then all the rest similarly."},{"Start":"08:25.720 ","End":"08:27.890","Text":"I think that\u0027s enough for now."},{"Start":"08:27.890 ","End":"08:30.960","Text":"We\u0027ll take a break and continue after."}],"ID":10496},{"Watched":false,"Name":"Multiplication III","Duration":"6m 26s","ChapterTopicVideoID":9542,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9542.jpeg","UploadDate":"2017-07-26T08:23:32.8800000","DurationForVideoObject":"PT6M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.025","Text":"Now let\u0027s discuss some of the rules of matrix multiplication."},{"Start":"00:05.025 ","End":"00:10.125","Text":"If we have 3 matrices to multiply together,"},{"Start":"00:10.125 ","End":"00:14.775","Text":"then it doesn\u0027t matter how we bunch them as long as we don\u0027t change the order."},{"Start":"00:14.775 ","End":"00:16.980","Text":"As we\u0027ll see, the order is important."},{"Start":"00:16.980 ","End":"00:19.140","Text":"If I have A times B times C,"},{"Start":"00:19.140 ","End":"00:23.280","Text":"I can multiply B times C first and then A times that,"},{"Start":"00:23.280 ","End":"00:25.470","Text":"or A times B first,"},{"Start":"00:25.470 ","End":"00:28.395","Text":"and the result of that by C."},{"Start":"00:28.395 ","End":"00:34.370","Text":"This rule is really 2 rules that separate but they\u0027re similar."},{"Start":"00:34.370 ","End":"00:39.350","Text":"If I have a matrix and I multiply it by the sum of 2 matrices,"},{"Start":"00:39.350 ","End":"00:43.315","Text":"I\u0027m assuming they\u0027re the same size and everything\u0027s compatible,"},{"Start":"00:43.315 ","End":"00:49.280","Text":"then this is the same as if I multiply A by the first,"},{"Start":"00:49.280 ","End":"00:52.475","Text":"then I multiply A by the second, and then I add."},{"Start":"00:52.475 ","End":"00:56.000","Text":"This is just like the distributive law for arithmetic."},{"Start":"00:56.000 ","End":"00:59.490","Text":"I\u0027ll even write the word distributive."},{"Start":"01:00.200 ","End":"01:02.955","Text":"It works also the other way,"},{"Start":"01:02.955 ","End":"01:08.975","Text":"that if I have a sum and then multiply it on the right by another matrix,"},{"Start":"01:08.975 ","End":"01:14.290","Text":"I can take A times C plus B times C. Notice that in all of these cases,"},{"Start":"01:14.290 ","End":"01:16.400","Text":"I haven\u0027t changed the order of things."},{"Start":"01:16.400 ","End":"01:19.235","Text":"The order matters with matrices."},{"Start":"01:19.235 ","End":"01:24.170","Text":"If I had a constant or scalar multiply it by a product,"},{"Start":"01:24.170 ","End":"01:28.070","Text":"it doesn\u0027t matter if instead of that,"},{"Start":"01:28.070 ","End":"01:31.205","Text":"I multiply k by the first 1,"},{"Start":"01:31.205 ","End":"01:35.400","Text":"or if I multiply k by the second 1."},{"Start":"01:36.380 ","End":"01:40.715","Text":"We also sometimes write the constant on the right."},{"Start":"01:40.715 ","End":"01:44.750","Text":"By definition, this just means the same thing as this."},{"Start":"01:44.750 ","End":"01:48.650","Text":"Another rule involves the identity matrix."},{"Start":"01:48.650 ","End":"01:51.770","Text":"Well, identity matrices come in different sizes."},{"Start":"01:51.770 ","End":"01:57.120","Text":"Let\u0027s say of size n and we have a square matrix A which is n by"},{"Start":"01:57.120 ","End":"02:05.310","Text":"n. Then the matrix A times the identity or the identity times the matrix,"},{"Start":"02:05.310 ","End":"02:09.605","Text":"in both cases is just the matrix itself."},{"Start":"02:09.605 ","End":"02:15.615","Text":"This identity matrix is like the number 1 in regular multiplication,"},{"Start":"02:15.615 ","End":"02:18.720","Text":"1 times anything is that thing."},{"Start":"02:18.720 ","End":"02:24.080","Text":"Besides rules, I want to talk about non-rules."},{"Start":"02:24.080 ","End":"02:27.830","Text":"Non-rules, they\u0027re common errors of people"},{"Start":"02:27.830 ","End":"02:32.275","Text":"supposing that certain rules exist where in fact they don\u0027t."},{"Start":"02:32.275 ","End":"02:36.695","Text":"Although it\u0027s normally bad teaching practice to say what not to do,"},{"Start":"02:36.695 ","End":"02:40.070","Text":"I\u0027m going to do it and I\u0027m going to show you what not to do."},{"Start":"02:40.070 ","End":"02:44.510","Text":"The order of multiplication of matrices makes a difference."},{"Start":"02:44.510 ","End":"02:49.955","Text":"Let\u0027s say they\u0027re both square matrices so you can multiply them in any order."},{"Start":"02:49.955 ","End":"02:54.470","Text":"Then the answer is not generally the same."},{"Start":"02:54.470 ","End":"02:57.815","Text":"It could by fluke for a particular pair,"},{"Start":"02:57.815 ","End":"02:59.240","Text":"come out the same,"},{"Start":"02:59.240 ","End":"03:02.740","Text":"but you can\u0027t say that it\u0027s true in general."},{"Start":"03:02.740 ","End":"03:05.375","Text":"I\u0027d like to give an example of that."},{"Start":"03:05.375 ","End":"03:09.560","Text":"Let\u0027s say this is our matrix A and this is our matrix B."},{"Start":"03:09.560 ","End":"03:12.020","Text":"If we multiply, this is what we get."},{"Start":"03:12.020 ","End":"03:13.880","Text":"I\u0027ll just do 1 example."},{"Start":"03:13.880 ","End":"03:19.100","Text":"Let\u0027s say the first row here times the first column here should give us this."},{"Start":"03:19.100 ","End":"03:24.169","Text":"Well, let\u0027s see, 1 times 5 plus 2 times 7 is indeed 19,"},{"Start":"03:24.169 ","End":"03:26.200","Text":"and so on for the rest."},{"Start":"03:26.200 ","End":"03:32.030","Text":"If I reverse the orders so that this 1 B is first and then A in second,"},{"Start":"03:32.030 ","End":"03:39.270","Text":"then that first element is 5 times 1 plus 6 times 3."},{"Start":"03:39.270 ","End":"03:42.990","Text":"5 plus 18 is 23, not the same."},{"Start":"03:42.990 ","End":"03:47.035","Text":"In general, you see that the elements are not the same."},{"Start":"03:47.035 ","End":"03:51.755","Text":"Another non-rule that you might think is a rule because it"},{"Start":"03:51.755 ","End":"03:56.600","Text":"works for numbers is that with numbers,"},{"Start":"03:56.600 ","End":"03:59.360","Text":"if you multiply 2 non-zero numbers together,"},{"Start":"03:59.360 ","End":"04:01.085","Text":"you can\u0027t get 0."},{"Start":"04:01.085 ","End":"04:05.615","Text":"If your product is 0, it must be that 1 is 0 or the other is 0."},{"Start":"04:05.615 ","End":"04:09.800","Text":"This is simply not true with matrices."},{"Start":"04:09.800 ","End":"04:13.210","Text":"It could be, but not necessarily."},{"Start":"04:13.210 ","End":"04:15.305","Text":"Here\u0027s an example."},{"Start":"04:15.305 ","End":"04:18.890","Text":"Suppose this is my matrix A and this is my matrix B."},{"Start":"04:18.890 ","End":"04:21.465","Text":"If I multiply them together,"},{"Start":"04:21.465 ","End":"04:25.540","Text":"let\u0027s see, first row of the first,"},{"Start":"04:25.540 ","End":"04:32.545","Text":"first column of the second. 1 times 1 plus 1 times negative 1 is 0, and so on."},{"Start":"04:32.545 ","End":"04:35.655","Text":"This times this is the same thing."},{"Start":"04:35.655 ","End":"04:39.670","Text":"They all come out to be 1 minus 1, which is 0."},{"Start":"04:39.670 ","End":"04:44.435","Text":"Neither of these is 0, but the product is 0, the 0 matrix."},{"Start":"04:44.435 ","End":"04:46.200","Text":"If we write 0 like this,"},{"Start":"04:46.200 ","End":"04:48.840","Text":"we mean the 0 matrix."},{"Start":"04:48.840 ","End":"04:53.275","Text":"I\u0027m just repeating that each of these is not 0 of course."},{"Start":"04:53.275 ","End":"04:55.310","Text":"Another non-rule."},{"Start":"04:55.310 ","End":"05:02.065","Text":"Now, here, I have an A and I have an A. I\u0027d like to take it out the brackets."},{"Start":"05:02.065 ","End":"05:07.835","Text":"Well, I can\u0027t do it because here the A is on the left and here it\u0027s on the right."},{"Start":"05:07.835 ","End":"05:09.920","Text":"If they were both on the left,"},{"Start":"05:09.920 ","End":"05:11.990","Text":"then I could put A on the left."},{"Start":"05:11.990 ","End":"05:13.700","Text":"If they were both on the right,"},{"Start":"05:13.700 ","End":"05:16.085","Text":"I could take it out and put it on the right"},{"Start":"05:16.085 ","End":"05:19.020","Text":"but you can\u0027t have it when it\u0027s mixed."},{"Start":"05:19.020 ","End":"05:22.565","Text":"That\u0027s not a rule."},{"Start":"05:22.565 ","End":"05:25.100","Text":"This just repeats what I said,"},{"Start":"05:25.100 ","End":"05:27.200","Text":"that if A is on the left in both cases,"},{"Start":"05:27.200 ","End":"05:28.655","Text":"you can take it out on the left."},{"Start":"05:28.655 ","End":"05:31.370","Text":"If A is on the right in both cases, you can take it out on the right."},{"Start":"05:31.370 ","End":"05:33.820","Text":"That\u0027s the distributive law we saw."},{"Start":"05:33.820 ","End":"05:37.730","Text":"Another thing that doesn\u0027t work that works with numbers is"},{"Start":"05:37.730 ","End":"05:41.120","Text":"this binomial expansion with numbers."},{"Start":"05:41.120 ","End":"05:42.590","Text":"When we have A plus B squared,"},{"Start":"05:42.590 ","End":"05:45.889","Text":"it\u0027s A squared plus 2AB plus B squared,"},{"Start":"05:45.889 ","End":"05:49.775","Text":"but with matrices, if we use the distributive law twice,"},{"Start":"05:49.775 ","End":"05:57.155","Text":"what we\u0027ll get is AA plus AB plus BA plus BB."},{"Start":"05:57.155 ","End":"06:00.295","Text":"Each 1 from here times each 1 with here,"},{"Start":"06:00.295 ","End":"06:02.190","Text":"times each 1 from here"},{"Start":"06:02.190 ","End":"06:04.245","Text":"but keeping the order."},{"Start":"06:04.245 ","End":"06:07.470","Text":"You could write this as A squared, that\u0027s what we mean."},{"Start":"06:07.470 ","End":"06:09.495","Text":"You could write this is B squared"},{"Start":"06:09.495 ","End":"06:11.870","Text":"but this middle term is not the same."},{"Start":"06:11.870 ","End":"06:14.360","Text":"We can\u0027t combine them and say 2AB."},{"Start":"06:14.360 ","End":"06:17.760","Text":"It stays AB plus BA."},{"Start":"06:18.440 ","End":"06:23.920","Text":"That concludes matrix multiplication."},{"Start":"06:23.920 ","End":"06:26.850","Text":"That\u0027s the end of this clip."}],"ID":10497},{"Watched":false,"Name":"Transpose","Duration":"6m 13s","ChapterTopicVideoID":9543,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9543.jpeg","UploadDate":"2017-07-26T08:23:54.2130000","DurationForVideoObject":"PT6M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"In this clip, we\u0027ll introduce another concept,"},{"Start":"00:03.330 ","End":"00:06.405","Text":"the transpose of a matrix."},{"Start":"00:06.405 ","End":"00:09.825","Text":"Suppose we are given a matrix A,"},{"Start":"00:09.825 ","End":"00:14.310","Text":"then there\u0027s another matrix called the transpose matrix of A and"},{"Start":"00:14.310 ","End":"00:19.155","Text":"the notation is a with a superscript T, T for transpose."},{"Start":"00:19.155 ","End":"00:25.520","Text":"You just say A transpose when we see this."},{"Start":"00:25.520 ","End":"00:27.515","Text":"It\u0027s obtained from A"},{"Start":"00:27.515 ","End":"00:32.525","Text":"by writing the rows of A as the columns of A transpose."},{"Start":"00:32.525 ","End":"00:36.565","Text":"Basically, you switch rows and columns."},{"Start":"00:36.565 ","End":"00:40.520","Text":"If A has m rows and n columns,"},{"Start":"00:40.520 ","End":"00:43.370","Text":"then the transpose of A will have the opposite."},{"Start":"00:43.370 ","End":"00:46.610","Text":"We\u0027ll have n rows and m columns."},{"Start":"00:46.610 ","End":"00:49.655","Text":"I\u0027ll give some examples and here\u0027s our first."},{"Start":"00:49.655 ","End":"00:52.415","Text":"This 1 has 1 row and 2 columns."},{"Start":"00:52.415 ","End":"00:55.670","Text":"We write this row as a column,"},{"Start":"00:55.670 ","End":"00:58.400","Text":"and then this is the transposed matrix."},{"Start":"00:58.400 ","End":"00:59.960","Text":"I think you get the idea."},{"Start":"00:59.960 ","End":"01:03.155","Text":"Let\u0027s get onto some larger examples."},{"Start":"01:03.155 ","End":"01:04.550","Text":"The transpose of this,"},{"Start":"01:04.550 ","End":"01:07.070","Text":"we write the rows as columns."},{"Start":"01:07.070 ","End":"01:10.580","Text":"This row, first row becomes first column,"},{"Start":"01:10.580 ","End":"01:13.580","Text":"second row becomes second column,"},{"Start":"01:13.580 ","End":"01:15.615","Text":"and this is the transpose."},{"Start":"01:15.615 ","End":"01:20.180","Text":"Another example, first row becomes first column,"},{"Start":"01:20.180 ","End":"01:23.345","Text":"second row becomes second column."},{"Start":"01:23.345 ","End":"01:24.930","Text":"third row, 7, 8,"},{"Start":"01:24.930 ","End":"01:28.965","Text":"9 becomes third column."},{"Start":"01:28.965 ","End":"01:31.355","Text":"We could have said it the other way around,"},{"Start":"01:31.355 ","End":"01:36.120","Text":"also that the columns here are the rows here, symmetrical."},{"Start":"01:36.400 ","End":"01:42.300","Text":"Another example, this 1 is 4 by 2, 4 rows 2 columns."},{"Start":"01:42.300 ","End":"01:46.015","Text":"The transpose is 2 rows, 4 columns."},{"Start":"01:46.015 ","End":"01:48.860","Text":"Like this goes here,"},{"Start":"01:48.860 ","End":"01:51.860","Text":"this goes here, this row,"},{"Start":"01:51.860 ","End":"01:55.405","Text":"this column, this row is this column."},{"Start":"01:55.405 ","End":"02:00.504","Text":"Just formally introducing something,"},{"Start":"02:00.504 ","End":"02:03.694","Text":"that if I take the transpose of this,"},{"Start":"02:03.694 ","End":"02:09.185","Text":"then this row could become this column, and this row, this column."},{"Start":"02:09.185 ","End":"02:12.980","Text":"If I do the transpose of the transpose, I\u0027m back to the original."},{"Start":"02:12.980 ","End":"02:16.660","Text":"But never mind, I\u0027m getting ahead of myself."},{"Start":"02:16.660 ","End":"02:20.900","Text":"Let\u0027s say we use the letter A for the matrix."},{"Start":"02:20.900 ","End":"02:29.375","Text":"Then notice that a_ij is a_ji transpose or vice versa."},{"Start":"02:29.375 ","End":"02:36.130","Text":"So that if I want, let\u0027s say, row 2, column 3 is this."},{"Start":"02:36.130 ","End":"02:44.355","Text":"Here, it\u0027s row 3, column 2 is this."},{"Start":"02:44.355 ","End":"02:46.610","Text":"If you basically reverse the i and"},{"Start":"02:46.610 ","End":"02:52.150","Text":"the j, row number and column number backwards in the transpose."},{"Start":"02:52.150 ","End":"02:55.870","Text":"There are certain rules for the transpose."},{"Start":"02:55.870 ","End":"02:58.820","Text":"Well, the first 1 I already mentioned,"},{"Start":"02:58.820 ","End":"03:02.690","Text":"if you take the transpose and then take the transpose again,"},{"Start":"03:02.690 ","End":"03:04.051","Text":"you\u0027re back to the original."},{"Start":"03:04.051 ","End":"03:06.830","Text":"If you switch rows and columns and then you switch, again,"},{"Start":"03:06.830 ","End":"03:10.210","Text":"it\u0027s like switching back, you get the original."},{"Start":"03:10.210 ","End":"03:15.905","Text":"If you add 2 matrices of the same size and then you transpose them,"},{"Start":"03:15.905 ","End":"03:20.785","Text":"it\u0027s the same as transposing each 1 separately and then adding."},{"Start":"03:20.785 ","End":"03:25.175","Text":"With multiplication, it\u0027s a little bit different."},{"Start":"03:25.175 ","End":"03:27.920","Text":"If you have a product of matrices, I chose 3,"},{"Start":"03:27.920 ","End":"03:29.150","Text":"but there could be any number,"},{"Start":"03:29.150 ","End":"03:31.070","Text":"it could be just AB or ABCD,"},{"Start":"03:31.070 ","End":"03:34.415","Text":"but when you transpose the product,"},{"Start":"03:34.415 ","End":"03:37.610","Text":"you have to inverse the order."},{"Start":"03:37.610 ","End":"03:39.799","Text":"You transpose each 1 separately,"},{"Start":"03:39.799 ","End":"03:42.575","Text":"but you also reverse the order."},{"Start":"03:42.575 ","End":"03:45.140","Text":"You have to remember that."},{"Start":"03:45.140 ","End":"03:50.450","Text":"After multiplication with a scalar, doesn\u0027t matter if you first multiply by"},{"Start":"03:50.450 ","End":"03:55.040","Text":"the scalar and then transpose or first transpose and then multiply by the scalar."},{"Start":"03:55.040 ","End":"03:56.965","Text":"That makes no difference."},{"Start":"03:56.965 ","End":"03:59.960","Text":"Here\u0027s another 1 where the order of operations doesn\u0027t"},{"Start":"03:59.960 ","End":"04:02.990","Text":"matter if you first transpose and then take"},{"Start":"04:02.990 ","End":"04:09.259","Text":"the inverse, it\u0027s the same as doing the inverse and taking the transpose."},{"Start":"04:09.259 ","End":"04:13.790","Text":"We haven\u0027t really discussed the inverse much,"},{"Start":"04:13.790 ","End":"04:17.440","Text":"but you have this for future reference."},{"Start":"04:17.440 ","End":"04:20.960","Text":"The determinant for those who know what that is, again,"},{"Start":"04:20.960 ","End":"04:24.515","Text":"this has not been fully discussed,"},{"Start":"04:24.515 ","End":"04:27.900","Text":"but there is something called the determinant."},{"Start":"04:27.900 ","End":"04:34.585","Text":"The determinant of the transpose is the same as the determinant of the original."},{"Start":"04:34.585 ","End":"04:38.235","Text":"These are the 6 rules I want to present."},{"Start":"04:38.235 ","End":"04:45.330","Text":"This concept of transpose gives rise to certain special matrices."},{"Start":"04:45.520 ","End":"04:49.310","Text":"Remember when we talked about special matrices,"},{"Start":"04:49.310 ","End":"04:51.680","Text":"there was something called a symmetric matrix,"},{"Start":"04:51.680 ","End":"04:58.834","Text":"which had a reflection symmetry about the main diagonal for a square matrix."},{"Start":"04:58.834 ","End":"05:02.060","Text":"Well, we can now redefine it as just saying it\u0027s"},{"Start":"05:02.060 ","End":"05:07.325","Text":"a matrix with the property that it\u0027s equal to its own transpose."},{"Start":"05:07.325 ","End":"05:10.369","Text":"In other words, if we reverse rows and columns"},{"Start":"05:10.369 ","End":"05:14.195","Text":"like this row becomes this column the way around,"},{"Start":"05:14.195 ","End":"05:17.485","Text":"then that means that it\u0027s symmetric."},{"Start":"05:17.485 ","End":"05:22.280","Text":"This is now going to be the new definition of a symmetric matrix."},{"Start":"05:22.280 ","End":"05:24.750","Text":"It has this property."},{"Start":"05:25.400 ","End":"05:32.345","Text":"Similarly, we can define an anti-symmetric matrix in terms of the transpose."},{"Start":"05:32.345 ","End":"05:35.195","Text":"Earlier we had this example"},{"Start":"05:35.195 ","End":"05:37.490","Text":"which is not a mirror image."},{"Start":"05:37.490 ","End":"05:39.200","Text":"When it goes through the mirror,"},{"Start":"05:39.200 ","End":"05:42.125","Text":"it comes out negated."},{"Start":"05:42.125 ","End":"05:43.550","Text":"If it\u0027s a minus,"},{"Start":"05:43.550 ","End":"05:46.820","Text":"then in the mirror it comes out to plus."},{"Start":"05:46.820 ","End":"05:51.425","Text":"If it\u0027s a plus, it comes out minus and it had this property."},{"Start":"05:51.425 ","End":"05:58.760","Text":"Well, this can be defined more simply as the matrix is minus"},{"Start":"05:58.760 ","End":"06:06.485","Text":"the transpose or the transpose is minus the original and that is more succinctly"},{"Start":"06:06.485 ","End":"06:09.710","Text":"or the condensed form."},{"Start":"06:09.710 ","End":"06:14.760","Text":"This is the definition of an anti-symmetric matrix."}],"ID":10498},{"Watched":false,"Name":"Trace","Duration":"1m 49s","ChapterTopicVideoID":9544,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9544.jpeg","UploadDate":"2017-07-26T08:24:05.2070000","DurationForVideoObject":"PT1M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.309","Text":"1 more new concept,"},{"Start":"00:02.309 ","End":"00:04.230","Text":"the Trace of a Matrix."},{"Start":"00:04.230 ","End":"00:07.020","Text":"From a matrix, we get a single number."},{"Start":"00:07.020 ","End":"00:09.509","Text":"It only applies to a square matrix."},{"Start":"00:09.509 ","End":"00:13.950","Text":"What we do is add all the entries along the diagonal."},{"Start":"00:13.950 ","End":"00:18.685","Text":"I should say that it\u0027s the main diagonal."},{"Start":"00:18.685 ","End":"00:20.420","Text":"When I just say diagonal,"},{"Start":"00:20.420 ","End":"00:21.965","Text":"I mean the main diagonal,"},{"Start":"00:21.965 ","End":"00:26.470","Text":"and it\u0027s called tr, tr [inaudible] for trace, of A."},{"Start":"00:26.470 ","End":"00:33.060","Text":"For example, the trace of this square matrix is a diagonal,"},{"Start":"00:33.060 ","End":"00:36.085","Text":"add them up, is 5."},{"Start":"00:36.085 ","End":"00:39.800","Text":"Another 1, the trace of this matrix,"},{"Start":"00:39.800 ","End":"00:41.705","Text":"there\u0027s the diagonal,"},{"Start":"00:41.705 ","End":"00:44.315","Text":"add them up, 15."},{"Start":"00:44.315 ","End":"00:47.935","Text":"Not very complicated."},{"Start":"00:47.935 ","End":"00:52.670","Text":"The trace also has some properties or rules."},{"Start":"00:52.670 ","End":"00:58.220","Text":"If we multiply a matrix by a scalar k and then take the trace,"},{"Start":"00:58.220 ","End":"01:04.010","Text":"or first take the trace and then multiply by the scalar, same thing."},{"Start":"01:04.010 ","End":"01:09.050","Text":"If I take the trace of the sum of 2 matrices,"},{"Start":"01:09.050 ","End":"01:12.140","Text":"they both have to be square of the same size,"},{"Start":"01:12.140 ","End":"01:15.260","Text":"then it\u0027s the same as taking the trace of the first"},{"Start":"01:15.260 ","End":"01:18.715","Text":"one separately and adding it to the trace of the second."},{"Start":"01:18.715 ","End":"01:21.210","Text":"The last 1,"},{"Start":"01:21.210 ","End":"01:26.600","Text":"the trace of AB is the same as the trace of BA."},{"Start":"01:26.600 ","End":"01:32.540","Text":"Now, BA and AB are in general not the same thing."},{"Start":"01:32.540 ","End":"01:35.180","Text":"Remember with matrices that the order of"},{"Start":"01:35.180 ","End":"01:41.210","Text":"multiplication matters even though there might be different matrices,"},{"Start":"01:41.210 ","End":"01:45.570","Text":"AB and BA, they still have the same trace."},{"Start":"01:46.330 ","End":"01:50.070","Text":"That concludes this clip."}],"ID":10499},{"Watched":false,"Name":"Exercise 1","Duration":"6m 46s","ChapterTopicVideoID":9555,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9555.jpeg","UploadDate":"2017-07-26T08:30:18.3400000","DurationForVideoObject":"PT6M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.590","Text":"In this exercise,"},{"Start":"00:01.590 ","End":"00:08.130","Text":"there\u0027s actually going to be 10 separate calculations with matrices,"},{"Start":"00:08.130 ","End":"00:13.005","Text":"and we have to say whether each 1 is undefined,"},{"Start":"00:13.005 ","End":"00:15.644","Text":"but if it is defined to give the order,"},{"Start":"00:15.644 ","End":"00:17.730","Text":"that is the size of the result,"},{"Start":"00:17.730 ","End":"00:20.505","Text":"how many rows by columns."},{"Start":"00:20.505 ","End":"00:23.745","Text":"The first 1 is an addition."},{"Start":"00:23.745 ","End":"00:27.210","Text":"For addition, we have to make sure that they\u0027re both of"},{"Start":"00:27.210 ","End":"00:30.000","Text":"the same order, and indeed they are."},{"Start":"00:30.000 ","End":"00:31.455","Text":"This is a 4 by 6,"},{"Start":"00:31.455 ","End":"00:34.020","Text":"and this is a 4 by 6."},{"Start":"00:34.020 ","End":"00:37.950","Text":"The result, lets give it a letter, say C,"},{"Start":"00:37.950 ","End":"00:40.350","Text":"will also be 4 by 6."},{"Start":"00:40.350 ","End":"00:43.150","Text":"That\u0027s for addition and subtraction."},{"Start":"00:43.910 ","End":"00:46.640","Text":"Number 2 is a multiplication,"},{"Start":"00:46.640 ","End":"00:48.410","Text":"and there we have a different rule."},{"Start":"00:48.410 ","End":"00:51.020","Text":"We have to have that a number of columns of the"},{"Start":"00:51.020 ","End":"00:54.395","Text":"first has got to be equal number of rows in the second."},{"Start":"00:54.395 ","End":"01:03.115","Text":"We see that number of columns here, it\u0027s not equal."},{"Start":"01:03.115 ","End":"01:05.255","Text":"Because it\u0027s not equal,"},{"Start":"01:05.255 ","End":"01:09.224","Text":"that means that we cannot do this multiplication."},{"Start":"01:09.224 ","End":"01:11.720","Text":"If not, doesn\u0027t make sense."},{"Start":"01:11.720 ","End":"01:15.110","Text":"Now here\u0027s 1 where we have a multiplication to do,"},{"Start":"01:15.110 ","End":"01:16.895","Text":"and then a subtraction also."},{"Start":"01:16.895 ","End":"01:19.990","Text":"Let\u0027s first of all look and see if the multiplication is okay."},{"Start":"01:19.990 ","End":"01:24.770","Text":"This time we are fine because 6 is equal to 6."},{"Start":"01:24.770 ","End":"01:29.900","Text":"But more than that, we know that the result is going to have a size,"},{"Start":"01:29.900 ","End":"01:32.510","Text":"the outer numbers are 4 by 2."},{"Start":"01:32.510 ","End":"01:34.310","Text":"Now this is a 4 by 2"},{"Start":"01:34.310 ","End":"01:36.500","Text":"and this is a 4 by 2."},{"Start":"01:36.500 ","End":"01:38.690","Text":"When we subtract them,"},{"Start":"01:38.690 ","End":"01:41.855","Text":"let me give it a letter, B,"},{"Start":"01:41.855 ","End":"01:46.190","Text":"it\u0027s also going to be 4 by 2, because like I said,"},{"Start":"01:46.190 ","End":"01:54.265","Text":"this times this gives us a 4 by 2 matrix minus another 4 by 2."},{"Start":"01:54.265 ","End":"01:59.065","Text":"Next 1 is also 1 of those compound problems of multiplication and a subtraction."},{"Start":"01:59.065 ","End":"02:01.205","Text":"First, the product here,"},{"Start":"02:01.205 ","End":"02:04.740","Text":"we check the middle guys, 6 equals 6,"},{"Start":"02:04.740 ","End":"02:11.210","Text":"that\u0027s fine, and that means that the result is going to be a 4 by 4."},{"Start":"02:11.210 ","End":"02:13.685","Text":"Now, for 4 by 4 is here"},{"Start":"02:13.685 ","End":"02:15.410","Text":"and a 4 by 6 is here,"},{"Start":"02:15.410 ","End":"02:16.685","Text":"they don\u0027t mix,"},{"Start":"02:16.685 ","End":"02:18.140","Text":"it\u0027s not the same kind,"},{"Start":"02:18.140 ","End":"02:22.690","Text":"so we say that this is undefined."},{"Start":"02:22.690 ","End":"02:25.640","Text":"Let\u0027s continue."},{"Start":"02:25.640 ","End":"02:28.170","Text":"Again, we have 2 operations,"},{"Start":"02:28.170 ","End":"02:29.690","Text":"a multiplication, and an addition."},{"Start":"02:29.690 ","End":"02:31.640","Text":"We have to do the multiplication first,"},{"Start":"02:31.640 ","End":"02:33.320","Text":"they come before addition."},{"Start":"02:33.320 ","End":"02:38.160","Text":"Middle 6 here and a 4 here."},{"Start":"02:38.160 ","End":"02:40.425","Text":"The product\u0027s already no good,"},{"Start":"02:40.425 ","End":"02:44.970","Text":"so nope. Next 1."},{"Start":"02:44.970 ","End":"02:47.340","Text":"Well, there\u0027s a brackets here,"},{"Start":"02:47.340 ","End":"02:50.000","Text":"that\u0027s why we do the addition before the multiplication."},{"Start":"02:50.000 ","End":"02:53.645","Text":"Do the addition, we check that the both of the same order,"},{"Start":"02:53.645 ","End":"02:57.240","Text":"4 times 6 equals 4 times 6."},{"Start":"02:57.800 ","End":"03:01.470","Text":"Right here we have a 4 by 6,"},{"Start":"03:01.470 ","End":"03:03.745","Text":"and here we have a 6 by 4."},{"Start":"03:03.745 ","End":"03:09.420","Text":"This is where we check if the middle guys are equal."},{"Start":"03:09.420 ","End":"03:16.460","Text":"The last number of columns of the first equals number of rows of the second,"},{"Start":"03:16.460 ","End":"03:18.955","Text":"and then if you eliminate those,"},{"Start":"03:18.955 ","End":"03:23.850","Text":"the 2 outer ones tell us it\u0027s going to be a 6 by 6."},{"Start":"03:23.850 ","End":"03:27.953","Text":"Let\u0027s see what letter we can use. I\u0027ll take it as C."},{"Start":"03:27.953 ","End":"03:31.505","Text":"C is going to be a 6 by 6."},{"Start":"03:31.505 ","End":"03:35.000","Text":"I guess if I was being pedantic and asked us to say what is"},{"Start":"03:35.000 ","End":"03:38.690","Text":"the size of the result if it\u0027s defined,"},{"Start":"03:38.690 ","End":"03:40.070","Text":"here it\u0027s 6 by 6."},{"Start":"03:40.070 ","End":"03:42.835","Text":"Let me just go back and fix those."},{"Start":"03:42.835 ","End":"03:45.060","Text":"First 1 was a 4 by 6,"},{"Start":"03:45.060 ","End":"03:46.825","Text":"second 1 undefined,"},{"Start":"03:46.825 ","End":"03:49.010","Text":"X will mean undefined,"},{"Start":"03:49.010 ","End":"03:52.190","Text":"4 by 2, 6 by 6."},{"Start":"03:52.190 ","End":"03:54.350","Text":"I\u0027ll either say the size"},{"Start":"03:54.350 ","End":"03:56.545","Text":"or put an X next to it."},{"Start":"03:56.545 ","End":"03:59.690","Text":"Now, in the next question,"},{"Start":"03:59.690 ","End":"04:01.415","Text":"I have a transpose."},{"Start":"04:01.415 ","End":"04:05.060","Text":"The transpose interchanges rows and columns."},{"Start":"04:05.060 ","End":"04:07.970","Text":"Now, if A is 4 by 6,"},{"Start":"04:07.970 ","End":"04:11.350","Text":"then it\u0027s transpose is going to be 6 by 4."},{"Start":"04:11.350 ","End":"04:13.785","Text":"That takes care of the transpose."},{"Start":"04:13.785 ","End":"04:16.410","Text":"Now you can have this plus this,"},{"Start":"04:16.410 ","End":"04:19.810","Text":"and they are both 6 by 4,"},{"Start":"04:20.600 ","End":"04:24.209","Text":"6 by 4 will be the answer."},{"Start":"04:24.209 ","End":"04:27.420","Text":"I\u0027ll write that here."},{"Start":"04:27.420 ","End":"04:30.600","Text":"I\u0027ll get a 6 by 4 as the result of this."},{"Start":"04:30.600 ","End":"04:34.120","Text":"Here I have a 4 by 2."},{"Start":"04:35.540 ","End":"04:39.300","Text":"The ones towards the middle are the same,"},{"Start":"04:39.300 ","End":"04:40.640","Text":"4 and 4,"},{"Start":"04:40.640 ","End":"04:44.430","Text":"so we know the answer\u0027s going to be a 6 by 2."},{"Start":"04:44.430 ","End":"04:46.460","Text":"Let\u0027s see what letter is unused."},{"Start":"04:46.460 ","End":"04:51.410","Text":"I\u0027ll use B. The answer will be B_6 by 6."},{"Start":"04:51.410 ","End":"04:56.350","Text":"Well, the answer is that the size or order is 6 by 6."},{"Start":"04:56.350 ","End":"05:03.410","Text":"Next 1, we have a transpose of a 6 by 4 matrix that makes it a 4 by 6."},{"Start":"05:03.410 ","End":"05:07.885","Text":"Here we have a 4 by 6 also."},{"Start":"05:07.885 ","End":"05:14.105","Text":"Since these are not compatible,"},{"Start":"05:14.105 ","End":"05:18.710","Text":"then we cannot do the multiplication."},{"Start":"05:18.710 ","End":"05:21.283","Text":"If you didn\u0027t notice the transpose, you might have thought,"},{"Start":"05:21.283 ","End":"05:23.360","Text":"\"Very well. 4 and 4 matched,"},{"Start":"05:23.360 ","End":"05:28.569","Text":"so the answer will be 6 by 6,\" but the transpose reverses rows and columns."},{"Start":"05:28.569 ","End":"05:31.415","Text":"Now we have a product of 3 things."},{"Start":"05:31.415 ","End":"05:33.560","Text":"We can\u0027t rearrange the order,"},{"Start":"05:33.560 ","End":"05:35.510","Text":"but we can group them how we like."},{"Start":"05:35.510 ","End":"05:36.890","Text":"We can do this times this"},{"Start":"05:36.890 ","End":"05:38.225","Text":"and then times this,"},{"Start":"05:38.225 ","End":"05:42.305","Text":"or this times the result of this times this. It doesn\u0027t really matter."},{"Start":"05:42.305 ","End":"05:45.560","Text":"When you have a chain of products, in each case,"},{"Start":"05:45.560 ","End":"05:47.960","Text":"the middle ones are going to be the same"},{"Start":"05:47.960 ","End":"05:53.960","Text":"and the answer comes out to be like the first and the last."},{"Start":"05:53.960 ","End":"05:56.750","Text":"We know it\u0027s going to be something that\u0027s 6 by 2,"},{"Start":"05:56.750 ","End":"05:59.870","Text":"let\u0027s call it B_6 by 2,"},{"Start":"05:59.870 ","End":"06:04.420","Text":"and the answer to the question is the order of 6 by 2."},{"Start":"06:04.420 ","End":"06:06.435","Text":"Now, 1 last 1."},{"Start":"06:06.435 ","End":"06:09.485","Text":"Because of the brackets, we have to do the subtraction first."},{"Start":"06:09.485 ","End":"06:11.150","Text":"We have a 4 by 6"},{"Start":"06:11.150 ","End":"06:12.890","Text":"and we take away a 4 by 6."},{"Start":"06:12.890 ","End":"06:16.805","Text":"Addition and subtraction of the same order, no problem."},{"Start":"06:16.805 ","End":"06:20.870","Text":"This here is also going to be a 4 by 6."},{"Start":"06:20.870 ","End":"06:24.990","Text":"Now look, we have a 6 by 4 times a 4 by 6."},{"Start":"06:28.070 ","End":"06:31.820","Text":"The rightmost of this equals the leftmost of that,"},{"Start":"06:31.820 ","End":"06:35.390","Text":"and then we know the result will be a 6 by 6."},{"Start":"06:35.390 ","End":"06:40.230","Text":"I\u0027ll use the letter C and say that it\u0027s a 6 by 6,"},{"Start":"06:40.230 ","End":"06:44.025","Text":"that what I highlighted is the order, 6 by 6."},{"Start":"06:44.025 ","End":"06:46.690","Text":"That\u0027s the last 1, we\u0027re done."}],"ID":10500},{"Watched":false,"Name":"Exercise 2","Duration":"4m 20s","ChapterTopicVideoID":9556,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9556.jpeg","UploadDate":"2019-01-13T08:02:18.2170000","DurationForVideoObject":"PT4M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise, we have to solve the matrix equation."},{"Start":"00:04.590 ","End":"00:06.570","Text":"This matrix equals this matrix."},{"Start":"00:06.570 ","End":"00:09.450","Text":"Now, they\u0027re both of the same order, 2 by 2."},{"Start":"00:09.450 ","End":"00:12.854","Text":"The way to tackle this is to assign"},{"Start":"00:12.854 ","End":"00:17.400","Text":"each element separately in the corresponding position."},{"Start":"00:17.400 ","End":"00:19.260","Text":"They\u0027ve all got to equal."},{"Start":"00:19.260 ","End":"00:26.025","Text":"I\u0027ll show you what I mean. This element here has to equal this element here."},{"Start":"00:26.025 ","End":"00:30.015","Text":"This one here has to equal this one here."},{"Start":"00:30.015 ","End":"00:33.960","Text":"This one here has to equal this one here."},{"Start":"00:33.960 ","End":"00:38.115","Text":"Lastly, this one here equals this one here."},{"Start":"00:38.115 ","End":"00:42.470","Text":"If we write those 4 equations nicely,"},{"Start":"00:42.470 ","End":"00:44.225","Text":"this is what we get."},{"Start":"00:44.225 ","End":"00:47.180","Text":"It\u0027s actually 4 equations in 3 unknowns,"},{"Start":"00:47.180 ","End":"00:49.410","Text":"x, y, and z."},{"Start":"00:49.490 ","End":"00:52.535","Text":"Before we solve it, we want to straighten it out,"},{"Start":"00:52.535 ","End":"00:55.400","Text":"bring everything with variables on the left"},{"Start":"00:55.400 ","End":"00:57.290","Text":"and constants on the right."},{"Start":"00:57.290 ","End":"00:59.160","Text":"For example, in the first equation,"},{"Start":"00:59.160 ","End":"01:04.190","Text":"I bring the 2z over to the left and we get this and so on for all the rest."},{"Start":"01:04.190 ","End":"01:09.240","Text":"What we\u0027re going to do is solve this using matrices."},{"Start":"01:09.310 ","End":"01:14.140","Text":"Here\u0027s the matrix representing this, the augmented matrix."},{"Start":"01:14.140 ","End":"01:17.500","Text":"Sometimes we put a vertical bar here."},{"Start":"01:17.500 ","End":"01:20.050","Text":"But it\u0027s okay to leave it out too,"},{"Start":"01:20.050 ","End":"01:22.300","Text":"it\u0027s not always necessary."},{"Start":"01:22.300 ","End":"01:25.540","Text":"Now we want to take this matrix and bring it into"},{"Start":"01:25.540 ","End":"01:29.330","Text":"row echelon form using elementary row operations."},{"Start":"01:29.330 ","End":"01:32.365","Text":"It\u0027s nice that we have a 1 here in the top left"},{"Start":"01:32.365 ","End":"01:35.750","Text":"that will help us to 0 out the rest of the column."},{"Start":"01:35.750 ","End":"01:39.310","Text":"What we\u0027ll do is subtract 3 times this row from this row,"},{"Start":"01:39.310 ","End":"01:40.810","Text":"twice this row from this row,"},{"Start":"01:40.810 ","End":"01:44.875","Text":"and twice this row from this row as written here."},{"Start":"01:44.875 ","End":"01:48.670","Text":"If we do all that and I\u0027ll leave you to check the calculations,"},{"Start":"01:48.670 ","End":"01:50.410","Text":"you can always pause the clip,"},{"Start":"01:50.410 ","End":"01:52.060","Text":"we should get this."},{"Start":"01:52.060 ","End":"01:53.740","Text":"Now we have 0s here."},{"Start":"01:53.740 ","End":"02:02.670","Text":"Now the next problem is to get 0s in these 2 positions."},{"Start":"02:02.670 ","End":"02:10.400","Text":"Now, normally I would divide this by minus 8 perhaps and get a 1 here and then do that."},{"Start":"02:10.400 ","End":"02:12.950","Text":"But sometimes you look for opportunities."},{"Start":"02:12.950 ","End":"02:16.070","Text":"Notice the last row is all divisible by 4."},{"Start":"02:16.070 ","End":"02:17.330","Text":"If I divide it by 4,"},{"Start":"02:17.330 ","End":"02:18.340","Text":"I\u0027ll get a 1 here,"},{"Start":"02:18.340 ","End":"02:22.170","Text":"and then I\u0027ll be able to swap rows. That\u0027s what we\u0027ll do."},{"Start":"02:22.630 ","End":"02:26.285","Text":"As I said, I\u0027ll divide the last 1 by 4."},{"Start":"02:26.285 ","End":"02:28.670","Text":"Everything\u0027s the same except the last row."},{"Start":"02:28.670 ","End":"02:35.180","Text":"Notice that the new entries 1/4 of these."},{"Start":"02:35.480 ","End":"02:40.670","Text":"As I mentioned, the next thing I\u0027ll do is to swap this row with this row,"},{"Start":"02:40.670 ","End":"02:43.530","Text":"row 4 with row 2 as written here."},{"Start":"02:43.530 ","End":"02:45.840","Text":"Now, this same row is here."},{"Start":"02:45.840 ","End":"02:49.400","Text":"This is good because I have this one here."},{"Start":"02:49.400 ","End":"02:56.065","Text":"Now I can subtract multiples of that second row from the other 2 rows."},{"Start":"02:56.065 ","End":"02:58.720","Text":"Well, in this case, it\u0027s add, not subtract."},{"Start":"02:58.720 ","End":"03:03.260","Text":"I add 9 times this to this and 8 times this to this."},{"Start":"03:03.260 ","End":"03:07.965","Text":"After we do that, we get the 0s here and we already had the 0s here,"},{"Start":"03:07.965 ","End":"03:11.155","Text":"but it\u0027s still not quite up to echelon form."},{"Start":"03:11.155 ","End":"03:13.840","Text":"I would also like a 0 here."},{"Start":"03:13.840 ","End":"03:15.910","Text":"But again, looking for opportunities,"},{"Start":"03:15.910 ","End":"03:23.330","Text":"I noticed that the 3rd row is divisible by 17 and the 4th row by 9, so let\u0027s simplify."},{"Start":"03:23.330 ","End":"03:26.090","Text":"If I do these 2 divisions,"},{"Start":"03:26.090 ","End":"03:28.925","Text":"then we get these 2 rows."},{"Start":"03:28.925 ","End":"03:35.970","Text":"Now all I have to do to get a 0 in the position here,"},{"Start":"03:35.970 ","End":"03:40.160","Text":"it\u0027s to subtract this row from this row."},{"Start":"03:40.160 ","End":"03:44.840","Text":"This is what it is in row notation. This is what we get."},{"Start":"03:44.840 ","End":"03:46.820","Text":"Really we don\u0027t even need the last row,"},{"Start":"03:46.820 ","End":"03:50.480","Text":"so we really have 3 equations and 3 unknowns."},{"Start":"03:50.480 ","End":"03:53.750","Text":"If I go back from matrix to x, y, z,"},{"Start":"03:53.750 ","End":"03:59.975","Text":"then this is the system of linear equations that we get."},{"Start":"03:59.975 ","End":"04:01.460","Text":"But because of the row echelon,"},{"Start":"04:01.460 ","End":"04:03.350","Text":"we can use back substitution."},{"Start":"04:03.350 ","End":"04:05.960","Text":"z, we already have, minus 1."},{"Start":"04:05.960 ","End":"04:07.340","Text":"Plug it in here."},{"Start":"04:07.340 ","End":"04:10.715","Text":"y minus 2 is minus 1,"},{"Start":"04:10.715 ","End":"04:13.535","Text":"so y is 1."},{"Start":"04:13.535 ","End":"04:18.395","Text":"Then go ahead and substitute z and y in here and you\u0027ll get x equals 2."},{"Start":"04:18.395 ","End":"04:21.450","Text":"That\u0027s the answer and we\u0027re done."}],"ID":10501},{"Watched":false,"Name":"Exercise 3 Parts 1-4","Duration":"6m 35s","ChapterTopicVideoID":9561,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9561.jpeg","UploadDate":"2017-07-26T08:32:01.3330000","DurationForVideoObject":"PT6M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.569","Text":"In this exercise, we\u0027re given 3 matrices at different sizes."},{"Start":"00:06.569 ","End":"00:09.075","Text":"This 1 is a 2 by 3."},{"Start":"00:09.075 ","End":"00:11.415","Text":"This is a 3 by 3."},{"Start":"00:11.415 ","End":"00:13.920","Text":"This is also a 3 by 3."},{"Start":"00:13.920 ","End":"00:21.905","Text":"They were practicing arithmetic operations on matrices:"},{"Start":"00:21.905 ","End":"00:29.575","Text":"addition, subtraction, multiplication, and taking of the trace."},{"Start":"00:29.575 ","End":"00:32.480","Text":"Let\u0027s start with number 1."},{"Start":"00:32.480 ","End":"00:37.010","Text":"E plus D, that\u0027s this plus this."},{"Start":"00:37.010 ","End":"00:39.965","Text":"Notice that the both of the same size,"},{"Start":"00:39.965 ","End":"00:42.770","Text":"same order, and if they\u0027re the same order,"},{"Start":"00:42.770 ","End":"00:45.950","Text":"then it\u0027s okay to add them."},{"Start":"00:45.950 ","End":"00:49.400","Text":"When we add them, we add them element-wise,"},{"Start":"00:49.400 ","End":"00:53.300","Text":"meaning in the corresponding position like 4 plus"},{"Start":"00:53.300 ","End":"00:58.400","Text":"1 would give me 5, and that\u0027s this 5 here."},{"Start":"00:58.400 ","End":"01:04.105","Text":"Similarly, 1 plus 4 will give me this 5 here."},{"Start":"01:04.105 ","End":"01:08.595","Text":"1 plus 2 gives me 3."},{"Start":"01:08.595 ","End":"01:12.724","Text":"Minus 1 and 1 is 0."},{"Start":"01:12.724 ","End":"01:14.345","Text":"I think you get the idea."},{"Start":"01:14.345 ","End":"01:15.980","Text":"I\u0027ll leave you to check all the rest."},{"Start":"01:15.980 ","End":"01:17.900","Text":"Maybe I\u0027ll just do them orally."},{"Start":"01:17.900 ","End":"01:19.550","Text":"0 plus 0 is 0,"},{"Start":"01:19.550 ","End":"01:21.575","Text":"1 minus 1 is 0,"},{"Start":"01:21.575 ","End":"01:23.330","Text":"4 and 4 is 8,"},{"Start":"01:23.330 ","End":"01:24.665","Text":"1 and 2 is 3,"},{"Start":"01:24.665 ","End":"01:27.775","Text":"minus 1 and 10 is 9."},{"Start":"01:27.775 ","End":"01:30.240","Text":"Onto part 2."},{"Start":"01:30.240 ","End":"01:34.620","Text":"Part 2 is E minus D plus I_3."},{"Start":"01:34.620 ","End":"01:35.670","Text":"What does I_3 mean?"},{"Start":"01:35.670 ","End":"01:37.700","Text":"It\u0027s the identity matrix,"},{"Start":"01:37.700 ","End":"01:40.000","Text":"but for a 3 by 3."},{"Start":"01:40.000 ","End":"01:42.585","Text":"Here\u0027s E from here,"},{"Start":"01:42.585 ","End":"01:46.770","Text":"D goes here, and this is I_3."},{"Start":"01:46.770 ","End":"01:54.220","Text":"It\u0027s the identity matrix means 1s on the main diagonal and 0s elsewhere."},{"Start":"01:54.770 ","End":"01:57.880","Text":"We could do them 2 at a time,"},{"Start":"01:57.880 ","End":"02:00.370","Text":"but for addition and subtraction,"},{"Start":"02:00.370 ","End":"02:04.585","Text":"we could do them addition and subtraction all at once."},{"Start":"02:04.585 ","End":"02:12.480","Text":"For example, I could take 4 minus 1 plus 1 gives me"},{"Start":"02:12.480 ","End":"02:21.855","Text":"4 and 1 minus 4 plus 0 is minus 3."},{"Start":"02:21.855 ","End":"02:27.040","Text":"1 minus 2 plus 0 is minus 1 and so on."},{"Start":"02:27.040 ","End":"02:30.445","Text":"I\u0027ll just do the last 1."},{"Start":"02:30.445 ","End":"02:36.285","Text":"Minus 1 minus 10 is minus 11"},{"Start":"02:36.285 ","End":"02:43.160","Text":"plus 1, and that gives me minus 10."},{"Start":"02:43.160 ","End":"02:46.040","Text":"There\u0027s a typo, sorry about that."},{"Start":"02:46.040 ","End":"02:49.200","Text":"That will be minus 10."},{"Start":"02:49.220 ","End":"02:52.680","Text":"Again, minus 1 minus 10 is minus 11,"},{"Start":"02:52.680 ","End":"02:54.170","Text":"plus 1 is minus 10."},{"Start":"02:54.170 ","End":"02:56.615","Text":"Maybe you\u0027d better check some of the others."},{"Start":"02:56.615 ","End":"03:01.530","Text":"1 and minus 2 is minus 1 plus 0 is minus 1."},{"Start":"03:01.530 ","End":"03:04.220","Text":"Minus 1, minus 1, and minus 2."},{"Start":"03:04.220 ","End":"03:07.175","Text":"0 minus 0 plus 1 is 1,"},{"Start":"03:07.175 ","End":"03:11.315","Text":"1 minus minus 1 is 2 plus 0."},{"Start":"03:11.315 ","End":"03:15.300","Text":"Looks like the rest are okay."},{"Start":"03:17.420 ","End":"03:20.400","Text":"Onto the next part."},{"Start":"03:20.400 ","End":"03:25.910","Text":"Part 3, 5 times C. There is C. Now a"},{"Start":"03:25.910 ","End":"03:29.000","Text":"scalar or constant times any matrix means that"},{"Start":"03:29.000 ","End":"03:32.330","Text":"we just multiply it by each of the entries."},{"Start":"03:32.330 ","End":"03:34.340","Text":"Like 5 times 1 is 5,"},{"Start":"03:34.340 ","End":"03:36.505","Text":"5 times 4 is 20."},{"Start":"03:36.505 ","End":"03:38.360","Text":"I wrote an intermediate step."},{"Start":"03:38.360 ","End":"03:39.530","Text":"5 times 1, 5 times 4,"},{"Start":"03:39.530 ","End":"03:41.450","Text":"5 times 2, 5 times 4,"},{"Start":"03:41.450 ","End":"03:42.860","Text":"5 times 1, 5 times 5."},{"Start":"03:42.860 ","End":"03:45.355","Text":"Then 5 times 1 is 5, 5 times 4 is 20,"},{"Start":"03:45.355 ","End":"03:47.430","Text":"5 times 2 is 10, 5 times 4 is 20,"},{"Start":"03:47.430 ","End":"03:50.800","Text":"5 times 5 is 25."},{"Start":"03:51.740 ","End":"03:53.580","Text":"That\u0027s straightforward."},{"Start":"03:53.580 ","End":"03:56.960","Text":"Onto next 1, that will be number 4."},{"Start":"03:56.960 ","End":"04:03.900","Text":"Here we are. 2D plus 4E times I_3."},{"Start":"04:03.900 ","End":"04:07.705","Text":"We already had I_3 as the identity 3 by 3 matrix."},{"Start":"04:07.705 ","End":"04:10.485","Text":"There, that\u0027s I_3."},{"Start":"04:10.485 ","End":"04:15.300","Text":"That\u0027s the exercise and now we have to evaluate it."},{"Start":"04:15.890 ","End":"04:18.570","Text":"Several things we need to do."},{"Start":"04:18.570 ","End":"04:20.750","Text":"This 1, we need to multiply by 2."},{"Start":"04:20.750 ","End":"04:22.550","Text":"Now here we have a product of 3 things,"},{"Start":"04:22.550 ","End":"04:24.215","Text":"4 times this times this."},{"Start":"04:24.215 ","End":"04:25.520","Text":"We have 2 choices."},{"Start":"04:25.520 ","End":"04:28.970","Text":"Either multiply 4 times this and then the result by"},{"Start":"04:28.970 ","End":"04:35.165","Text":"this or multiply these 2 first and then 4 times the answer."},{"Start":"04:35.165 ","End":"04:40.430","Text":"Now I\u0027d much rather do this first because this I is not"},{"Start":"04:40.430 ","End":"04:43.340","Text":"just any old matrix, it\u0027s the identity and identity"},{"Start":"04:43.340 ","End":"04:46.570","Text":"means it\u0027s like the number 1 in arithmetic."},{"Start":"04:46.570 ","End":"04:47.810","Text":"If I said to you,"},{"Start":"04:47.810 ","End":"04:49.535","Text":"what is 7 times 1?"},{"Start":"04:49.535 ","End":"04:51.640","Text":"You\u0027d say, 7."},{"Start":"04:51.640 ","End":"04:58.750","Text":"Likewise with matrices, something times the identity matrix is just that something."},{"Start":"04:58.750 ","End":"05:01.640","Text":"I\u0027ll just do this multiplication first,"},{"Start":"05:01.640 ","End":"05:05.375","Text":"which means basically just keeping this and throwing this out."},{"Start":"05:05.375 ","End":"05:07.115","Text":"If you\u0027re not sure,"},{"Start":"05:07.115 ","End":"05:10.010","Text":"you can always take a row and a column like"},{"Start":"05:10.010 ","End":"05:13.640","Text":"this first row with this first column gives us 4 times 1,"},{"Start":"05:13.640 ","End":"05:16.835","Text":"1 times 0, 1 times 0 is 4."},{"Start":"05:16.835 ","End":"05:22.070","Text":"But it\u0027s known that the identity matrix on the left or on the right,"},{"Start":"05:22.070 ","End":"05:24.730","Text":"if you multiply, you get the same thing."},{"Start":"05:24.730 ","End":"05:28.550","Text":"Now we have a couple of scalar times matrix."},{"Start":"05:28.550 ","End":"05:30.230","Text":"We have to multiply 2 times 1,"},{"Start":"05:30.230 ","End":"05:31.850","Text":"2 times 4, 2 times 2, and so on."},{"Start":"05:31.850 ","End":"05:35.050","Text":"Then 4 times 4, 4 times 1, 4 times 1."},{"Start":"05:35.050 ","End":"05:39.260","Text":"It gets tedious. 2 times 1 is 2,"},{"Start":"05:39.260 ","End":"05:41.030","Text":"2 times 0 is 0, 2 times minus 1,"},{"Start":"05:41.030 ","End":"05:42.800","Text":"minus 2, and so on up to the last 1."},{"Start":"05:42.800 ","End":"05:44.570","Text":"2 times 10 is 20."},{"Start":"05:44.570 ","End":"05:46.880","Text":"Here, 4 times minus 1 is minus 4."},{"Start":"05:46.880 ","End":"05:49.144","Text":"I\u0027ll leave you to check that everyone is correct."},{"Start":"05:49.144 ","End":"05:54.095","Text":"Now we have the sum of 2 matrices and they\u0027re both of the same size."},{"Start":"05:54.095 ","End":"05:57.950","Text":"I mean this is a 3 by 3 and this is a 3 by 3."},{"Start":"05:57.950 ","End":"06:00.890","Text":"We\u0027ll add 2 plus 16 is 18,"},{"Start":"06:00.890 ","End":"06:05.220","Text":"8 plus 4 is 12, 4 plus 4."},{"Start":"06:05.810 ","End":"06:09.690","Text":"There\u0027s a typo, this should be 8."},{"Start":"06:09.690 ","End":"06:10.467","Text":"Perhaps I better keep checking."},{"Start":"06:10.467 ","End":"06:13.395","Text":"2 and minus 4 is minus 2, 0 and 0 is 0,"},{"Start":"06:13.395 ","End":"06:18.210","Text":"minus 2 plus 4 is 2,"},{"Start":"06:18.210 ","End":"06:21.135","Text":"8 plus 16 is 24,"},{"Start":"06:21.135 ","End":"06:26.610","Text":"4 and 4 is 8, and 20 minus 4 is 16."},{"Start":"06:26.610 ","End":"06:29.910","Text":"We\u0027re okay except for 1 typo."},{"Start":"06:29.910 ","End":"06:35.970","Text":"Time for a break, I\u0027ll do number 5 in the next clip."}],"ID":10502},{"Watched":false,"Name":"Exercise 3 Part 5","Duration":"5m 19s","ChapterTopicVideoID":9557,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9557.jpeg","UploadDate":"2017-07-26T08:30:49.2030000","DurationForVideoObject":"PT5M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.875","Text":"Continuing from the previous clip where we did Part 1 through 4,"},{"Start":"00:04.875 ","End":"00:12.480","Text":"what remains is number 5 and number 5 contains 2 strange letters tr,"},{"Start":"00:12.480 ","End":"00:17.190","Text":"just want to remind you that means the trace of a matrix."},{"Start":"00:17.190 ","End":"00:19.125","Text":"In case you\u0027ve forgotten,"},{"Start":"00:19.125 ","End":"00:22.650","Text":"to give you quick reminder what the trace is let\u0027s say we have,"},{"Start":"00:22.650 ","End":"00:24.540","Text":"it has to be a square matrix."},{"Start":"00:24.540 ","End":"00:27.089","Text":"Let\u0027s just take a 3 by 3 example."},{"Start":"00:27.089 ","End":"00:32.085","Text":"1, 2, 3, 4, 5, 6, 7, 8, 9."},{"Start":"00:32.085 ","End":"00:36.920","Text":"The trace means the sum of the elements along the diagonal."},{"Start":"00:36.920 ","End":"00:40.415","Text":"If I put tr in front of this,"},{"Start":"00:40.415 ","End":"00:45.215","Text":"it just means 1 plus 5 plus 9,"},{"Start":"00:45.215 ","End":"00:48.380","Text":"which in this case happens to be 15."},{"Start":"00:48.380 ","End":"00:51.320","Text":"So that\u0027s just a reminder of what the trace is."},{"Start":"00:51.320 ","End":"00:53.645","Text":"Now, to do this exercise,"},{"Start":"00:53.645 ","End":"00:54.890","Text":"we have several steps."},{"Start":"00:54.890 ","End":"01:00.760","Text":"We, first of all have to compute inside the brackets D squared minus 2e."},{"Start":"01:00.760 ","End":"01:05.715","Text":"This itself involves squaring D and then multiplying E by 2."},{"Start":"01:05.715 ","End":"01:11.100","Text":"When we got that, then we take the trace and finally we multiply by 2."},{"Start":"01:11.240 ","End":"01:15.720","Text":"So first we\u0027ll do the inside the D squared minus 2E."},{"Start":"01:15.720 ","End":"01:19.725","Text":"Now D squared means D times D. This is D,"},{"Start":"01:19.725 ","End":"01:24.990","Text":"and this is D, and this here is E, and there\u0027s the 2."},{"Start":"01:24.990 ","End":"01:30.790","Text":"So we have to do a matrix multiplication."},{"Start":"01:31.040 ","End":"01:37.385","Text":"For example, the top-left entry will be first row with the first column,"},{"Start":"01:37.385 ","End":"01:39.530","Text":"this times this plus this times this,"},{"Start":"01:39.530 ","End":"01:40.820","Text":"plus this times this."},{"Start":"01:40.820 ","End":"01:44.195","Text":"Here we just multiply everything by 2."},{"Start":"01:44.195 ","End":"01:45.800","Text":"Now if we do this product,"},{"Start":"01:45.800 ","End":"01:49.115","Text":"1 times 1 is 1, 4 times 1 is 4,"},{"Start":"01:49.115 ","End":"01:50.360","Text":"2 times 4 is 8."},{"Start":"01:50.360 ","End":"01:54.835","Text":"If we add them,1 plus 4 plus 8 is 13."},{"Start":"01:54.835 ","End":"02:02.910","Text":"Here, 2 times 4 gives me 8."},{"Start":"02:03.860 ","End":"02:07.980","Text":"Next element, this with this column,"},{"Start":"02:07.980 ","End":"02:18.240","Text":"1 times 4 plus 4 times 0 plus 2 times 2 is 4 plus 0 plus 4 is 8, and minus 2."},{"Start":"02:18.240 ","End":"02:21.975","Text":"Sorry, the minus 2 with 1 is 2."},{"Start":"02:21.975 ","End":"02:24.510","Text":"Have 2 down 7 to go."},{"Start":"02:24.510 ","End":"02:34.520","Text":"Next, this with this 1 times 2 minus 4 times minus 4 times 1,"},{"Start":"02:34.520 ","End":"02:39.320","Text":"it\u0027s a 2 minus 4 is minus 2 plus 20 is 18."},{"Start":"02:39.320 ","End":"02:43.410","Text":"Here 2 times 1 is 2."},{"Start":"02:44.500 ","End":"02:47.675","Text":"Let\u0027s just skip some."},{"Start":"02:47.675 ","End":"02:49.475","Text":"I\u0027ll do another couple."},{"Start":"02:49.475 ","End":"02:57.360","Text":"Maybe let\u0027s do this 1."},{"Start":"02:57.360 ","End":"03:01.800","Text":"So for the middle element we\u0027ll need this with this."},{"Start":"03:01.800 ","End":"03:04.005","Text":"Over here we\u0027ll need this."},{"Start":"03:04.005 ","End":"03:11.410","Text":"Let\u0027s see, 1 times 4 is 4, 0 minus 2."},{"Start":"03:11.930 ","End":"03:15.435","Text":"4 minus 2 is 2."},{"Start":"03:15.435 ","End":"03:17.840","Text":"Over here, the middle is 0,"},{"Start":"03:17.840 ","End":"03:22.160","Text":"it stays 0 even if I multiply by 2. Let\u0027s do 1 more."},{"Start":"03:22.160 ","End":"03:24.395","Text":"The last bottom right,"},{"Start":"03:24.395 ","End":"03:27.135","Text":"the bottom right, third row first."},{"Start":"03:27.135 ","End":"03:29.300","Text":"Third column will need the third row here,"},{"Start":"03:29.300 ","End":"03:30.530","Text":"the third column here,"},{"Start":"03:30.530 ","End":"03:31.940","Text":"and here we\u0027ll need this element."},{"Start":"03:31.940 ","End":"03:41.040","Text":"So let\u0027s check. 4 times 2 is 8 minus 2 plus a 100."},{"Start":"03:41.040 ","End":"03:44.100","Text":"8 minus 2 plus a 100 is indeed a 106,"},{"Start":"03:44.100 ","End":"03:47.765","Text":"and here 2 times minus 1 is indeed minus 2."},{"Start":"03:47.765 ","End":"03:53.195","Text":"I\u0027ll leave you to check the remaining few that are left."},{"Start":"03:53.195 ","End":"03:55.865","Text":"Now let\u0027s do the subtraction."},{"Start":"03:55.865 ","End":"03:58.165","Text":"Of course, I mean,"},{"Start":"03:58.165 ","End":"04:02.165","Text":"normally mentally that this is the same size as this."},{"Start":"04:02.165 ","End":"04:05.460","Text":"So we can do a subtraction."},{"Start":"04:06.350 ","End":"04:11.765","Text":"The first element would be 13 minus 8 is 5."},{"Start":"04:11.765 ","End":"04:15.160","Text":"Here\u0027s that 5."},{"Start":"04:15.560 ","End":"04:18.490","Text":"Let\u0027s just check the ones in yellow."},{"Start":"04:18.490 ","End":"04:25.350","Text":"8 minus 2 is 6, 18 plus minus 2 is 16,"},{"Start":"04:25.350 ","End":"04:28.590","Text":"2 and 0 is 2."},{"Start":"04:28.640 ","End":"04:35.125","Text":"106 minus 2 is a 108,"},{"Start":"04:35.125 ","End":"04:36.910","Text":"and so on and so on."},{"Start":"04:36.910 ","End":"04:39.115","Text":"I\u0027ll leave you to check the others."},{"Start":"04:39.115 ","End":"04:45.295","Text":"Now we have to do the matter of the trace and then to double it."},{"Start":"04:45.295 ","End":"04:49.740","Text":"First of all, just take the trace of this means the diagonal."},{"Start":"04:49.740 ","End":"04:52.775","Text":"So I just left the highlight on the diagonal."},{"Start":"04:52.775 ","End":"04:55.595","Text":"So 5 plus 2 plus a 108,"},{"Start":"04:55.595 ","End":"04:57.800","Text":"which is a 115."},{"Start":"04:57.800 ","End":"05:02.480","Text":"Then finally, I can say that twice this trace is"},{"Start":"05:02.480 ","End":"05:11.490","Text":"twice 115 and this is equal to 230."},{"Start":"05:11.840 ","End":"05:15.405","Text":"That\u0027s the final answer,"},{"Start":"05:15.405 ","End":"05:19.410","Text":"and we\u0027re done with part 5 and with the whole exercise."}],"ID":10503},{"Watched":false,"Name":"Exercise 3 Parts 6-7","Duration":"5m 9s","ChapterTopicVideoID":9562,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9562.jpeg","UploadDate":"2017-07-26T08:32:20.9100000","DurationForVideoObject":"PT5M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.880","Text":"This exercise is really a continuation of the previous 5 exercises."},{"Start":"00:05.880 ","End":"00:09.045","Text":"It\u0027s the last 5 and a 10-part series,"},{"Start":"00:09.045 ","End":"00:11.835","Text":"that\u0027s why they\u0027re numbered from 6 through 10,"},{"Start":"00:11.835 ","End":"00:15.780","Text":"and we even have the same matrices as before."},{"Start":"00:15.780 ","End":"00:18.210","Text":"We\u0027re just not going to be using E here,"},{"Start":"00:18.210 ","End":"00:20.500","Text":"so I didn\u0027t include it."},{"Start":"00:23.150 ","End":"00:25.545","Text":"The first one is number 6."},{"Start":"00:25.545 ","End":"00:33.550","Text":"We want 4 times the transpose of C plus A. Let\u0027s see."},{"Start":"00:34.280 ","End":"00:37.990","Text":"I\u0027m going to copy it first."},{"Start":"00:38.450 ","End":"00:40.950","Text":"We need 4 times,"},{"Start":"00:40.950 ","End":"00:43.010","Text":"now, what is C transpose?"},{"Start":"00:43.010 ","End":"00:47.555","Text":"C is a matrix which is 2 by 3,"},{"Start":"00:47.555 ","End":"00:52.945","Text":"so the transpose has got to be something which is 3 by 2."},{"Start":"00:52.945 ","End":"00:56.100","Text":"In other words, 3 rows, 2 columns."},{"Start":"00:56.100 ","End":"01:03.610","Text":"What\u0027s going to happen is that the first row here will become the first column here,"},{"Start":"01:03.610 ","End":"01:07.865","Text":"and the second row will become the second column,"},{"Start":"01:07.865 ","End":"01:11.480","Text":"so here I can write the 1, 4, 2,"},{"Start":"01:11.480 ","End":"01:12.935","Text":"which I got from here,"},{"Start":"01:12.935 ","End":"01:16.985","Text":"and then the 4, 1, 5, I can write here,"},{"Start":"01:16.985 ","End":"01:23.145","Text":"and then plus and then A is just as is,"},{"Start":"01:23.145 ","End":"01:28.080","Text":"is 4, 0, 1, 2, minus 1,1,"},{"Start":"01:28.080 ","End":"01:35.000","Text":"and note that this is a 3 by 2 because they\u0027re both with the same size,"},{"Start":"01:35.000 ","End":"01:37.370","Text":"the same order, I can add them."},{"Start":"01:37.370 ","End":"01:41.450","Text":"If I didn\u0027t have a transpose here, I couldn\u0027t add them."},{"Start":"01:41.450 ","End":"01:45.515","Text":"I couldn\u0027t add C plus A, for example."},{"Start":"01:45.515 ","End":"01:48.510","Text":"Now at this point,"},{"Start":"01:48.730 ","End":"01:51.375","Text":"2 things to do, actually."},{"Start":"01:51.375 ","End":"01:55.605","Text":"We have to first do the multiplication,"},{"Start":"01:55.605 ","End":"01:59.360","Text":"and what we do is we multiply it element-wise,"},{"Start":"01:59.360 ","End":"02:01.670","Text":"meaning 4 times each of the elements."},{"Start":"02:01.670 ","End":"02:04.020","Text":"4 times 1 is 4,"},{"Start":"02:04.020 ","End":"02:05.849","Text":"4 times 4, 16,"},{"Start":"02:05.849 ","End":"02:07.245","Text":"4 times 4, 16,"},{"Start":"02:07.245 ","End":"02:08.985","Text":"4 times 1 is 4,"},{"Start":"02:08.985 ","End":"02:10.320","Text":"4 times 2 is 8,"},{"Start":"02:10.320 ","End":"02:12.330","Text":"and 4 times 5 is 10,"},{"Start":"02:12.330 ","End":"02:16.995","Text":"plus now just a copy paste,"},{"Start":"02:16.995 ","End":"02:20.970","Text":"I should be writing the equals."},{"Start":"02:20.970 ","End":"02:22.985","Text":"Finally, we\u0027ll get the answer,"},{"Start":"02:22.985 ","End":"02:25.975","Text":"which will also be since it\u0027s a 3 by 2,"},{"Start":"02:25.975 ","End":"02:27.280","Text":"this is 3 by 2,"},{"Start":"02:27.280 ","End":"02:29.560","Text":"we\u0027re also going to get a 3 by 2 here,"},{"Start":"02:29.560 ","End":"02:34.320","Text":"meaning something here element-wise."},{"Start":"02:34.320 ","End":"02:35.865","Text":"4 plus 4 is 8,"},{"Start":"02:35.865 ","End":"02:38.250","Text":"16 and 0, 16,"},{"Start":"02:38.250 ","End":"02:40.350","Text":"16 and 1, 17,"},{"Start":"02:40.350 ","End":"02:42.300","Text":"4 and 2, 6,"},{"Start":"02:42.300 ","End":"02:44.760","Text":"8 and minus 1, 7,"},{"Start":"02:44.760 ","End":"02:47.070","Text":"10 and 1, 11."},{"Start":"02:47.070 ","End":"02:50.925","Text":"I just noticed that 4 times 5 is actually 20,"},{"Start":"02:50.925 ","End":"02:52.665","Text":"believe it or not,"},{"Start":"02:52.665 ","End":"02:55.530","Text":"and therefore, 20 plus 1 is 21,"},{"Start":"02:55.530 ","End":"03:01.005","Text":"and now is the answer to part 6."},{"Start":"03:01.005 ","End":"03:02.170","Text":"Let\u0027s move on."},{"Start":"03:02.170 ","End":"03:10.155","Text":"This time we want 1/2 of A transpose and a 1/4 of C as is."},{"Start":"03:10.155 ","End":"03:13.875","Text":"What we get is 1/2."},{"Start":"03:13.875 ","End":"03:18.950","Text":"Now, A transpose will be to reverse rows and columns here,"},{"Start":"03:18.950 ","End":"03:24.320","Text":"so actually what we\u0027ll get is instead of a 3 by 2,"},{"Start":"03:24.320 ","End":"03:28.370","Text":"which is this we\u0027ll get a 2 by 3."},{"Start":"03:28.370 ","End":"03:32.585","Text":"The first column becomes the first row,"},{"Start":"03:32.585 ","End":"03:35.120","Text":"and the second column, 0, 2, 1"},{"Start":"03:35.120 ","End":"03:37.565","Text":"becomes the second column here."},{"Start":"03:37.565 ","End":"03:42.714","Text":"Now a 1/4 and then C is,"},{"Start":"03:42.714 ","End":"03:46.620","Text":"let\u0027s just copy 1, 4, 2,"},{"Start":"03:46.620 ","End":"03:51.410","Text":"4, 1, 5, and this is also 2 by 3,"},{"Start":"03:51.410 ","End":"03:54.515","Text":"which is good because now we can add them,"},{"Start":"03:54.515 ","End":"03:57.245","Text":"but we\u0027ll do the multiplications first,"},{"Start":"03:57.245 ","End":"03:59.360","Text":"and those don\u0027t change the size,"},{"Start":"03:59.360 ","End":"04:01.395","Text":"we just apply them element-wise."},{"Start":"04:01.395 ","End":"04:06.440","Text":"1/2 of 4 is 2, a 1/2, minus a 1/2,"},{"Start":"04:06.440 ","End":"04:15.410","Text":"0, 1, 1/2 plus here a 1/4, so it\u0027s 1/4."},{"Start":"04:15.410 ","End":"04:19.820","Text":"1/4 times 2 is a 1/2 times 4 is 1,"},{"Start":"04:19.820 ","End":"04:23.260","Text":"1/4, 5 over 4."},{"Start":"04:23.260 ","End":"04:26.460","Text":"These are both 2 by 3 matrices,"},{"Start":"04:26.460 ","End":"04:27.705","Text":"so we can add them,"},{"Start":"04:27.705 ","End":"04:32.535","Text":"and what we\u0027ll get will also be a 2 by 3 matrix."},{"Start":"04:32.535 ","End":"04:35.925","Text":"Element-wise we add 2 plus 1/4,"},{"Start":"04:35.925 ","End":"04:38.115","Text":"2 and 1/4,"},{"Start":"04:38.115 ","End":"04:41.565","Text":"1/2 plus 1, 1 and a 1/2."},{"Start":"04:41.565 ","End":"04:44.835","Text":"Minus a 1/2 and a 1/2, 0."},{"Start":"04:44.835 ","End":"04:47.505","Text":"0 plus 1 is 1."},{"Start":"04:47.505 ","End":"04:50.040","Text":"1 plus 1/4, 1 and 1/4."},{"Start":"04:50.040 ","End":"04:55.155","Text":"A 1/2 plus, this is 1 and 1/4,"},{"Start":"04:55.155 ","End":"04:57.945","Text":"so that\u0027s 1 and 3/4."},{"Start":"04:57.945 ","End":"05:00.285","Text":"Leave it in mixed numbers,"},{"Start":"05:00.285 ","End":"05:06.265","Text":"and this is the answer to number 7."},{"Start":"05:06.265 ","End":"05:10.080","Text":"Take a break now, continue in the next."}],"ID":10504},{"Watched":false,"Name":"Exercise 3 Part 8","Duration":"4m 17s","ChapterTopicVideoID":9558,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9558.jpeg","UploadDate":"2017-07-26T08:31:09.8170000","DurationForVideoObject":"PT4M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"Continuing with the same exercise,"},{"Start":"00:02.580 ","End":"00:05.940","Text":"this time we\u0027re up to number 8."},{"Start":"00:05.940 ","End":"00:10.215","Text":"We want the identity of size 2,"},{"Start":"00:10.215 ","End":"00:14.340","Text":"I_2 times B times C. We know what this is."},{"Start":"00:14.340 ","End":"00:16.035","Text":"It\u0027s not in this list, of course."},{"Start":"00:16.035 ","End":"00:20.070","Text":"It just means the identity matrix,"},{"Start":"00:20.070 ","End":"00:24.510","Text":"which is 1, 0, 0, 1."},{"Start":"00:24.510 ","End":"00:27.120","Text":"Identity for size 2 by 2,"},{"Start":"00:27.120 ","End":"00:28.830","Text":"1 is on the diagonal,"},{"Start":"00:28.830 ","End":"00:32.260","Text":"0 is elsewhere on a 2 by 2."},{"Start":"00:33.920 ","End":"00:41.385","Text":"B, I copy from here, 4, 1, 0, minus 2."},{"Start":"00:41.385 ","End":"00:45.905","Text":"C, from here, it\u0027s going to be 1,"},{"Start":"00:45.905 ","End":"00:51.605","Text":"4, 2, 4, 1, 5. There we are."},{"Start":"00:51.605 ","End":"00:56.795","Text":"Now, we should really do a check first to see everything is okay."},{"Start":"00:56.795 ","End":"01:02.190","Text":"I mean about the size, 2 by 2,"},{"Start":"01:02.190 ","End":"01:09.090","Text":"this 1 is 2 by 2 and this 1 is 2 by 3."},{"Start":"01:09.090 ","End":"01:11.870","Text":"Now, when we do a multiplication,"},{"Start":"01:11.870 ","End":"01:14.269","Text":"we have to make sure that the number of columns"},{"Start":"01:14.269 ","End":"01:16.790","Text":"in the first is equal to number of rows in the second."},{"Start":"01:16.790 ","End":"01:19.100","Text":"This has to be throughout."},{"Start":"01:19.100 ","End":"01:23.390","Text":"For example, this 2 and this 2 are the same, we\u0027re okay."},{"Start":"01:23.390 ","End":"01:27.170","Text":"This 2 and this 2 are same, we\u0027re okay."},{"Start":"01:27.170 ","End":"01:34.105","Text":"We even know that the answer is going to be 2 by 3."},{"Start":"01:34.105 ","End":"01:36.500","Text":"We can do them in any order,"},{"Start":"01:36.500 ","End":"01:41.600","Text":"the multiplications, but because we have an identity matrix here,"},{"Start":"01:41.600 ","End":"01:44.390","Text":"it\u0027s easiest to multiply these 2 first,"},{"Start":"01:44.390 ","End":"01:46.220","Text":"which is probably what you would have d1 anyway,"},{"Start":"01:46.220 ","End":"01:50.060","Text":"because an identity times something is that something itself."},{"Start":"01:50.060 ","End":"01:54.575","Text":"It\u0027s like the 1 in numbers, 1 times 5 equals 5."},{"Start":"01:54.575 ","End":"01:57.630","Text":"This times this is just this."},{"Start":"01:59.810 ","End":"02:08.670","Text":"What we get is this times this first it doesn\u0027t matter which you multiply first,"},{"Start":"02:08.670 ","End":"02:11.115","Text":"this times this and this times this, but you can\u0027t change the order."},{"Start":"02:11.115 ","End":"02:12.515","Text":"I\u0027m doing this with this."},{"Start":"02:12.515 ","End":"02:15.970","Text":"Like I said, it\u0027s going to be just 4, 1,"},{"Start":"02:15.970 ","End":"02:20.285","Text":"0, minus 2 because of the special properties of the identity."},{"Start":"02:20.285 ","End":"02:23.905","Text":"Of course, you could multiply it out again if you forgot that,"},{"Start":"02:23.905 ","End":"02:29.140","Text":"1, 0 with 4, 0 is 1 times 4 plus 0 times 0 and so on."},{"Start":"02:32.090 ","End":"02:34.620","Text":"The other 1 I\u0027ll just write."},{"Start":"02:34.620 ","End":"02:39.715","Text":"Again, the 1, 4, 2, 4, 1, 5."},{"Start":"02:39.715 ","End":"02:41.630","Text":"We don\u0027t need to check anymore,"},{"Start":"02:41.630 ","End":"02:45.740","Text":"but this is a 2 by 2 and this is a 2 by 3 and a number"},{"Start":"02:45.740 ","End":"02:51.595","Text":"of columns here is equal to the number of rows here."},{"Start":"02:51.595 ","End":"02:55.705","Text":"We just have to do the actual calculations."},{"Start":"02:55.705 ","End":"02:59.820","Text":"We know we\u0027re going to get a 2 by 3. Maybe I\u0027ll do that again."},{"Start":"02:59.820 ","End":"03:04.950","Text":"This is 2 by 2."},{"Start":"03:04.950 ","End":"03:07.460","Text":"This 1 is 2 by 3."},{"Start":"03:07.460 ","End":"03:10.615","Text":"The middle 1s are equal."},{"Start":"03:10.615 ","End":"03:13.690","Text":"The answer comes out to be the first and the last."},{"Start":"03:13.690 ","End":"03:16.250","Text":"We\u0027re looking at a 2 by 3."},{"Start":"03:16.250 ","End":"03:18.325","Text":"Just leave space."},{"Start":"03:18.325 ","End":"03:21.610","Text":"This is going to be a 2 by 3. Let\u0027s get started."},{"Start":"03:21.610 ","End":"03:23.260","Text":"The first 1,"},{"Start":"03:23.260 ","End":"03:26.035","Text":"we get this times this,"},{"Start":"03:26.035 ","End":"03:29.900","Text":"4 times 1 plus 1 times 4 is 8."},{"Start":"03:29.900 ","End":"03:36.945","Text":"Next is 4, 1 with 4, 1, 16 plus 1 is 17."},{"Start":"03:36.945 ","End":"03:41.460","Text":"Now, 4, 1 with 2, 5, so it\u0027s 4 times 2 is 8,"},{"Start":"03:41.460 ","End":"03:44.910","Text":"plus 5 is 13."},{"Start":"03:44.910 ","End":"03:50.835","Text":"Next, this with this minus 8,"},{"Start":"03:50.835 ","End":"03:57.840","Text":"this with this 0 times 4 minus 2 times 1 minus 2."},{"Start":"03:57.840 ","End":"04:01.635","Text":"Finally, this with this is just"},{"Start":"04:01.635 ","End":"04:09.780","Text":"minus 2 times 5."},{"Start":"04:09.780 ","End":"04:12.940","Text":"It\u0027s minus 10."},{"Start":"04:14.030 ","End":"04:17.800","Text":"This is the answer to number 8."}],"ID":10505},{"Watched":false,"Name":"Exercise 3 Part 9","Duration":"3m 26s","ChapterTopicVideoID":9559,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9559.jpeg","UploadDate":"2017-07-26T08:31:19.5830000","DurationForVideoObject":"PT3M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"We\u0027re now up to number 9,"},{"Start":"00:02.670 ","End":"00:09.615","Text":"which wants the trace of C transpose times C. I\u0027ll just write it in brackets for clarity."},{"Start":"00:09.615 ","End":"00:12.630","Text":"To remind you also that tr,"},{"Start":"00:12.630 ","End":"00:14.864","Text":"which is pronounced trace,"},{"Start":"00:14.864 ","End":"00:21.010","Text":"is just the sum of the elements along the diagonal."},{"Start":"00:21.890 ","End":"00:25.440","Text":"Let\u0027s see now if it even makes sense,"},{"Start":"00:25.440 ","End":"00:27.510","Text":"as far as size goes,"},{"Start":"00:27.510 ","End":"00:32.130","Text":"C is a matrix which is 2 by 3."},{"Start":"00:32.130 ","End":"00:40.375","Text":"That means that C transpose will be a matrix that\u0027s 3 by 2."},{"Start":"00:40.375 ","End":"00:42.860","Text":"Thus this product makes sense."},{"Start":"00:42.860 ","End":"00:48.830","Text":"Well, C trace is a matrix which is 3 by 2,"},{"Start":"00:48.830 ","End":"00:53.190","Text":"just saying it\u0027s size and C has size,"},{"Start":"00:53.190 ","End":"00:55.765","Text":"we already said 2 by 3."},{"Start":"00:55.765 ","End":"00:57.925","Text":"Now if we multiply them,"},{"Start":"00:57.925 ","End":"01:02.000","Text":"we\u0027ll be okay because the middle numbers are the same,"},{"Start":"01:02.000 ","End":"01:04.240","Text":"the 2 with the 2 is the same."},{"Start":"01:04.240 ","End":"01:07.035","Text":"The result comes out the first and the last."},{"Start":"01:07.035 ","End":"01:09.210","Text":"We\u0027re going to get a 3 by 3."},{"Start":"01:09.210 ","End":"01:12.260","Text":"If this thing is a 3 by 3 for 1 thing,"},{"Start":"01:12.260 ","End":"01:14.875","Text":"that means that it\u0027s a square matrix."},{"Start":"01:14.875 ","End":"01:19.025","Text":"If it\u0027s a square matrix and it has a diagonal and we can do the trace."},{"Start":"01:19.025 ","End":"01:21.950","Text":"The sizes are all fine, the orders."},{"Start":"01:21.950 ","End":"01:24.590","Text":"Let\u0027s just do the computation."},{"Start":"01:24.590 ","End":"01:28.640","Text":"Now, C trace is going to be just the opposite of"},{"Start":"01:28.640 ","End":"01:33.020","Text":"C. The rows become columns. We\u0027ve seen this before."},{"Start":"01:33.020 ","End":"01:37.010","Text":"1, 4, 2 becomes 1, 4, 2, 4,"},{"Start":"01:37.010 ","End":"01:41.415","Text":"1, 5, 4, 1, 5."},{"Start":"01:41.415 ","End":"01:44.760","Text":"Now we multiply it by as is 1,"},{"Start":"01:44.760 ","End":"01:48.680","Text":"4, 2, 4, 1, 5."},{"Start":"01:48.680 ","End":"01:51.805","Text":"Now we said our answer is going to be 3 by 3."},{"Start":"01:51.805 ","End":"01:53.250","Text":"Well, let\u0027s see."},{"Start":"01:53.250 ","End":"01:55.755","Text":"I\u0027ll leave room for 3 by 3."},{"Start":"01:55.755 ","End":"01:59.985","Text":"First of all, do this with this."},{"Start":"01:59.985 ","End":"02:04.005","Text":"We get 1 plus 16 is 17."},{"Start":"02:04.005 ","End":"02:09.420","Text":"Next, this with this we get 4 plus 4 is 8."},{"Start":"02:09.420 ","End":"02:15.880","Text":"This with this which is 2 plus 20 is 22."},{"Start":"02:15.880 ","End":"02:19.540","Text":"Second row, it\u0027s going to be this with each of the 3."},{"Start":"02:19.540 ","End":"02:24.135","Text":"First of all, with this 4 plus 4 is 8."},{"Start":"02:24.135 ","End":"02:29.760","Text":"Then 16 and 1 is 17."},{"Start":"02:29.760 ","End":"02:33.600","Text":"Then 8 and 5 is 13."},{"Start":"02:33.600 ","End":"02:37.410","Text":"To the last row quickly 2, 5 with 1,"},{"Start":"02:37.410 ","End":"02:44.109","Text":"4 is 2 plus 20 is 22, it\u0027s the next 1."},{"Start":"02:44.109 ","End":"02:46.770","Text":"8 and 5 is 13,"},{"Start":"02:46.770 ","End":"02:52.265","Text":"and then 4 and 25 is 29."},{"Start":"02:52.265 ","End":"02:56.650","Text":"Okay, that\u0027s just C transpose times C. I\u0027ll remind you,"},{"Start":"02:56.650 ","End":"03:00.905","Text":"C transpose C. But what we want is the trace."},{"Start":"03:00.905 ","End":"03:05.900","Text":"The trace of C transpose C is equal to,"},{"Start":"03:05.900 ","End":"03:08.420","Text":"remember the sum of the diagonal."},{"Start":"03:08.420 ","End":"03:10.510","Text":"Here\u0027s the diagonal."},{"Start":"03:10.510 ","End":"03:17.000","Text":"What do we get? 17 plus 17 plus 29,"},{"Start":"03:17.000 ","End":"03:20.135","Text":"that\u0027s 34 plus 29,"},{"Start":"03:20.135 ","End":"03:23.450","Text":"that should be 63."},{"Start":"03:23.450 ","End":"03:27.150","Text":"That\u0027s the answer to Part 9."}],"ID":10506},{"Watched":false,"Name":"Exercise 3 Part 10","Duration":"7m 49s","ChapterTopicVideoID":9560,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9560.jpeg","UploadDate":"2017-07-26T08:31:42.2370000","DurationForVideoObject":"PT7M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:03.540","Text":"Now we\u0027re at number 10,"},{"Start":"00:03.540 ","End":"00:05.685","Text":"which is the last 1 in the series."},{"Start":"00:05.685 ","End":"00:08.565","Text":"We have to multiply 4 matrices,"},{"Start":"00:08.565 ","End":"00:11.835","Text":"D times A times B times C. Now,"},{"Start":"00:11.835 ","End":"00:14.385","Text":"we could do it in many ways,"},{"Start":"00:14.385 ","End":"00:16.230","Text":"as long as we don\u0027t change the order,"},{"Start":"00:16.230 ","End":"00:17.700","Text":"we can change the grouping."},{"Start":"00:17.700 ","End":"00:20.910","Text":"We could do D times A and the result of that times B,"},{"Start":"00:20.910 ","End":"00:22.680","Text":"and the result of that times C,"},{"Start":"00:22.680 ","End":"00:24.315","Text":"that\u0027s probably how I\u0027m going to do it,"},{"Start":"00:24.315 ","End":"00:29.670","Text":"but we could also do it B times C and then A times the result of that,"},{"Start":"00:29.670 ","End":"00:31.365","Text":"D times a result of that,"},{"Start":"00:31.365 ","End":"00:33.770","Text":"or we could do D times A separately,"},{"Start":"00:33.770 ","End":"00:35.090","Text":"B times C separately,"},{"Start":"00:35.090 ","End":"00:39.020","Text":"and then multiply the result of this by that and any kind of combinations."},{"Start":"00:39.020 ","End":"00:48.160","Text":"But it all has to match with the sizes let\u0027s see D is a 3 by 3."},{"Start":"00:48.160 ","End":"00:55.320","Text":"That\u0027s the D. A is 3 by 2,"},{"Start":"00:55.320 ","End":"00:58.200","Text":"and what else do we have?"},{"Start":"00:58.200 ","End":"01:00.765","Text":"B is 2 by 2,"},{"Start":"01:00.765 ","End":"01:02.370","Text":"and then we have C,"},{"Start":"01:02.370 ","End":"01:04.770","Text":"which is 2 by 3,"},{"Start":"01:04.770 ","End":"01:07.265","Text":"2 rows, 3 columns."},{"Start":"01:07.265 ","End":"01:09.760","Text":"Now anytime you multiply,"},{"Start":"01:09.760 ","End":"01:11.740","Text":"the middle numbers have to be the same."},{"Start":"01:11.740 ","End":"01:14.185","Text":"This matches with this, that\u0027s fine."},{"Start":"01:14.185 ","End":"01:16.944","Text":"This matches with this, that\u0027s fine."},{"Start":"01:16.944 ","End":"01:18.460","Text":"This matches with this,"},{"Start":"01:18.460 ","End":"01:21.460","Text":"that\u0027s fine, and you know that at the end,"},{"Start":"01:21.460 ","End":"01:25.795","Text":"we\u0027re going to get an answer that\u0027s 3 by 3,"},{"Start":"01:25.795 ","End":"01:28.970","Text":"and doesn\u0027t matter in what order you do it."},{"Start":"01:29.090 ","End":"01:34.170","Text":"As I said, let\u0027s just multiply each time,"},{"Start":"01:34.170 ","End":"01:36.760","Text":"we\u0027ll start with D and add 1 more on the right,"},{"Start":"01:36.760 ","End":"01:43.125","Text":"D times A and the result of that times B. I\u0027m going to start with D, A,"},{"Start":"01:43.125 ","End":"01:46.935","Text":"and D is 1, 4,"},{"Start":"01:46.935 ","End":"01:48.840","Text":"2,1, 0,"},{"Start":"01:48.840 ","End":"01:52.455","Text":"minus 1, 4, 2, 10."},{"Start":"01:52.455 ","End":"01:54.435","Text":"From here I copied, of course,"},{"Start":"01:54.435 ","End":"01:57.105","Text":"and then times A,"},{"Start":"01:57.105 ","End":"02:03.390","Text":"and A is a 3 by 2 from here 4,"},{"Start":"02:03.390 ","End":"02:06.975","Text":"0, 1, 2,"},{"Start":"02:06.975 ","End":"02:09.060","Text":"minus 1, 1,"},{"Start":"02:09.060 ","End":"02:17.380","Text":"and 3 by 3 and then 3 by 2 is going to equal also 3 by 2."},{"Start":"02:17.620 ","End":"02:21.155","Text":"It\u0027s going to equal something like this."},{"Start":"02:21.155 ","End":"02:26.405","Text":"Because, well, you can see it from here from the first 2 and you get a 3 by 2."},{"Start":"02:26.405 ","End":"02:34.245","Text":"Now, here goes this with this gives me let\u0027s see 4,"},{"Start":"02:34.245 ","End":"02:40.450","Text":"and 4 minus 2, 6."},{"Start":"02:40.450 ","End":"02:42.710","Text":"Next this with this,"},{"Start":"02:42.710 ","End":"02:47.745","Text":"so 0 plus 8 plus 2 is 10."},{"Start":"02:47.745 ","End":"02:50.990","Text":"Now this with each of the 2 columns."},{"Start":"02:50.990 ","End":"03:00.050","Text":"First of all, with this 1 we\u0027ve got 4 and 0 and 1, that\u0027s 5."},{"Start":"03:00.050 ","End":"03:04.945","Text":"Then 0 and 0 and minus 1."},{"Start":"03:04.945 ","End":"03:12.285","Text":"Then 4 with 4 is 16 and 2 is 18,"},{"Start":"03:12.285 ","End":"03:15.540","Text":"minus 10 is 8,"},{"Start":"03:15.540 ","End":"03:23.500","Text":"and finally, 0 and 4 and 10 is 14."},{"Start":"03:23.660 ","End":"03:28.335","Text":"To remind you, this is D times A, this is DA."},{"Start":"03:28.335 ","End":"03:30.390","Text":"Now I want to compute DA,"},{"Start":"03:30.390 ","End":"03:36.385","Text":"I\u0027ll take the result of this,"},{"Start":"03:36.385 ","End":"03:39.405","Text":"which is the 6, 10,"},{"Start":"03:39.405 ","End":"03:41.565","Text":"5, minus 1,"},{"Start":"03:41.565 ","End":"03:47.625","Text":"8, 14, and multiply it by B,"},{"Start":"03:47.625 ","End":"03:51.270","Text":"which is, I\u0027ve lost it."},{"Start":"03:51.270 ","End":"03:52.725","Text":"There it is 4,"},{"Start":"03:52.725 ","End":"03:55.930","Text":"1, 0 minus 2."},{"Start":"04:00.090 ","End":"04:02.725","Text":"Is this the right size?"},{"Start":"04:02.725 ","End":"04:04.060","Text":"Yeah, 2 columns here,"},{"Start":"04:04.060 ","End":"04:07.165","Text":"2 rows here. That\u0027s okay."},{"Start":"04:07.165 ","End":"04:12.944","Text":"This is going to equal also,"},{"Start":"04:12.944 ","End":"04:16.810","Text":"this is a 3 by 2 times 2 times 2."},{"Start":"04:16.810 ","End":"04:19.825","Text":"It\u0027s going to be also a 3 by 2."},{"Start":"04:19.825 ","End":"04:21.750","Text":"I\u0027ll do it a bit quicker."},{"Start":"04:21.750 ","End":"04:26.765","Text":"This row with this column 24 and 0,"},{"Start":"04:26.765 ","End":"04:30.340","Text":"sorry, that\u0027s a 10."},{"Start":"04:30.410 ","End":"04:34.905","Text":"That\u0027s right, yeah 10 with 0 is 24."},{"Start":"04:34.905 ","End":"04:39.185","Text":"Then the first row with second column,"},{"Start":"04:39.185 ","End":"04:45.530","Text":"6 minus 20 is minus 14."},{"Start":"04:45.530 ","End":"04:47.405","Text":"Then this middle row,"},{"Start":"04:47.405 ","End":"04:51.420","Text":"with this, it\u0027s 20 minus 0,"},{"Start":"04:51.620 ","End":"04:57.600","Text":"and with this it\u0027s going to be 5 plus 2,"},{"Start":"04:57.600 ","End":"05:00.315","Text":"and then 8, 14,"},{"Start":"05:00.315 ","End":"05:05.130","Text":"with this, it\u0027s going to be just 32 and 0,"},{"Start":"05:05.130 ","End":"05:11.415","Text":"and with this it\u0027s going to be 8 minus 28 is minus 20."},{"Start":"05:11.415 ","End":"05:15.630","Text":"What we have here now is DAB,"},{"Start":"05:15.630 ","End":"05:18.335","Text":"so all we have to do is multiply this,"},{"Start":"05:18.335 ","End":"05:19.725","Text":"because we\u0027ve got DAB,"},{"Start":"05:19.725 ","End":"05:21.855","Text":"multiply that by C,"},{"Start":"05:21.855 ","End":"05:23.610","Text":"and we\u0027ll have our answer."},{"Start":"05:23.610 ","End":"05:26.775","Text":"Now C is,"},{"Start":"05:26.775 ","End":"05:29.160","Text":"I see it there, 1,"},{"Start":"05:29.160 ","End":"05:32.730","Text":"4, 2, 4, 1, 5."},{"Start":"05:32.730 ","End":"05:34.710","Text":"Just remember that."},{"Start":"05:34.710 ","End":"05:37.070","Text":"That is 1,"},{"Start":"05:37.070 ","End":"05:40.490","Text":"4, 2, 4, 1, 5,"},{"Start":"05:40.490 ","End":"05:46.585","Text":"that was C. Now I have to take the DAB and I just copy it from here,"},{"Start":"05:46.585 ","End":"05:50.790","Text":"24 minus 14,"},{"Start":"05:50.790 ","End":"05:55.125","Text":"27, 32, minus 20."},{"Start":"05:55.125 ","End":"05:56.340","Text":"A 3 by 2,"},{"Start":"05:56.340 ","End":"06:00.430","Text":"then a 2 by 3 the answer is going to be 3 by 3."},{"Start":"06:01.040 ","End":"06:09.165","Text":"Let\u0027s see, 24 times 1 minus 14 times 4 is 24"},{"Start":"06:09.165 ","End":"06:17.540","Text":"minus 56 is minus 32."},{"Start":"06:17.540 ","End":"06:21.430","Text":"I\u0027ll keep going with the first row next times 4,"},{"Start":"06:21.430 ","End":"06:25.315","Text":"1 to give me 96 minus 14,"},{"Start":"06:25.315 ","End":"06:29.660","Text":"that would be 82,"},{"Start":"06:30.350 ","End":"06:37.570","Text":"and then this row with this column gives me 48 minus"},{"Start":"06:37.570 ","End":"06:47.200","Text":"70 that makes it minus 22."},{"Start":"06:47.970 ","End":"06:52.630","Text":"Next, I\u0027ll take this row with each of these 3 columns with 1,"},{"Start":"06:52.630 ","End":"06:58.125","Text":"4, it gives me 20 plus 28, and that\u0027s 48."},{"Start":"06:58.125 ","End":"06:59.670","Text":"Here, with this column,"},{"Start":"06:59.670 ","End":"07:04.395","Text":"it gives me 80 plus 7, 87."},{"Start":"07:04.395 ","End":"07:09.845","Text":"With the last column I\u0027ve got 40 plus 35, 75."},{"Start":"07:09.845 ","End":"07:12.580","Text":"Now the last row with the first column,"},{"Start":"07:12.580 ","End":"07:17.530","Text":"it\u0027s 32 minus 80,"},{"Start":"07:17.530 ","End":"07:21.585","Text":"which will be minus 48,"},{"Start":"07:21.585 ","End":"07:23.925","Text":"and then with the second column,"},{"Start":"07:23.925 ","End":"07:30.930","Text":"it\u0027s 128 minus 20 is 108,"},{"Start":"07:30.930 ","End":"07:41.490","Text":"and then with this 1 it\u0027s 64 minus 100 is minus 36."},{"Start":"07:41.490 ","End":"07:43.990","Text":"This is DABC,"},{"Start":"07:43.990 ","End":"07:46.370","Text":"and this is the answer,"},{"Start":"07:46.370 ","End":"07:49.860","Text":"and we are done."}],"ID":10507},{"Watched":false,"Name":"Exercise 4","Duration":"4m 12s","ChapterTopicVideoID":9563,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9563.jpeg","UploadDate":"2017-07-26T08:32:37.4870000","DurationForVideoObject":"PT4M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.550","Text":"In this exercise, we\u0027re given a couple of systems of linear equations."},{"Start":"00:05.550 ","End":"00:08.190","Text":"We want to rewrite them in matrix form,"},{"Start":"00:08.190 ","End":"00:13.590","Text":"where the matrix form in some matrix times this will turn out to be a column vector."},{"Start":"00:13.590 ","End":"00:19.110","Text":"That\u0027s why it\u0027s a lowercase with underscore and it\u0027s equal to some other column vector."},{"Start":"00:19.110 ","End":"00:21.360","Text":"Now, what we do is"},{"Start":"00:21.360 ","End":"00:30.645","Text":"we just notice that there are coefficients that are not written like 1y here,"},{"Start":"00:30.645 ","End":"00:34.005","Text":"1 minus 1z, and so on."},{"Start":"00:34.005 ","End":"00:36.350","Text":"If something\u0027s missing, it would be a 0."},{"Start":"00:36.350 ","End":"00:37.550","Text":"Well, that will be in the next 1."},{"Start":"00:37.550 ","End":"00:40.730","Text":"For example, here we would have 0x."},{"Start":"00:44.230 ","End":"00:46.745","Text":"Here\u0027s again."},{"Start":"00:46.745 ","End":"00:52.770","Text":"Now let\u0027s take the coefficients 2 like here\u0027s the 1,"},{"Start":"00:52.770 ","End":"00:53.970","Text":"here\u0027s the 1,"},{"Start":"00:53.970 ","End":"00:56.670","Text":"here\u0027s the 1, so we have 2,"},{"Start":"00:56.670 ","End":"00:58.680","Text":"1 minus 1, 1,"},{"Start":"00:58.680 ","End":"01:00.690","Text":"2 minus 4, 6,"},{"Start":"01:00.690 ","End":"01:01.980","Text":"4, and then another 1."},{"Start":"01:01.980 ","End":"01:05.020","Text":"I can actually write it. We just think it usually."},{"Start":"01:05.020 ","End":"01:10.190","Text":"Then we write the column vector of variables x,"},{"Start":"01:10.190 ","End":"01:15.050","Text":"y, z, that\u0027s what I call x vector underscore."},{"Start":"01:15.050 ","End":"01:19.175","Text":"Then the vector of constants call that B."},{"Start":"01:19.175 ","End":"01:21.650","Text":"You can actually check."},{"Start":"01:21.650 ","End":"01:23.810","Text":"If we do a matrix multiplication,"},{"Start":"01:23.810 ","End":"01:28.625","Text":"what this says is that this times this is this,"},{"Start":"01:28.625 ","End":"01:35.195","Text":"and if you do it, you see it\u0027s 2 times x plus 1 times y minus 1 times z equals 3."},{"Start":"01:35.195 ","End":"01:37.510","Text":"That\u0027s exactly the first row."},{"Start":"01:37.510 ","End":"01:41.930","Text":"Similarly, we do the second row with this,"},{"Start":"01:41.930 ","End":"01:45.830","Text":"we get 1 times x plus 2 times y minus 4 times z."},{"Start":"01:45.830 ","End":"01:51.090","Text":"That\u0027s exactly this, equals 5, which is here."},{"Start":"01:51.090 ","End":"01:55.310","Text":"I\u0027ll leave you to check the last row that 6, 4,1 times x,"},{"Start":"01:55.310 ","End":"01:58.250","Text":"y, z is really 2,"},{"Start":"01:58.250 ","End":"02:00.350","Text":"you can see it."},{"Start":"02:00.350 ","End":"02:03.460","Text":"That was Number 1."},{"Start":"02:03.460 ","End":"02:07.085","Text":"Number 2 is pretty much the same."},{"Start":"02:07.085 ","End":"02:09.620","Text":"Well this time this 4 equations."},{"Start":"02:09.620 ","End":"02:12.815","Text":"But there\u0027s a little bit of trickiness here."},{"Start":"02:12.815 ","End":"02:15.140","Text":"Because here we have some that are missing,"},{"Start":"02:15.140 ","End":"02:18.155","Text":"not just the ones like here."},{"Start":"02:18.155 ","End":"02:23.420","Text":"Honestly, we don\u0027t usually write them here, here, here, here."},{"Start":"02:23.420 ","End":"02:27.605","Text":"But there are also hidden zeros."},{"Start":"02:27.605 ","End":"02:31.015","Text":"For example, here there\u0027s no t,"},{"Start":"02:31.015 ","End":"02:33.350","Text":"so I would just think it,"},{"Start":"02:33.350 ","End":"02:36.730","Text":"I wouldn\u0027t normally write it I\u0027d write plus 0t,"},{"Start":"02:36.730 ","End":"02:42.005","Text":"and like here in the beginning I\u0027d have a 0x."},{"Start":"02:42.005 ","End":"02:45.365","Text":"Here is doubled trick."},{"Start":"02:45.365 ","End":"02:47.210","Text":"First of all, the orders change,"},{"Start":"02:47.210 ","End":"02:54.330","Text":"so I have to take x minus 2y minus 4z and then plus 0t."},{"Start":"02:54.330 ","End":"02:56.675","Text":"You have to be careful the order can change,"},{"Start":"02:56.675 ","End":"03:00.365","Text":"could be missing elements and the ones we know about."},{"Start":"03:00.365 ","End":"03:02.140","Text":"From the first 1,"},{"Start":"03:02.140 ","End":"03:05.100","Text":"we get just 2 minus 3,"},{"Start":"03:05.100 ","End":"03:06.975","Text":"1, 1, which is here."},{"Start":"03:06.975 ","End":"03:08.550","Text":"Then 4,1, 2,"},{"Start":"03:08.550 ","End":"03:13.338","Text":"0, 0, 1, 1, 1."},{"Start":"03:13.338 ","End":"03:16.690","Text":"1 minus 4 minus 2,"},{"Start":"03:17.750 ","End":"03:20.925","Text":"0 because there\u0027s a 0 missing."},{"Start":"03:20.925 ","End":"03:23.460","Text":"Then the right-hand sides, 1, 4,"},{"Start":"03:23.460 ","End":"03:26.265","Text":"1, 10, 1, 4, 1, 10."},{"Start":"03:26.265 ","End":"03:29.385","Text":"As before, we can verify."},{"Start":"03:29.385 ","End":"03:30.790","Text":"I\u0027ll take 1 example."},{"Start":"03:30.790 ","End":"03:36.800","Text":"Let\u0027s take the third and see what it means that this times this is equal to this."},{"Start":"03:36.800 ","End":"03:39.140","Text":"We see that 0x,"},{"Start":"03:39.140 ","End":"03:41.120","Text":"which is just nothing,"},{"Start":"03:41.120 ","End":"03:45.210","Text":"1y plus 1z plus 1t,"},{"Start":"03:45.210 ","End":"03:48.240","Text":"y plus z plus t equals 1."},{"Start":"03:48.240 ","End":"03:50.925","Text":"You can check the other 3."},{"Start":"03:50.925 ","End":"03:53.450","Text":"There it is in matrix notation."},{"Start":"03:53.450 ","End":"03:55.385","Text":"If you let A be this,"},{"Start":"03:55.385 ","End":"03:57.575","Text":"vector x be this,"},{"Start":"03:57.575 ","End":"03:59.000","Text":"and vector B is this,"},{"Start":"03:59.000 ","End":"04:03.020","Text":"then a times x equals B in matrices."},{"Start":"04:03.020 ","End":"04:08.650","Text":"Vectors is of course also a matrix with 1 column or 1 row."},{"Start":"04:08.650 ","End":"04:12.010","Text":"That\u0027s it for this exercise."}],"ID":10508},{"Watched":false,"Name":"Exercise 5 Part 1","Duration":"2m 8s","ChapterTopicVideoID":10190,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10190.jpeg","UploadDate":"2017-08-22T08:36:37.8270000","DurationForVideoObject":"PT2M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.495","Text":"In this exercise, we\u0027re given a matrix,"},{"Start":"00:03.495 ","End":"00:07.530","Text":"A 3 by 3, a vector x,"},{"Start":"00:07.530 ","End":"00:12.870","Text":"which is a 3 by 1 matrix,"},{"Start":"00:12.870 ","End":"00:17.190","Text":"or just 1 column and b also a column."},{"Start":"00:17.190 ","End":"00:22.020","Text":"It\u0027s a vector. We have to express each of the following,"},{"Start":"00:22.020 ","End":"00:23.460","Text":"there\u0027s 5 parts,"},{"Start":"00:23.460 ","End":"00:25.950","Text":"as a system of linear equations."},{"Start":"00:25.950 ","End":"00:29.220","Text":"We start out easy with this 1."},{"Start":"00:29.220 ","End":"00:33.360","Text":"Basically just copy a is here, that\u0027s a."},{"Start":"00:33.360 ","End":"00:36.525","Text":"Then I need vector x, which is this."},{"Start":"00:36.525 ","End":"00:41.770","Text":"Then here I need equals, and then here\u0027s B."},{"Start":"00:41.770 ","End":"00:46.970","Text":"Now, this matrix multiplication will work."},{"Start":"00:46.970 ","End":"00:48.965","Text":"If you look at the sizes,"},{"Start":"00:48.965 ","End":"00:53.600","Text":"I\u0027ve written all the sizes here and really a 3 by 3 times a 3 by 1."},{"Start":"00:53.600 ","End":"00:55.370","Text":"The middle bit is equal,"},{"Start":"00:55.370 ","End":"00:58.595","Text":"then it drops out so we get a 3 by 1."},{"Start":"00:58.595 ","End":"01:05.245","Text":"Let\u0027s actually do the product of A with x and see what we get."},{"Start":"01:05.245 ","End":"01:10.670","Text":"The first entry is going to be this times this."},{"Start":"01:10.670 ","End":"01:17.870","Text":"4 times x minus 2 times y plus 4 times z, which is this."},{"Start":"01:17.870 ","End":"01:23.670","Text":"Similarly for the other 2."},{"Start":"01:24.410 ","End":"01:30.140","Text":"We can see it 1 times x minus 1 times y plus 1 times z."},{"Start":"01:30.140 ","End":"01:32.195","Text":"Of course, we don\u0027t write the ones here."},{"Start":"01:32.195 ","End":"01:37.615","Text":"But it sometimes helps if you imagine or you actually write in the 1s everywhere."},{"Start":"01:37.615 ","End":"01:39.270","Text":"Like the last row also,"},{"Start":"01:39.270 ","End":"01:42.770","Text":"1 times x minus 6 times y plus 3 times z."},{"Start":"01:42.770 ","End":"01:47.960","Text":"Now we have 2 matrices of the same size."},{"Start":"01:47.960 ","End":"01:49.415","Text":"They\u0027re both 3 by 1."},{"Start":"01:49.415 ","End":"01:55.160","Text":"If they\u0027re equal, then they\u0027re equal element-wise that this equals this,"},{"Start":"01:55.160 ","End":"01:58.525","Text":"and then this will equal this and this will equal this."},{"Start":"01:58.525 ","End":"02:01.700","Text":"That will just give us these 3 equations."},{"Start":"02:01.700 ","End":"02:05.210","Text":"I put a curly brace and now it\u0027s a system of linear equations."},{"Start":"02:05.210 ","End":"02:08.100","Text":"That\u0027s the end of part 1."}],"ID":10509},{"Watched":false,"Name":"Exercise 5 Part 2","Duration":"3m 36s","ChapterTopicVideoID":10191,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10191.jpeg","UploadDate":"2017-08-22T08:36:45.0800000","DurationForVideoObject":"PT3M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"Continuing from the previous exercise,"},{"Start":"00:03.300 ","End":"00:05.430","Text":"now in Part 2,"},{"Start":"00:05.430 ","End":"00:09.525","Text":"Ax equals 4x plus b."},{"Start":"00:09.525 ","End":"00:11.655","Text":"Here I just copied it,"},{"Start":"00:11.655 ","End":"00:15.390","Text":"and now I bring the 4x over to the other side,"},{"Start":"00:15.390 ","End":"00:18.315","Text":"so Ax minus 4x is b."},{"Start":"00:18.315 ","End":"00:25.460","Text":"Now, what we\u0027d like to do would be to take x out of the bracket on the right and say,"},{"Start":"00:25.460 ","End":"00:28.175","Text":"\"A minus 4x is b.\""},{"Start":"00:28.175 ","End":"00:30.050","Text":"Now, why is this not okay?"},{"Start":"00:30.050 ","End":"00:31.475","Text":"Well, you can\u0027t mix."},{"Start":"00:31.475 ","End":"00:34.890","Text":"A is a matrix and 4 is a scalar,"},{"Start":"00:34.890 ","End":"00:42.205","Text":"so you can\u0027t do this distributive law."},{"Start":"00:42.205 ","End":"00:45.830","Text":"You have to have either both of them constants,"},{"Start":"00:45.830 ","End":"00:48.050","Text":"scalars, or both of the matrices."},{"Start":"00:48.050 ","End":"00:51.305","Text":"What we do, there\u0027s a way around that."},{"Start":"00:51.305 ","End":"00:54.800","Text":"What we can do is instead of x,"},{"Start":"00:54.800 ","End":"00:57.395","Text":"we can write Ix."},{"Start":"00:57.395 ","End":"01:01.160","Text":"Actually, this I is I_3,"},{"Start":"01:01.160 ","End":"01:04.770","Text":"it\u0027s the identity of order 3 but doesn\u0027t matter"},{"Start":"01:04.770 ","End":"01:10.940","Text":"because the identity matrix has the property that times anything is itself,"},{"Start":"01:10.940 ","End":"01:14.325","Text":"so we get this."},{"Start":"01:14.325 ","End":"01:19.405","Text":"Now, 4I is a matrix."},{"Start":"01:19.405 ","End":"01:22.840","Text":"What we\u0027ve done actually is change the brackets."},{"Start":"01:22.840 ","End":"01:28.240","Text":"We had 4 and then I times x."},{"Start":"01:28.240 ","End":"01:34.660","Text":"What we did is that this is the same as 4I times x."},{"Start":"01:34.660 ","End":"01:38.865","Text":"Now, 4I is a matrix."},{"Start":"01:38.865 ","End":"01:42.315","Text":"Now, we can take x out of the brackets."},{"Start":"01:42.315 ","End":"01:45.255","Text":"Here, we have A minus 4I,"},{"Start":"01:45.255 ","End":"01:54.480","Text":"all of these times x equals b. I is just 1s along the diagonal."},{"Start":"01:54.480 ","End":"02:00.500","Text":"This is I noticed the 1s here and everything else is 0."},{"Start":"02:00.500 ","End":"02:02.570","Text":"If I multiply this by 4,"},{"Start":"02:02.570 ","End":"02:06.410","Text":"then I just get 4s along the diagonal."},{"Start":"02:06.410 ","End":"02:08.570","Text":"Now, I\u0027m just going to substitute."},{"Start":"02:08.570 ","End":"02:11.765","Text":"Now, A is here, I copied it."},{"Start":"02:11.765 ","End":"02:15.260","Text":"This part here is what we said is 4I,"},{"Start":"02:15.260 ","End":"02:18.575","Text":"vector x is this,"},{"Start":"02:18.575 ","End":"02:20.795","Text":"vector b is this."},{"Start":"02:20.795 ","End":"02:23.000","Text":"That\u0027s what we have now."},{"Start":"02:23.000 ","End":"02:30.575","Text":"Then the next thing to do will be the subtraction, this minus this."},{"Start":"02:30.575 ","End":"02:35.180","Text":"I just have to take 4 off the diagonal here."},{"Start":"02:35.180 ","End":"02:38.090","Text":"4 minus 4 is 0,"},{"Start":"02:38.090 ","End":"02:41.380","Text":"minus 1 minus 4 is minus 5,"},{"Start":"02:41.380 ","End":"02:44.035","Text":"and 3 minus 4 is minus 1,"},{"Start":"02:44.035 ","End":"02:46.355","Text":"and the other entries are the same."},{"Start":"02:46.355 ","End":"02:49.405","Text":"Here\u0027s the x, y, and z and the 1, 2, 3."},{"Start":"02:49.405 ","End":"02:55.295","Text":"As before, a 3 by 3 times a 3 by 1 is going to give me also a 3 by 1,"},{"Start":"02:55.295 ","End":"02:59.350","Text":"the first entry will be this by this."},{"Start":"02:59.350 ","End":"03:03.605","Text":"That\u0027s going to be this. Notice I didn\u0027t bother to write 0x,"},{"Start":"03:03.605 ","End":"03:06.740","Text":"just minus 2 with the y and 4 with the z."},{"Start":"03:06.740 ","End":"03:08.960","Text":"Then this 1 with this 1,"},{"Start":"03:08.960 ","End":"03:10.280","Text":"then this 1 with this 1."},{"Start":"03:10.280 ","End":"03:14.990","Text":"Notice that just basically copying the coefficients with x, y, and z."},{"Start":"03:14.990 ","End":"03:18.980","Text":"Now that I have 2 column vectors that are equal,"},{"Start":"03:18.980 ","End":"03:20.540","Text":"then each entry is equal."},{"Start":"03:20.540 ","End":"03:23.630","Text":"This 1 is equal to this 1 and the second 1,"},{"Start":"03:23.630 ","End":"03:26.290","Text":"the second 1, so on."},{"Start":"03:26.290 ","End":"03:30.210","Text":"Here is our system of linear equations."},{"Start":"03:30.210 ","End":"03:33.255","Text":"That finishes Part 2."},{"Start":"03:33.255 ","End":"03:35.920","Text":"On to the next clip."}],"ID":10510},{"Watched":false,"Name":"Exercise 5 Part 3","Duration":"2m 6s","ChapterTopicVideoID":10187,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10187.jpeg","UploadDate":"2017-08-22T08:36:20.6470000","DurationForVideoObject":"PT2M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"Now Part 3."},{"Start":"00:02.550 ","End":"00:04.680","Text":"It\u0027s actually quite similar to Part 2."},{"Start":"00:04.680 ","End":"00:06.870","Text":"Instead of the 4, we have a minus case,"},{"Start":"00:06.870 ","End":"00:08.550","Text":"so we proceed similarly."},{"Start":"00:08.550 ","End":"00:10.170","Text":"Copied it."},{"Start":"00:10.170 ","End":"00:15.210","Text":"Now, we take the minus kx to the left, becomes plus kx."},{"Start":"00:15.210 ","End":"00:19.560","Text":"Now remember we can\u0027t just take x out of the brackets"},{"Start":"00:19.560 ","End":"00:21.530","Text":"because we can\u0027t have A plus 4,"},{"Start":"00:21.530 ","End":"00:23.540","Text":"so we have this trick with the I."},{"Start":"00:23.540 ","End":"00:26.690","Text":"I actually skipped the step"},{"Start":"00:26.690 ","End":"00:31.610","Text":"because we did it in the previous part."},{"Start":"00:31.610 ","End":"00:34.490","Text":"But in general, when you take x out of a bracket,"},{"Start":"00:34.490 ","End":"00:40.440","Text":"you can do it provided that scalar you glue an I to it."},{"Start":"00:40.440 ","End":"00:42.425","Text":"You put an I on the right of it,"},{"Start":"00:42.425 ","End":"00:45.035","Text":"and then this is what we get."},{"Start":"00:45.035 ","End":"00:49.445","Text":"Of course, k times I is k times all the 1s,"},{"Start":"00:49.445 ","End":"00:53.100","Text":"which is just k is along the diagonal."},{"Start":"00:54.880 ","End":"01:09.320","Text":"A is this, kI is this bit, and then x and b."},{"Start":"01:09.320 ","End":"01:11.090","Text":"Here\u0027s the result of the addition."},{"Start":"01:11.090 ","End":"01:12.680","Text":"The addition is what we had to do."},{"Start":"01:12.680 ","End":"01:17.765","Text":"4 plus k is 4 plus k minus 2 plus 0 is minus 2."},{"Start":"01:17.765 ","End":"01:20.960","Text":"Well, it\u0027s all going to be the same except along the diagonal."},{"Start":"01:20.960 ","End":"01:24.575","Text":"Notice there\u0027s a k here, k here, and a plus k here."},{"Start":"01:24.575 ","End":"01:26.375","Text":"That\u0027s basically what we did."},{"Start":"01:26.375 ","End":"01:30.570","Text":"We still have x, y, z, and 1, 2, 3."},{"Start":"01:31.580 ","End":"01:34.890","Text":"We\u0027re going to get also a column vector."},{"Start":"01:34.890 ","End":"01:38.560","Text":"This times this will give me the top element."},{"Start":"01:38.560 ","End":"01:49.510","Text":"That\u0027s this, 4 plus k with x minus 2 with y and 1 with z multiplying and adding."},{"Start":"01:49.760 ","End":"01:53.010","Text":"That\u0027s got to equal 1."},{"Start":"01:53.010 ","End":"01:55.380","Text":"Similarly for the other 2,"},{"Start":"01:55.380 ","End":"01:57.735","Text":"and we end up with this."},{"Start":"01:57.735 ","End":"02:01.910","Text":"This is our system of linear equations with the parameter k."},{"Start":"02:01.910 ","End":"02:04.325","Text":"That\u0027s this part."},{"Start":"02:04.325 ","End":"02:07.050","Text":"Next part in the next clip."}],"ID":10511},{"Watched":false,"Name":"Exercise 5 Part 4","Duration":"2m 26s","ChapterTopicVideoID":10188,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10188.jpeg","UploadDate":"2017-08-22T08:36:25.7700000","DurationForVideoObject":"PT2M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.635","Text":"Continuing."},{"Start":"00:01.635 ","End":"00:04.050","Text":"We\u0027re now in Part 4,"},{"Start":"00:04.050 ","End":"00:09.120","Text":"which has some similarities to both the 2 and 3"},{"Start":"00:09.120 ","End":"00:13.455","Text":"and here I just copied it."},{"Start":"00:13.455 ","End":"00:16.200","Text":"Let me bring the x over to the other side,"},{"Start":"00:16.200 ","End":"00:18.525","Text":"but I\u0027ll write it as 1x,"},{"Start":"00:18.525 ","End":"00:21.040","Text":"just like we had 4x."},{"Start":"00:21.410 ","End":"00:28.530","Text":"Then just like before when we wrote this as minus 4x,"},{"Start":"00:28.530 ","End":"00:30.450","Text":"we had minus 4Ix"},{"Start":"00:30.450 ","End":"00:32.630","Text":"and we\u0027re doing the same thing just with the 1,"},{"Start":"00:32.630 ","End":"00:36.220","Text":"just prefer to keep the 1 in for the moment."},{"Start":"00:36.220 ","End":"00:38.060","Text":"Now, I get rid of that 1,"},{"Start":"00:38.060 ","End":"00:42.410","Text":"just write it as A minus Ix after I took it out the brackets."},{"Start":"00:42.410 ","End":"00:48.679","Text":"1I is just I so I is 1s along the diagonal,"},{"Start":"00:48.679 ","End":"00:53.235","Text":"0s elsewhere and here is I."},{"Start":"00:53.235 ","End":"00:56.470","Text":"Strictly speaking, we could call this I3,"},{"Start":"00:56.470 ","End":"00:58.130","Text":"but if it doesn\u0027t matter,"},{"Start":"00:58.130 ","End":"01:01.250","Text":"we don\u0027t have to write the subscript."},{"Start":"01:01.250 ","End":"01:06.500","Text":"This is A so we\u0027re going to need to subtract A minus I,"},{"Start":"01:06.500 ","End":"01:08.390","Text":"and then there\u0027s our x,"},{"Start":"01:08.390 ","End":"01:12.090","Text":"and this is the vector 0,"},{"Start":"01:12.090 ","End":"01:14.630","Text":"as opposed to a number 0."},{"Start":"01:14.630 ","End":"01:17.270","Text":"It\u0027s a vector consisting of all 0s."},{"Start":"01:17.270 ","End":"01:20.585","Text":"Like I said, you want to do the subtraction first,"},{"Start":"01:20.585 ","End":"01:23.470","Text":"which means taking 1s off the diagonal here,"},{"Start":"01:23.470 ","End":"01:26.475","Text":"so 4 minus 1 is 3."},{"Start":"01:26.475 ","End":"01:29.215","Text":"This 1 take away 1 is this."},{"Start":"01:29.215 ","End":"01:31.475","Text":"Sorry, from here, take away 1."},{"Start":"01:31.475 ","End":"01:33.050","Text":"3 minus 1 is 2"},{"Start":"01:33.050 ","End":"01:41.520","Text":"and then we have this matrix equation and as before,"},{"Start":"01:41.520 ","End":"01:46.225","Text":"this product will give us a 3 by 1."},{"Start":"01:46.225 ","End":"01:49.880","Text":"The top element will be this times this"},{"Start":"01:49.880 ","End":"01:58.025","Text":"and then 3 times x minus 2 times y plus 4 times z is this and the other 2,"},{"Start":"01:58.025 ","End":"01:59.990","Text":"I\u0027ll leave you to check on your own,"},{"Start":"01:59.990 ","End":"02:01.910","Text":"this with this and this with this,"},{"Start":"02:01.910 ","End":"02:03.125","Text":"give me this and this."},{"Start":"02:03.125 ","End":"02:06.440","Text":"Now, this column vector equals"},{"Start":"02:06.440 ","End":"02:10.860","Text":"this column vector so they\u0027re equal component or element-wise."},{"Start":"02:10.860 ","End":"02:13.275","Text":"This will equal this, this will equal this,"},{"Start":"02:13.275 ","End":"02:18.390","Text":"and we\u0027ll get 3 equations and 3 unknowns as follows."},{"Start":"02:18.390 ","End":"02:20.730","Text":"This equals 0s here, similarly the other 2"},{"Start":"02:20.730 ","End":"02:26.710","Text":"and that\u0027s it for this part and the next part in next clip."}],"ID":10512},{"Watched":false,"Name":"Exercise 5 Part 5","Duration":"3m 21s","ChapterTopicVideoID":10189,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10189.jpeg","UploadDate":"2017-08-22T08:36:32.9070000","DurationForVideoObject":"PT3M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"Continuing now, Part 5,"},{"Start":"00:02.850 ","End":"00:05.850","Text":"the last part, this 1 has a bit of a twist."},{"Start":"00:05.850 ","End":"00:09.225","Text":"It has a transpose in it."},{"Start":"00:09.225 ","End":"00:13.140","Text":"Anyway, we start out similar to Part 2."},{"Start":"00:13.140 ","End":"00:18.280","Text":"Let\u0027s say, where we bring the x to the left."},{"Start":"00:18.710 ","End":"00:21.510","Text":"From here, if we bring the 2x over,"},{"Start":"00:21.510 ","End":"00:26.160","Text":"we get A transpose x minus 2x is 3b,"},{"Start":"00:26.160 ","End":"00:28.215","Text":"and then we can take out of the brackets."},{"Start":"00:28.215 ","End":"00:30.135","Text":"Remember, we have just a number,"},{"Start":"00:30.135 ","End":"00:32.430","Text":"we have to put an I after it."},{"Start":"00:32.430 ","End":"00:36.450","Text":"We saw this in all the previous parts,"},{"Start":"00:36.450 ","End":"00:38.850","Text":"doing it a bit shorter here."},{"Start":"00:38.850 ","End":"00:41.075","Text":"We have to compute this,"},{"Start":"00:41.075 ","End":"00:43.430","Text":"that I put it in blue."},{"Start":"00:43.430 ","End":"00:46.860","Text":"A transpose minus 2I."},{"Start":"00:46.880 ","End":"00:50.300","Text":"I\u0027m going to actually start with the right-hand side,"},{"Start":"00:50.300 ","End":"00:55.430","Text":"3b is 3 times 1, 2, 3."},{"Start":"00:55.430 ","End":"00:58.825","Text":"When you take a number by a vector,"},{"Start":"00:58.825 ","End":"01:01.820","Text":"just multiply it element-wise."},{"Start":"01:01.820 ","End":"01:03.150","Text":"3 times 1 is 3,"},{"Start":"01:03.150 ","End":"01:06.540","Text":"3 times 2 is 6, 3 times 3 is 9."},{"Start":"01:06.540 ","End":"01:11.595","Text":"Now, I\u0027ll do this calculation of A transpose minus 2I."},{"Start":"01:11.595 ","End":"01:14.710","Text":"This is A transpose."},{"Start":"01:14.710 ","End":"01:18.920","Text":"Suppose I could go back and look at A,"},{"Start":"01:18.920 ","End":"01:22.370","Text":"here I just copied it so we have it handy."},{"Start":"01:22.370 ","End":"01:25.445","Text":"What we can do is swap the rows and the columns like,"},{"Start":"01:25.445 ","End":"01:29.525","Text":"this row will become this column,"},{"Start":"01:29.525 ","End":"01:33.275","Text":"and this row becomes this column,"},{"Start":"01:33.275 ","End":"01:38.660","Text":"and this row becomes this column,"},{"Start":"01:38.660 ","End":"01:42.005","Text":"and that\u0027s the transpose of A."},{"Start":"01:42.005 ","End":"01:45.035","Text":"2I, well, we know how to do that."},{"Start":"01:45.035 ","End":"01:46.910","Text":"I is just 1s on the diagonal,"},{"Start":"01:46.910 ","End":"01:49.765","Text":"so we pop 2s on the diagonal."},{"Start":"01:49.765 ","End":"01:51.650","Text":"Now, the subtraction,"},{"Start":"01:51.650 ","End":"01:54.680","Text":"the only thing that changes is the diagonal,"},{"Start":"01:54.680 ","End":"02:00.470","Text":"4 minus 2 is 2 minus 1,"},{"Start":"02:00.470 ","End":"02:03.545","Text":"minus 2, minus 3."},{"Start":"02:03.545 ","End":"02:07.180","Text":"3 minus 2 is 1."},{"Start":"02:07.180 ","End":"02:09.560","Text":"The rest of them are the same here,"},{"Start":"02:09.560 ","End":"02:15.400","Text":"1, 1 and 6, 1,1 and 6 minus 2, 4, 1, minus 3, 4, 1."},{"Start":"02:15.400 ","End":"02:21.220","Text":"Next, I need to plug these into this equation."},{"Start":"02:22.250 ","End":"02:24.930","Text":"In place of this, I\u0027ll put this,"},{"Start":"02:24.930 ","End":"02:27.790","Text":"and in place of this, I\u0027ll put this."},{"Start":"02:27.800 ","End":"02:30.820","Text":"Here we are,"},{"Start":"02:32.300 ","End":"02:35.234","Text":"this I copied from here,"},{"Start":"02:35.234 ","End":"02:42.280","Text":"is the x, y, z and the 3b I copied from here,"},{"Start":"02:43.190 ","End":"02:47.560","Text":"now we just have to multiply out."},{"Start":"02:50.240 ","End":"02:56.330","Text":"This times this will give us the top entry in the column vector."},{"Start":"02:56.330 ","End":"03:00.290","Text":"Then the result will be assigned to 3."},{"Start":"03:00.290 ","End":"03:02.675","Text":"I\u0027m doing 2 steps in 1."},{"Start":"03:02.675 ","End":"03:07.375","Text":"We\u0027ll get this, that will be the first of 3."},{"Start":"03:07.375 ","End":"03:09.340","Text":"Similarly, the other 2,"},{"Start":"03:09.340 ","End":"03:12.440","Text":"this with this will give me 6."},{"Start":"03:12.440 ","End":"03:17.195","Text":"This with this which is 4x plus 1y plus z is 9,"},{"Start":"03:17.195 ","End":"03:21.540","Text":"and that gives us a system of linear equations."}],"ID":10513},{"Watched":false,"Name":"Exercise 6","Duration":"1m 28s","ChapterTopicVideoID":9564,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9564.jpeg","UploadDate":"2017-07-26T08:32:42.9700000","DurationForVideoObject":"PT1M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"This clip is just to remind you of the rules for the transpose,"},{"Start":"00:04.830 ","End":"00:07.440","Text":"which we\u0027ve already covered in an earlier clip."},{"Start":"00:07.440 ","End":"00:11.310","Text":"But what follow are some exercises on the transpose"},{"Start":"00:11.310 ","End":"00:13.740","Text":"and although you should have them handy,"},{"Start":"00:13.740 ","End":"00:16.140","Text":"I won\u0027t go through them again 1 by 1."},{"Start":"00:16.140 ","End":"00:20.370","Text":"You can go back to an earlier tutorial, but meanwhile,"},{"Start":"00:20.370 ","End":"00:26.055","Text":"I remembered another rule, a 7th rule, that is going to be useful."},{"Start":"00:26.055 ","End":"00:28.110","Text":"Each of these equalities you can read"},{"Start":"00:28.110 ","End":"00:29.910","Text":"from left to right or from right to left."},{"Start":"00:29.910 ","End":"00:33.150","Text":"In any event, if I have a transpose of a matrix"},{"Start":"00:33.150 ","End":"00:35.490","Text":"and then I raise it to the power of n,"},{"Start":"00:35.490 ","End":"00:38.400","Text":"well it just means multiply it by itself, by itself, by itself,"},{"Start":"00:38.400 ","End":"00:40.340","Text":"until there are n factors."},{"Start":"00:40.340 ","End":"00:42.920","Text":"It\u0027s the same as taking A^n,"},{"Start":"00:42.920 ","End":"00:44.530","Text":"which is A times A times A,"},{"Start":"00:44.530 ","End":"00:48.135","Text":"N factors and then taking the transpose."},{"Start":"00:48.135 ","End":"00:50.220","Text":"This actually follows from this rule,"},{"Start":"00:50.220 ","End":"00:52.905","Text":"instead of ABC, if I put AAA,"},{"Start":"00:52.905 ","End":"00:56.060","Text":"I\u0027d get A cubed transpose,"},{"Start":"00:56.060 ","End":"00:58.019","Text":"then here I\u0027d get A transpose,"},{"Start":"00:58.019 ","End":"00:59.720","Text":"A transpose, A transpose,"},{"Start":"00:59.720 ","End":"01:01.425","Text":"which is A transpose cubed."},{"Start":"01:01.425 ","End":"01:06.450","Text":"It\u0027s really just like this but with all of the factors being matrix A."},{"Start":"01:06.450 ","End":"01:10.460","Text":"I\u0027m not going to be using all of them in the exercises."},{"Start":"01:10.460 ","End":"01:13.310","Text":"Probably will not be using maybe these 2,"},{"Start":"01:13.310 ","End":"01:14.750","Text":"I don\u0027t know if we\u0027ll be using,"},{"Start":"01:14.750 ","End":"01:16.435","Text":"but probably most of the rest."},{"Start":"01:16.435 ","End":"01:18.410","Text":"In any event here you have them handy"},{"Start":"01:18.410 ","End":"01:23.880","Text":"and this is going to be,"},{"Start":"01:23.880 ","End":"01:26.300","Text":"as I said, useful for the following exercises."},{"Start":"01:26.300 ","End":"01:28.080","Text":"That\u0027s all."}],"ID":10514},{"Watched":false,"Name":"Exercise 6 Part a","Duration":"7m 32s","ChapterTopicVideoID":9550,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9550.jpeg","UploadDate":"2017-07-26T08:27:41.8800000","DurationForVideoObject":"PT7M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"Here we have an exercise in 3 parts, 1, 2, 3"},{"Start":"00:05.070 ","End":"00:11.760","Text":"and it relates to the concept of symmetric and antisymmetric matrices,"},{"Start":"00:11.760 ","End":"00:17.565","Text":"which I will remind you are defined in terms of the transpose operation."},{"Start":"00:17.565 ","End":"00:23.580","Text":"Symmetric, if the matrix is equal to its own transpose,"},{"Start":"00:23.580 ","End":"00:29.309","Text":"and antisymmetric if it\u0027s minus its own transpose,"},{"Start":"00:29.309 ","End":"00:32.140","Text":"or the transpose is minus it."},{"Start":"00:32.720 ","End":"00:35.820","Text":"Those are the 2 concepts."},{"Start":"00:35.820 ","End":"00:41.705","Text":"We are given a specific matrix A, which is square."},{"Start":"00:41.705 ","End":"00:44.420","Text":"Transpose is defined on any size matrix,"},{"Start":"00:44.420 ","End":"00:46.415","Text":"but here we have a square matrix."},{"Start":"00:46.415 ","End":"00:51.265","Text":"We want to see which of the following 3 must be true."},{"Start":"00:51.265 ","End":"00:55.100","Text":"I guess if it\u0027s true, we want to prove it also."},{"Start":"00:55.100 ","End":"00:57.605","Text":"We start off with Part 1,"},{"Start":"00:57.605 ","End":"01:01.315","Text":"A times a transpose is symmetric."},{"Start":"01:01.315 ","End":"01:04.600","Text":"Let\u0027s work off the definition."},{"Start":"01:06.140 ","End":"01:07.860","Text":"I\u0027ll just write that."},{"Start":"01:07.860 ","End":"01:09.615","Text":"We\u0027re doing Part 1 now."},{"Start":"01:09.615 ","End":"01:13.790","Text":"What we have to check for symmetric is,"},{"Start":"01:13.790 ","End":"01:16.190","Text":"is its own transpose equal to itself?"},{"Start":"01:16.190 ","End":"01:21.680","Text":"In other words, if I take this and take the transpose of this,"},{"Start":"01:21.680 ","End":"01:26.060","Text":"is this equal to the thing itself,"},{"Start":"01:26.060 ","End":"01:34.140","Text":"which was AA transpose at the moment it\u0027s a question mark."},{"Start":"01:34.270 ","End":"01:40.850","Text":"I say that the answer is yes and the several ways of proving equality,"},{"Start":"01:40.850 ","End":"01:42.290","Text":"I can work on both sides."},{"Start":"01:42.290 ","End":"01:44.930","Text":"It\u0027s often most convenient to start with 1 side"},{"Start":"01:44.930 ","End":"01:47.510","Text":"and through a series of changes, get to the other side."},{"Start":"01:47.510 ","End":"01:49.075","Text":"That\u0027s what I\u0027m going to do here."},{"Start":"01:49.075 ","End":"01:51.545","Text":"This is like the left-hand side,"},{"Start":"01:51.545 ","End":"01:53.240","Text":"and this is the right-hand side"},{"Start":"01:53.240 ","End":"01:56.105","Text":"and I\u0027m going to start with the left-hand side."},{"Start":"01:56.105 ","End":"01:59.705","Text":"What it equals is,"},{"Start":"01:59.705 ","End":"02:02.545","Text":"I\u0027m going to use 1 of the rules."},{"Start":"02:02.545 ","End":"02:05.150","Text":"I\u0027ll write it at the side, we did it with A and B,"},{"Start":"02:05.150 ","End":"02:07.340","Text":"but I don\u0027t want to get mixed up with the letters,"},{"Start":"02:07.340 ","End":"02:08.990","Text":"I\u0027ll do it with M and N."},{"Start":"02:08.990 ","End":"02:14.930","Text":"If I have M times N and I take the transpose of a product,"},{"Start":"02:14.930 ","End":"02:18.289","Text":"then I take the transpose of each,"},{"Start":"02:18.289 ","End":"02:20.930","Text":"but we have to reverse the order."},{"Start":"02:20.930 ","End":"02:23.125","Text":"This is important, sometimes you forget,"},{"Start":"02:23.125 ","End":"02:27.120","Text":"you have to take N transpose M transpose with matrices,"},{"Start":"02:27.120 ","End":"02:28.485","Text":"the order is important."},{"Start":"02:28.485 ","End":"02:31.700","Text":"In this case, if this is my M and this is my N,"},{"Start":"02:31.700 ","End":"02:35.650","Text":"then I get the second 1 transposed."},{"Start":"02:35.650 ","End":"02:43.970","Text":"A transpose, transpose and then the first 1 transpose."},{"Start":"02:43.970 ","End":"02:49.070","Text":"Now, there is another rule for a transpose of a transpose."},{"Start":"02:49.070 ","End":"02:51.760","Text":"I don\u0027t mind using the same letter again,"},{"Start":"02:51.760 ","End":"02:53.565","Text":"we actually had it with A,"},{"Start":"02:53.565 ","End":"02:55.950","Text":"A transpose transpose is A."},{"Start":"02:55.950 ","End":"02:58.250","Text":"The transpose of the transpose is the thing itself,"},{"Start":"02:58.250 ","End":"03:01.890","Text":"basically is because this is a reflection along the diagonal."},{"Start":"03:01.890 ","End":"03:04.070","Text":"If your reflect, something and you reflect it again,"},{"Start":"03:04.070 ","End":"03:06.660","Text":"and you\u0027re back to the original."},{"Start":"03:06.660 ","End":"03:09.545","Text":"Mirror image of the mirror image is like no mirror image."},{"Start":"03:09.545 ","End":"03:15.755","Text":"This is equal to A times a transpose,"},{"Start":"03:15.755 ","End":"03:18.320","Text":"and that is equal to the right-hand side."},{"Start":"03:18.320 ","End":"03:23.450","Text":"We started off with the left and got to the right and so this is true."},{"Start":"03:23.450 ","End":"03:27.335","Text":"Now let\u0027s go on to Part 2."},{"Start":"03:27.335 ","End":"03:31.150","Text":"Part 2 is a bit similar to Part 1 only here we had a product"},{"Start":"03:31.150 ","End":"03:32.410","Text":"and here we have a sum."},{"Start":"03:32.410 ","End":"03:34.420","Text":"We have to check if this is symmetric"},{"Start":"03:34.420 ","End":"03:36.865","Text":"and by the definition of symmetric,"},{"Start":"03:36.865 ","End":"03:40.135","Text":"we have to check if this thing,"},{"Start":"03:40.135 ","End":"03:42.310","Text":"if I take its transpose,"},{"Start":"03:42.310 ","End":"03:44.260","Text":"it\u0027s equal to the thing itself,"},{"Start":"03:44.260 ","End":"03:47.050","Text":"which is A plus A transpose,"},{"Start":"03:47.050 ","End":"03:49.180","Text":"but I put a question mark now"},{"Start":"03:49.180 ","End":"03:52.265","Text":"because that\u0027s what we\u0027re going to show or check."},{"Start":"03:52.265 ","End":"03:55.585","Text":"I know that the answer happens to be yes."},{"Start":"03:55.585 ","End":"03:56.830","Text":"I\u0027m going to prove it."},{"Start":"03:56.830 ","End":"03:59.140","Text":"I\u0027ll start with the left-hand side"},{"Start":"03:59.140 ","End":"04:01.660","Text":"and work my way to the right-hand side."},{"Start":"04:01.660 ","End":"04:05.750","Text":"This left-hand side is equal 2."},{"Start":"04:05.750 ","End":"04:09.805","Text":"Now, the rule is that if I have a sum,"},{"Start":"04:09.805 ","End":"04:12.955","Text":"say M and N transpose,"},{"Start":"04:12.955 ","End":"04:18.985","Text":"then it\u0027s just the transpose of each 1 separately and added."},{"Start":"04:18.985 ","End":"04:22.195","Text":"With multiplication, we have to reverse the order here,"},{"Start":"04:22.195 ","End":"04:28.905","Text":"we don\u0027t have to because addition makes no difference anyway."},{"Start":"04:28.905 ","End":"04:33.570","Text":"We get here A transpose plus,"},{"Start":"04:33.570 ","End":"04:38.423","Text":"and the second 1 transpose is A transpose"},{"Start":"04:38.423 ","End":"04:40.980","Text":"and the transpose of that."},{"Start":"04:40.980 ","End":"04:43.190","Text":"Now this is equal to,"},{"Start":"04:43.190 ","End":"04:44.705","Text":"I put my equals that way,"},{"Start":"04:44.705 ","End":"04:49.150","Text":"never mind. A transpose plus."},{"Start":"04:49.150 ","End":"04:50.720","Text":"Now we already had the rule that"},{"Start":"04:50.720 ","End":"04:54.360","Text":"the transpose of the transpose is the thing itself."},{"Start":"04:57.020 ","End":"04:59.895","Text":"Not quite the same as this,"},{"Start":"04:59.895 ","End":"05:03.080","Text":"I\u0027ll reverse the order because we can do that with addition."},{"Start":"05:03.080 ","End":"05:07.760","Text":"This is A plus A transpose and this is equal to this."},{"Start":"05:07.760 ","End":"05:10.940","Text":"We started with the left and ended up on the right"},{"Start":"05:10.940 ","End":"05:15.485","Text":"and as we met the condition for symmetric."},{"Start":"05:15.485 ","End":"05:21.205","Text":"That\u0027s Part 2 and now let\u0027s go on to Part 3."},{"Start":"05:21.205 ","End":"05:27.515","Text":"The first 2 involve the concept of symmetric in Part 3 is antisymmetric."},{"Start":"05:27.515 ","End":"05:31.115","Text":"We have to show that A minus A transpose is this."},{"Start":"05:31.115 ","End":"05:33.740","Text":"We have here the definition of antisymmetric"},{"Start":"05:33.740 ","End":"05:37.325","Text":"if the transpose is minus the original."},{"Start":"05:37.325 ","End":"05:46.440","Text":"Let\u0027s see what we have to show here is that A minus A transpose,"},{"Start":"05:46.440 ","End":"05:48.425","Text":"take the transpose of that."},{"Start":"05:48.425 ","End":"05:51.230","Text":"It\u0027s got to be equal minus the original,"},{"Start":"05:51.230 ","End":"05:55.205","Text":"which is A minus A transpose,"},{"Start":"05:55.205 ","End":"05:58.375","Text":"but for now it\u0027s question mark."},{"Start":"05:58.375 ","End":"06:00.890","Text":"I say the answer is yes,"},{"Start":"06:00.890 ","End":"06:05.330","Text":"and I\u0027ll start with the left-hand side"},{"Start":"06:05.330 ","End":"06:08.030","Text":"and then we\u0027ll reach the right-hand side like"},{"Start":"06:08.030 ","End":"06:12.170","Text":"this is what I call sometimes it\u0027s abbreviated left-hand side."},{"Start":"06:12.170 ","End":"06:14.285","Text":"Right-hand side of the equation,"},{"Start":"06:14.285 ","End":"06:18.845","Text":"so the left-hand side is equal to."},{"Start":"06:18.845 ","End":"06:22.310","Text":"Now, we already saw that if you have a sum"},{"Start":"06:22.310 ","End":"06:24.050","Text":"or a difference of a transpose,"},{"Start":"06:24.050 ","End":"06:26.675","Text":"we take the transpose of each 1 separately."},{"Start":"06:26.675 ","End":"06:30.230","Text":"It\u0027s this 1 transpose, on the right,"},{"Start":"06:30.230 ","End":"06:34.250","Text":"the rule each time we had this just a moment ago"},{"Start":"06:34.250 ","End":"06:42.505","Text":"with the plus or minus, minus A transpose transpose."},{"Start":"06:42.505 ","End":"06:47.690","Text":"This is equal to A transpose minus."},{"Start":"06:47.690 ","End":"06:51.570","Text":"Now the transpose of a transpose is the original."},{"Start":"06:52.580 ","End":"06:54.870","Text":"Is this the same as this?"},{"Start":"06:54.870 ","End":"06:57.800","Text":"Well, I can write it in reverse order."},{"Start":"06:57.800 ","End":"07:01.785","Text":"First of all, minus A plus A transpose."},{"Start":"07:01.785 ","End":"07:04.370","Text":"Now I can take a minus outside the brackets"},{"Start":"07:04.370 ","End":"07:05.660","Text":"and all the time looking at this"},{"Start":"07:05.660 ","End":"07:08.600","Text":"and I\u0027m trying to shape this to get it to be like that."},{"Start":"07:08.600 ","End":"07:10.625","Text":"So take a minus out."},{"Start":"07:10.625 ","End":"07:16.680","Text":"I\u0027ve got here A and here will have to be a minus A transpose"},{"Start":"07:16.680 ","End":"07:21.290","Text":"and that is equal to the right-hand side."},{"Start":"07:21.290 ","End":"07:23.425","Text":"The left-hand side is right-hand side,"},{"Start":"07:23.425 ","End":"07:25.500","Text":"that number 3 is proved also."},{"Start":"07:25.500 ","End":"07:29.310","Text":"So all 3 are true, all 3 of them."},{"Start":"07:29.310 ","End":"07:32.110","Text":"We\u0027re done."}],"ID":10515},{"Watched":false,"Name":"Exercise 6 Part b","Duration":"8m 33s","ChapterTopicVideoID":9551,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9551.jpeg","UploadDate":"2017-07-26T08:28:35.1700000","DurationForVideoObject":"PT8M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.975","Text":"This exercise, a bit similar to the previous one,"},{"Start":"00:03.975 ","End":"00:10.140","Text":"starts out with a reminder of the definition of symmetric and antisymmetric matrices,"},{"Start":"00:10.140 ","End":"00:12.480","Text":"I won\u0027t dwell on that."},{"Start":"00:12.480 ","End":"00:19.525","Text":"In this case, we\u0027re given 2 antisymmetric square matrices."},{"Start":"00:19.525 ","End":"00:23.300","Text":"Let\u0027s say n by n just to indicate they are square"},{"Start":"00:23.300 ","End":"00:29.015","Text":"and we want to know which of the following are true."},{"Start":"00:29.015 ","End":"00:31.190","Text":"But before we even get into that,"},{"Start":"00:31.190 ","End":"00:34.310","Text":"let me just interpret what the question means."},{"Start":"00:34.310 ","End":"00:38.180","Text":"What does it mean that A and B are antisymmetric?"},{"Start":"00:38.180 ","End":"00:40.160","Text":"For all 3 parts,"},{"Start":"00:40.160 ","End":"00:45.349","Text":"what I will know is that the transpose of A is minus"},{"Start":"00:45.349 ","End":"00:52.685","Text":"A and the transpose of B is minus B."},{"Start":"00:52.685 ","End":"00:56.885","Text":"That\u0027s from the definition of antisymmetric and we\u0027re given"},{"Start":"00:56.885 ","End":"01:02.310","Text":"that A and B are antisymmetric."},{"Start":"01:03.080 ","End":"01:07.160","Text":"Now let\u0027s get onto number 1."},{"Start":"01:07.160 ","End":"01:13.275","Text":"Now we have to show that B times A times B times A times B times A,"},{"Start":"01:13.275 ","End":"01:17.805","Text":"BABABA, whatever is antisymmetric."},{"Start":"01:17.805 ","End":"01:21.819","Text":"We have to show that it satisfies the definition."},{"Start":"01:21.819 ","End":"01:31.360","Text":"In other words, we have to show that BABABA transpose I don\u0027t know."},{"Start":"01:31.360 ","End":"01:32.545","Text":"I mean, I\u0027m going to check,"},{"Start":"01:32.545 ","End":"01:40.130","Text":"is it true that this is equal to minus BABABA."},{"Start":"01:41.720 ","End":"01:45.290","Text":"Let\u0027s see. Let\u0027s start with"},{"Start":"01:45.290 ","End":"01:52.590","Text":"the left-hand side and see what this equals and see if we get to the right-hand side."},{"Start":"01:52.910 ","End":"01:55.895","Text":"If I take the transpose of this,"},{"Start":"01:55.895 ","End":"01:59.090","Text":"I want to remind you that in general,"},{"Start":"01:59.090 ","End":"02:01.700","Text":"if I take the product doesn\u0027t have to be 3,"},{"Start":"02:01.700 ","End":"02:04.295","Text":"but just to be definite,"},{"Start":"02:04.295 ","End":"02:06.860","Text":"we say ABC transpose."},{"Start":"02:06.860 ","End":"02:08.315","Text":"We take the transpose of each,"},{"Start":"02:08.315 ","End":"02:15.840","Text":"but we also reverse the order and it can generalize to 2 or 3 or more."},{"Start":"02:15.840 ","End":"02:21.515","Text":"In this case, I write it in reverse order from right to left."},{"Start":"02:21.515 ","End":"02:26.060","Text":"From each time putting a transpose, A transpose, B transpose,"},{"Start":"02:26.060 ","End":"02:27.710","Text":"then this A transpose,"},{"Start":"02:27.710 ","End":"02:29.705","Text":"then this B transpose,"},{"Start":"02:29.705 ","End":"02:32.855","Text":"A transpose, B transpose."},{"Start":"02:32.855 ","End":"02:35.260","Text":"Now, what do we do with this?"},{"Start":"02:35.260 ","End":"02:37.500","Text":"Well, we look at this,"},{"Start":"02:37.500 ","End":"02:42.405","Text":"what we wrote here that A transpose is minus A and B transpose is minus B."},{"Start":"02:42.405 ","End":"02:47.040","Text":"We have minus A and B transpose is minus B,"},{"Start":"02:47.040 ","End":"02:53.715","Text":"minus A minus B minus A minus B."},{"Start":"02:53.715 ","End":"02:56.300","Text":"Now, how many minuses do we have?"},{"Start":"02:56.300 ","End":"02:59.810","Text":"We have 6 minuses, 1, 2, 3,"},{"Start":"02:59.810 ","End":"03:03.735","Text":"4, 5, 6, so 6 minuses."},{"Start":"03:03.735 ","End":"03:07.515","Text":"Minus 1 to the power of 6 is plus 1."},{"Start":"03:07.515 ","End":"03:10.110","Text":"This is just equal to"},{"Start":"03:10.110 ","End":"03:19.440","Text":"ABABAB and that in general,"},{"Start":"03:19.440 ","End":"03:23.960","Text":"this doesn\u0027t look the same,"},{"Start":"03:23.960 ","End":"03:27.950","Text":"I suppose in specific circumstances it could be."},{"Start":"03:27.950 ","End":"03:29.780","Text":"I mean, if one of them was 0,"},{"Start":"03:29.780 ","End":"03:31.775","Text":"then this would be equal to this."},{"Start":"03:31.775 ","End":"03:36.180","Text":"But in general, it\u0027s not going to be equal to that."},{"Start":"03:36.400 ","End":"03:39.270","Text":"If we were going to be mathematically strict,"},{"Start":"03:39.270 ","End":"03:41.900","Text":"we would really have to give a counter-example,"},{"Start":"03:41.900 ","End":"03:45.005","Text":"actual example with A and B that these are not equal."},{"Start":"03:45.005 ","End":"03:48.350","Text":"I\u0027m not going to do that but you could,"},{"Start":"03:48.350 ","End":"03:50.735","Text":"for example, take A equals B,"},{"Start":"03:50.735 ","End":"03:58.620","Text":"just find some antisymmetric matrix where A equals B and then here we\u0027d have A to the 6,"},{"Start":"03:58.620 ","End":"04:01.570","Text":"and here we have minus A to the 6."},{"Start":"04:01.640 ","End":"04:11.205","Text":"All you have to do is find some A with non-zero determinant and that\u0027s antisymmetric."},{"Start":"04:11.205 ","End":"04:13.160","Text":"I\u0027m not going to get into it more."},{"Start":"04:13.160 ","End":"04:14.825","Text":"We just say for the shape of this,"},{"Start":"04:14.825 ","End":"04:18.440","Text":"there\u0027s no reason why these 2 have to in general be equal so"},{"Start":"04:18.440 ","End":"04:23.165","Text":"the answer is this one is no."},{"Start":"04:23.165 ","End":"04:28.790","Text":"For part 1 not true."},{"Start":"04:28.790 ","End":"04:30.970","Text":"Let\u0027s try part 2."},{"Start":"04:30.970 ","End":"04:35.660","Text":"Here in part 2 the same beginning is in part 1,"},{"Start":"04:35.660 ","End":"04:39.230","Text":"we have to show that A squared minus B squared is symmetric,"},{"Start":"04:39.230 ","End":"04:41.840","Text":"or A square just means A times A and so on."},{"Start":"04:41.840 ","End":"04:46.400","Text":"Remember that we have that A transpose is"},{"Start":"04:46.400 ","End":"04:51.830","Text":"minus A and B transpose is minus B from the previous."},{"Start":"04:51.830 ","End":"04:58.200","Text":"That\u0027s what it means for A and B to be antisymmetric."},{"Start":"04:58.200 ","End":"05:00.710","Text":"Let\u0027s see if this is symmetric."},{"Start":"05:00.710 ","End":"05:04.165","Text":"Symmetric means that its transpose is equal to itself."},{"Start":"05:04.165 ","End":"05:05.820","Text":"We have to ask,"},{"Start":"05:05.820 ","End":"05:11.090","Text":"is A squared minus B squared transpose?"},{"Start":"05:11.090 ","End":"05:13.790","Text":"Is it equal to itself,"},{"Start":"05:13.790 ","End":"05:16.185","Text":"A squared minus B squared?"},{"Start":"05:16.185 ","End":"05:19.700","Text":"Well, let\u0027s try and see what the left-hand side,"},{"Start":"05:19.700 ","End":"05:22.385","Text":"which is this, is going to equal."},{"Start":"05:22.385 ","End":"05:28.130","Text":"The transpose of a sum or difference is each piece separately,"},{"Start":"05:28.130 ","End":"05:35.685","Text":"so we get A squared transpose minus B squared transpose."},{"Start":"05:35.685 ","End":"05:37.220","Text":"Now we have the rule,"},{"Start":"05:37.220 ","End":"05:43.535","Text":"I think it was number 7 in our list that we can exchange the exponent with the transpose."},{"Start":"05:43.535 ","End":"05:50.580","Text":"It\u0027s A transpose squared minus B transpose squared."},{"Start":"05:50.580 ","End":"05:52.675","Text":"Remember there was a rule,"},{"Start":"05:52.675 ","End":"05:58.340","Text":"A transpose^n is A^n transpose on here,"},{"Start":"05:58.340 ","End":"06:01.115","Text":"we could take n equals 2."},{"Start":"06:01.115 ","End":"06:09.030","Text":"Now, A transpose is minus A and B transpose is minus B."},{"Start":"06:09.030 ","End":"06:14.085","Text":"But minus A times minus A is the same as A times A so that\u0027s A squared,"},{"Start":"06:14.085 ","End":"06:19.170","Text":"and minus B times minus B is the same as B times B"},{"Start":"06:19.170 ","End":"06:26.220","Text":"that\u0027s B squared and that is equal to the right-hand side, RHS for short."},{"Start":"06:26.220 ","End":"06:29.030","Text":"Yeah, we started from the left-hand by a series of equals,"},{"Start":"06:29.030 ","End":"06:31.670","Text":"get to the right and so yes,"},{"Start":"06:31.670 ","End":"06:34.265","Text":"this one is true."},{"Start":"06:34.265 ","End":"06:37.375","Text":"Now on to the next part."},{"Start":"06:37.375 ","End":"06:41.840","Text":"Now part 3, and I want to remind you that we have that"},{"Start":"06:41.840 ","End":"06:47.120","Text":"A transpose is minus A and B transpose is minus B,"},{"Start":"06:47.120 ","End":"06:52.640","Text":"and we want to know if A squared plus B is symmetric and to be symmetric,"},{"Start":"06:52.640 ","End":"06:59.895","Text":"its transpose has to be equal to the thing itself and that\u0027s what we want to ask."},{"Start":"06:59.895 ","End":"07:01.550","Text":"Let\u0027s start. It\u0027s called this,"},{"Start":"07:01.550 ","End":"07:05.000","Text":"the left-hand side and the right-hand side and"},{"Start":"07:05.000 ","End":"07:08.540","Text":"the left-hand side is equal to the transpose of"},{"Start":"07:08.540 ","End":"07:11.930","Text":"the sum is the sum of the transpose so take"},{"Start":"07:11.930 ","End":"07:15.905","Text":"the transpose of this plus the transpose of this."},{"Start":"07:15.905 ","End":"07:19.640","Text":"But we already had the rule just in"},{"Start":"07:19.640 ","End":"07:23.930","Text":"this exercise that when I square something and take its transpose,"},{"Start":"07:23.930 ","End":"07:33.345","Text":"it\u0027s like the transpose squared plus B transpose now."},{"Start":"07:33.345 ","End":"07:36.930","Text":"I\u0027m going to use the fact that A transpose is"},{"Start":"07:36.930 ","End":"07:41.010","Text":"minus A and B transpose is minus B so we have"},{"Start":"07:41.010 ","End":"07:50.710","Text":"minus A squared plus minus B and if we multiply this out,"},{"Start":"07:50.710 ","End":"07:57.895","Text":"minus A times minus A is the same as A times A but here we have minus B."},{"Start":"07:57.895 ","End":"08:02.650","Text":"Now the right-hand side is the plus here,"},{"Start":"08:02.650 ","End":"08:04.780","Text":"and we have a minus here."},{"Start":"08:04.780 ","End":"08:08.320","Text":"Now, it could be the same if B was 0,"},{"Start":"08:08.320 ","End":"08:09.550","Text":"but when it says true,"},{"Start":"08:09.550 ","End":"08:10.750","Text":"it means true in general,"},{"Start":"08:10.750 ","End":"08:18.290","Text":"if B is not 0 then these things are not going to be equal and so in general,"},{"Start":"08:18.290 ","End":"08:23.590","Text":"this is not true."},{"Start":"08:23.590 ","End":"08:27.740","Text":"It could be for a specific pair of A and B,"},{"Start":"08:27.740 ","End":"08:30.005","Text":"but it\u0027s not true in general."},{"Start":"08:30.005 ","End":"08:33.900","Text":"That was part 3 and we are done."}],"ID":10516},{"Watched":false,"Name":"Exercise 6 Part c","Duration":"7m 34s","ChapterTopicVideoID":9552,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9552.jpeg","UploadDate":"2017-07-26T08:29:14.2970000","DurationForVideoObject":"PT7M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"This exercise has 3 parts."},{"Start":"00:02.820 ","End":"00:08.310","Text":"First of all, we\u0027re given the definition of symmetric"},{"Start":"00:08.310 ","End":"00:10.965","Text":"and antisymmetric which you should know by now."},{"Start":"00:10.965 ","End":"00:14.280","Text":"We\u0027re given 2 square matrices,"},{"Start":"00:14.280 ","End":"00:15.900","Text":"A and B of the same size."},{"Start":"00:15.900 ","End":"00:20.365","Text":"That\u0027s why I say n by n about square and of the same size."},{"Start":"00:20.365 ","End":"00:25.080","Text":"They\u0027re both symmetric and they both satisfy this condition."},{"Start":"00:25.080 ","End":"00:28.980","Text":"Then we have 2 for each of the following say which must be true."},{"Start":"00:28.980 ","End":"00:34.485","Text":"Let\u0027s start with number 1 and just want to go over again what we\u0027re given."},{"Start":"00:34.485 ","End":"00:44.595","Text":"We\u0027re given that A is symmetric which means that A transpose is equal to A."},{"Start":"00:44.595 ","End":"00:49.005","Text":"B is also symmetric which means B transpose equals B."},{"Start":"00:49.005 ","End":"00:51.030","Text":"We\u0027re given that third condition,"},{"Start":"00:51.030 ","End":"00:54.835","Text":"that AB is equal to minus BA."},{"Start":"00:54.835 ","End":"00:57.455","Text":"Remember that matrix multiplication,"},{"Start":"00:57.455 ","End":"01:00.320","Text":"the order makes a difference and this could happen."},{"Start":"01:00.320 ","End":"01:05.300","Text":"We have to show that AB cubed is antisymmetric."},{"Start":"01:05.300 ","End":"01:15.745","Text":"In other words, we have to show that AB cubed transpose antisymmetric is minus AB cubed."},{"Start":"01:15.745 ","End":"01:19.505","Text":"But here we need a question mark was what we have to show."},{"Start":"01:19.505 ","End":"01:22.040","Text":"What I\u0027m going to do is as usual,"},{"Start":"01:22.040 ","End":"01:26.675","Text":"I like to start and always been an often like to start with the left-hand side."},{"Start":"01:26.675 ","End":"01:31.460","Text":"Then work my way through a series of equals to get to the right-hand side."},{"Start":"01:31.460 ","End":"01:32.870","Text":"Well, first of all,"},{"Start":"01:32.870 ","End":"01:35.840","Text":"this is equal too because of the properties of the transpose"},{"Start":"01:35.840 ","End":"01:40.785","Text":"is equal to B cubed transpose A transpose."},{"Start":"01:40.785 ","End":"01:43.190","Text":"You don\u0027t keep writing the rules each time."},{"Start":"01:43.190 ","End":"01:44.510","Text":"But when you have a product,"},{"Start":"01:44.510 ","End":"01:46.880","Text":"transpose is the transpose of each,"},{"Start":"01:46.880 ","End":"01:48.920","Text":"but you reverse the order."},{"Start":"01:48.920 ","End":"01:52.670","Text":"Next thing is that the transpose with the an exponent,"},{"Start":"01:52.670 ","End":"01:54.110","Text":"you can reverse them."},{"Start":"01:54.110 ","End":"01:59.595","Text":"It\u0027s B transpose cubed A transpose."},{"Start":"01:59.595 ","End":"02:02.990","Text":"Now we use the fact that A and B are symmetric,"},{"Start":"02:02.990 ","End":"02:04.685","Text":"meaning these 2 are true."},{"Start":"02:04.685 ","End":"02:10.410","Text":"This just gives us B cubed times A."},{"Start":"02:10.970 ","End":"02:13.070","Text":"That\u0027s not what we want."},{"Start":"02:13.070 ","End":"02:15.030","Text":"We want minus AB cubed."},{"Start":"02:15.030 ","End":"02:17.660","Text":"Let\u0027s see, we can use this piece of information."},{"Start":"02:17.660 ","End":"02:19.295","Text":"We haven\u0027t used this yet."},{"Start":"02:19.295 ","End":"02:28.170","Text":"We could write this as B squared times BA or we can write it as BBBA."},{"Start":"02:28.170 ","End":"02:31.820","Text":"Now BA is minus AB."},{"Start":"02:31.820 ","End":"02:36.320","Text":"I could have written the other way around BA is minus AB."},{"Start":"02:36.320 ","End":"02:46.800","Text":"This is equal to B squared minus AB a minus I can bring to the front AB."},{"Start":"02:46.800 ","End":"02:49.670","Text":"I\u0027ll continue over here. Equals."},{"Start":"02:49.670 ","End":"02:57.185","Text":"Now use try and use the same trick again minus B and then I\u0027ll split this 1."},{"Start":"02:57.185 ","End":"03:02.155","Text":"I\u0027ve got another BA and then B."},{"Start":"03:02.155 ","End":"03:06.860","Text":"This equals because BA is minus AB,"},{"Start":"03:06.860 ","End":"03:16.360","Text":"I can write a minus minus which is a plus B and then AB and then B."},{"Start":"03:17.390 ","End":"03:22.545","Text":"Now I can use this 1 last time on this pair."},{"Start":"03:22.545 ","End":"03:24.390","Text":"BA is minus AB,"},{"Start":"03:24.390 ","End":"03:26.955","Text":"so it\u0027s minus AB,"},{"Start":"03:26.955 ","End":"03:32.490","Text":"BB which equals minus AB cubed."},{"Start":"03:32.490 ","End":"03:34.920","Text":"That is the right-hand side."},{"Start":"03:34.920 ","End":"03:36.510","Text":"Starting from here going here."},{"Start":"03:36.510 ","End":"03:37.770","Text":"That\u0027s okay."},{"Start":"03:37.770 ","End":"03:40.260","Text":"The answer is yes, it\u0027s true."},{"Start":"03:40.260 ","End":"03:44.980","Text":"Next part, I\u0027ll just write what we had before."},{"Start":"03:44.980 ","End":"03:49.450","Text":"A transpose is equal to A,"},{"Start":"03:49.450 ","End":"03:56.685","Text":"B transpose equals B and AB equals minus BA."},{"Start":"03:56.685 ","End":"04:00.360","Text":"Here we have to prove that AB squared is symmetric."},{"Start":"04:00.360 ","End":"04:04.220","Text":"In other words, we want to show that AB squared,"},{"Start":"04:04.220 ","End":"04:09.915","Text":"symmetric means it\u0027s transpose equals itself AB squared."},{"Start":"04:09.915 ","End":"04:11.565","Text":"This is what we have to prove."},{"Start":"04:11.565 ","End":"04:13.695","Text":"I\u0027m putting a question mark here."},{"Start":"04:13.695 ","End":"04:15.500","Text":"As usual, I\u0027m going to start with"},{"Start":"04:15.500 ","End":"04:18.800","Text":"the left-hand side and see if I can reach the right-hand side."},{"Start":"04:18.800 ","End":"04:21.020","Text":"I may be will, maybe won\u0027t."},{"Start":"04:21.020 ","End":"04:25.830","Text":"What we\u0027ll do is we\u0027ll start working on this."},{"Start":"04:25.830 ","End":"04:29.810","Text":"AB squared transpose, like we said,"},{"Start":"04:29.810 ","End":"04:31.595","Text":"the product, when it\u0027s transposed,"},{"Start":"04:31.595 ","End":"04:34.370","Text":"we transpose each of the pieces A separately,"},{"Start":"04:34.370 ","End":"04:41.045","Text":"B squared separately, but reversing the order first the B squared and then the A."},{"Start":"04:41.045 ","End":"04:45.995","Text":"Then, well, you know that we can exchange the exponent with the transpose."},{"Start":"04:45.995 ","End":"04:51.080","Text":"It\u0027s B transpose squared A transpose."},{"Start":"04:51.080 ","End":"04:53.870","Text":"But B transpose is B,"},{"Start":"04:53.870 ","End":"04:57.690","Text":"and A transpose is A."},{"Start":"04:57.880 ","End":"05:00.080","Text":"We\u0027ve got B squared A,"},{"Start":"05:00.080 ","End":"05:01.879","Text":"but we want AB squared,"},{"Start":"05:01.879 ","End":"05:03.695","Text":"so we have to keep going."},{"Start":"05:03.695 ","End":"05:07.800","Text":"This is equal to B, BA."},{"Start":"05:07.870 ","End":"05:12.470","Text":"Now BA is minus AB."},{"Start":"05:12.470 ","End":"05:18.845","Text":"I mean, I could have just written it the other way that BA is minus AB."},{"Start":"05:18.845 ","End":"05:23.100","Text":"This is equal to the minus I can pull out front,"},{"Start":"05:23.100 ","End":"05:26.770","Text":"it\u0027s minus B, AB."},{"Start":"05:26.770 ","End":"05:30.110","Text":"Now I can do the trick again,"},{"Start":"05:30.110 ","End":"05:32.495","Text":"and I can do it on this pair."},{"Start":"05:32.495 ","End":"05:34.460","Text":"BA is minus AB,"},{"Start":"05:34.460 ","End":"05:37.610","Text":"but the minus with the minus will come out plus."},{"Start":"05:37.610 ","End":"05:40.115","Text":"We\u0027ll get ABB."},{"Start":"05:40.115 ","End":"05:43.995","Text":"This is equal to AB squared,"},{"Start":"05:43.995 ","End":"05:46.755","Text":"and this is the right-hand side."},{"Start":"05:46.755 ","End":"05:48.735","Text":"Yes. That is also true."},{"Start":"05:48.735 ","End":"05:52.340","Text":"And now on to the next part and they\u0027ll just copy from"},{"Start":"05:52.340 ","End":"05:56.120","Text":"the previous part that we had that A transpose was A,"},{"Start":"05:56.120 ","End":"05:58.820","Text":"B transpose was equal to B."},{"Start":"05:58.820 ","End":"06:02.405","Text":"Then AB is minus BA."},{"Start":"06:02.405 ","End":"06:06.440","Text":"Or alternatively we could write it as"},{"Start":"06:06.440 ","End":"06:13.115","Text":"BA is equal to minus AB just by bringing the minus over and switching sides."},{"Start":"06:13.115 ","End":"06:18.020","Text":"Now we have to show that A minus B squared is symmetric."},{"Start":"06:18.020 ","End":"06:26.790","Text":"We have to show that A minus B squared transpose"},{"Start":"06:26.790 ","End":"06:31.860","Text":"is equal to A minus B squared."},{"Start":"06:31.860 ","End":"06:36.800","Text":"Let\u0027s call this the left-hand side and the right-hand side."},{"Start":"06:36.800 ","End":"06:39.560","Text":"I\u0027ll develop the left-hand side and"},{"Start":"06:39.560 ","End":"06:43.295","Text":"hopefully I\u0027ll reach the right-hand side or perhaps not."},{"Start":"06:43.295 ","End":"06:47.840","Text":"I\u0027m going to reverse the switch places with the 2 and the T."},{"Start":"06:47.840 ","End":"06:55.365","Text":"I\u0027ve got A minus B transpose squared."},{"Start":"06:55.365 ","End":"07:01.090","Text":"This is what the left-hand side equals."},{"Start":"07:02.090 ","End":"07:04.840","Text":"A minus B transpose."},{"Start":"07:04.840 ","End":"07:10.405","Text":"A transpose of a sum or a difference is just the sum or difference of the transposes."},{"Start":"07:10.405 ","End":"07:18.610","Text":"It\u0027s A transpose minus B transpose squared changes the square brackets around bracket,"},{"Start":"07:18.610 ","End":"07:23.465","Text":"but A transpose is A and B transpose is B."},{"Start":"07:23.465 ","End":"07:25.590","Text":"This is what we get."},{"Start":"07:25.590 ","End":"07:29.470","Text":"This is equal to the right-hand side."},{"Start":"07:29.750 ","End":"07:35.220","Text":"This 1 is being proved also when that\u0027s this exercise."}],"ID":10517},{"Watched":false,"Name":"Exercise 6 Part d","Duration":"3m 41s","ChapterTopicVideoID":9553,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9553.jpeg","UploadDate":"2017-07-26T08:29:33.7570000","DurationForVideoObject":"PT3M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"Here\u0027s another exercise with symmetric and antisymmetric."},{"Start":"00:04.080 ","End":"00:05.660","Text":"First of all, the definitions."},{"Start":"00:05.660 ","End":"00:07.410","Text":"Well, you know these already."},{"Start":"00:07.410 ","End":"00:10.050","Text":"Now we\u0027re given 2 square matrices,"},{"Start":"00:10.050 ","End":"00:14.280","Text":"A and B of the same size and by n, but 1 of them,"},{"Start":"00:14.280 ","End":"00:16.050","Text":"A is symmetric,"},{"Start":"00:16.050 ","End":"00:17.760","Text":"while the other 1, B,"},{"Start":"00:17.760 ","End":"00:23.360","Text":"is anti-symmetric and we\u0027re also told that AB is equal to BA."},{"Start":"00:23.360 ","End":"00:28.145","Text":"From these things, we have to prove that AB is anti-symmetric,"},{"Start":"00:28.145 ","End":"00:30.305","Text":"part 2 when we get to it."},{"Start":"00:30.305 ","End":"00:33.080","Text":"First of all, let me just rewrite what\u0027s given."},{"Start":"00:33.080 ","End":"00:39.785","Text":"A is symmetric, so A_transpose equals A."},{"Start":"00:39.785 ","End":"00:41.950","Text":"B is antisymmetric,"},{"Start":"00:41.950 ","End":"00:45.000","Text":"so B_transpose is minus B,"},{"Start":"00:45.000 ","End":"00:50.255","Text":"and we have this extra condition given that AB is equal to BA."},{"Start":"00:50.255 ","End":"00:51.260","Text":"That\u0027s what we\u0027re given."},{"Start":"00:51.260 ","End":"00:52.580","Text":"Now, in number 1,"},{"Start":"00:52.580 ","End":"00:55.115","Text":"we have to prove AB is antisymmetric."},{"Start":"00:55.115 ","End":"01:00.970","Text":"In other words, we have to prove that its transpose is minus itself."},{"Start":"01:00.970 ","End":"01:07.400","Text":"That\u0027s what we have to show or disprove as I put a question mark here."},{"Start":"01:07.400 ","End":"01:11.690","Text":"I\u0027m going to call this the left-hand side and this the right-hand side."},{"Start":"01:11.690 ","End":"01:14.570","Text":"I\u0027m going to start with the left-hand side and see if I can work"},{"Start":"01:14.570 ","End":"01:18.050","Text":"my way towards the right-hand side."},{"Start":"01:18.050 ","End":"01:20.825","Text":"First of all, the transpose of a product,"},{"Start":"01:20.825 ","End":"01:25.370","Text":"I reverse the order and put a transpose on each."},{"Start":"01:25.370 ","End":"01:26.930","Text":"Now I\u0027m going to interpret these."},{"Start":"01:26.930 ","End":"01:30.420","Text":"B_transpose is minus B,"},{"Start":"01:30.440 ","End":"01:36.130","Text":"and A_transpose is just A,"},{"Start":"01:36.170 ","End":"01:41.265","Text":"and because BA is equal to AB,"},{"Start":"01:41.265 ","End":"01:44.775","Text":"this is equal to minus AB,"},{"Start":"01:44.775 ","End":"01:50.250","Text":"which is the right-hand side and so that part 1 is done."},{"Start":"01:50.250 ","End":"01:52.740","Text":"Now, let\u0027s get onto part 2."},{"Start":"01:52.740 ","End":"01:57.155","Text":"I copied what was given, re-interpreted."},{"Start":"01:57.155 ","End":"02:01.360","Text":"We\u0027re going to show that AB plus B is antisymmetric."},{"Start":"02:01.360 ","End":"02:06.915","Text":"In other words, we have to show or disprove if it\u0027s not so,"},{"Start":"02:06.915 ","End":"02:09.925","Text":"that AB plus B antisymmetric,"},{"Start":"02:09.925 ","End":"02:15.100","Text":"means its transpose is minus itself."},{"Start":"02:15.100 ","End":"02:18.000","Text":"Now that\u0027s a question mark."},{"Start":"02:18.000 ","End":"02:21.200","Text":"As usual in these exercises,"},{"Start":"02:21.200 ","End":"02:23.975","Text":"I like to start from the left-hand side"},{"Start":"02:23.975 ","End":"02:28.025","Text":"and work my way and try and get to the right-hand side."},{"Start":"02:28.025 ","End":"02:31.850","Text":"The transpose of a sum is the sum of the transposes."},{"Start":"02:31.850 ","End":"02:40.255","Text":"This is AB and I need the brackets transpose plus B_transpose."},{"Start":"02:40.255 ","End":"02:43.660","Text":"Then this is equal to transpose of a product,"},{"Start":"02:43.660 ","End":"02:45.015","Text":"you reverse the order."},{"Start":"02:45.015 ","End":"02:50.780","Text":"B_transpose, A_transpose plus B_transpose."},{"Start":"02:50.780 ","End":"02:54.660","Text":"Now, B transpose is minus B,"},{"Start":"02:54.660 ","End":"02:57.300","Text":"so this is minus B,"},{"Start":"02:57.300 ","End":"03:04.090","Text":"and A_transpose is A and B_transpose is minus B."},{"Start":"03:06.500 ","End":"03:10.325","Text":"If I just take the minus out of the brackets,"},{"Start":"03:10.325 ","End":"03:19.500","Text":"it\u0027s minus BA plus B."},{"Start":"03:19.500 ","End":"03:22.170","Text":"Still not quite what we needed here."},{"Start":"03:22.170 ","End":"03:25.095","Text":"Then we remember, AB is BA,"},{"Start":"03:25.095 ","End":"03:32.420","Text":"so this is equal to minus AB plus B. BA is equal to AB,"},{"Start":"03:32.420 ","End":"03:38.420","Text":"and that is equal to the right-hand side and so that part is proven also,"},{"Start":"03:38.420 ","End":"03:41.700","Text":"so they\u0027re both true and we\u0027re done."}],"ID":10518},{"Watched":false,"Name":"Exercise 6 Part e","Duration":"4m 19s","ChapterTopicVideoID":9554,"CourseChapterTopicPlaylistID":18295,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9554.jpeg","UploadDate":"2017-07-26T08:29:52.4070000","DurationForVideoObject":"PT4M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.100","Text":"Here\u0027s just another 1 of those symmetric, antisymmetric questions."},{"Start":"00:05.100 ","End":"00:06.990","Text":"I\u0027ll go through it a bit quicker."},{"Start":"00:06.990 ","End":"00:11.625","Text":"It\u0027s the same setup definitions of symmetric antisymmetric."},{"Start":"00:11.625 ","End":"00:15.360","Text":"This time we\u0027re given the A, B and AB are symmetric"},{"Start":"00:15.360 ","End":"00:17.220","Text":"and we have to prove this."},{"Start":"00:17.220 ","End":"00:19.095","Text":"Let\u0027s start to the right what we\u0027re given."},{"Start":"00:19.095 ","End":"00:27.375","Text":"We\u0027re given that A transpose equals A, B transpose equals B,"},{"Start":"00:27.375 ","End":"00:31.980","Text":"and AB transpose is equal to AB."},{"Start":"00:31.980 ","End":"00:35.160","Text":"I can rewrite this condition but differently."},{"Start":"00:35.160 ","End":"00:37.010","Text":"If I take the left-hand side to this,"},{"Start":"00:37.010 ","End":"00:38.825","Text":"I\u0027ll do it at the side,"},{"Start":"00:38.825 ","End":"00:48.120","Text":"AB transpose is B transpose, A transpose."},{"Start":"00:48.120 ","End":"00:53.775","Text":"B transpose is equal to B and A transpose is equal to A."},{"Start":"00:53.775 ","End":"00:59.180","Text":"What this thing in effect says is that I could rewrite it"},{"Start":"00:59.180 ","End":"01:06.285","Text":"as BA equals AB or AB equals BA, it doesn\u0027t matter."},{"Start":"01:06.285 ","End":"01:08.120","Text":"Now let\u0027s see what we want to prove."},{"Start":"01:08.120 ","End":"01:16.525","Text":"We want to prove that A^4, B^4 is equal to?"},{"Start":"01:16.525 ","End":"01:20.830","Text":"That\u0027s the question, B^4, A^4."},{"Start":"01:23.270 ","End":"01:26.790","Text":"I\u0027m warning you this 1 you need a little bit of patience for it."},{"Start":"01:26.790 ","End":"01:30.070","Text":"It\u0027s very easy, but it\u0027s a bit lengthy."},{"Start":"01:30.260 ","End":"01:33.860","Text":"As usual, I\u0027ll start with the left-hand side"},{"Start":"01:33.860 ","End":"01:37.190","Text":"and see if I can reach the right-hand side."},{"Start":"01:37.190 ","End":"01:39.380","Text":"Now, the left-hand side,"},{"Start":"01:39.380 ","End":"01:47.140","Text":"I can just write it as AAAABBBB,"},{"Start":"01:47.140 ","End":"01:50.980","Text":"just by expanding what B^4 means."},{"Start":"01:53.330 ","End":"01:57.000","Text":"I want to get it so that all the B\u0027s are on the left"},{"Start":"01:57.000 ","End":"02:01.585","Text":"and the A\u0027s are on the right because I\u0027m 1 to get B^4, A^4."},{"Start":"02:01.585 ","End":"02:03.380","Text":"Now, what I can do is,"},{"Start":"02:03.380 ","End":"02:05.660","Text":"every time I see a pair AB,"},{"Start":"02:05.660 ","End":"02:09.110","Text":"I can replace it by BA."},{"Start":"02:09.110 ","End":"02:19.020","Text":"The first step I\u0027ve got is AAA and then BABBB."},{"Start":"02:19.020 ","End":"02:23.815","Text":"Now, here I have 2 pairs, AB, I can reverse each of them."},{"Start":"02:23.815 ","End":"02:26.600","Text":"It will gradually getting B\u0027s further to the left"},{"Start":"02:26.600 ","End":"02:27.890","Text":"and A is further to the right,"},{"Start":"02:27.890 ","End":"02:29.480","Text":"and we\u0027ll get there."},{"Start":"02:29.480 ","End":"02:37.400","Text":"This is equal to AABABA,"},{"Start":"02:37.400 ","End":"02:42.000","Text":"that\u0027s to above and then BB."},{"Start":"02:42.440 ","End":"02:46.770","Text":"Again, we have AB, we have AB and we have AB,"},{"Start":"02:46.770 ","End":"02:49.125","Text":"and we switch each of those."},{"Start":"02:49.125 ","End":"03:00.780","Text":"We\u0027ll get ABABABA, and then B."},{"Start":"03:00.780 ","End":"03:04.560","Text":"Now we have all 4 of them with AB."},{"Start":"03:04.560 ","End":"03:06.615","Text":"Actually, this is the halfway point."},{"Start":"03:06.615 ","End":"03:10.770","Text":"It\u0027s going backwards from here, but still okay."},{"Start":"03:10.770 ","End":"03:13.220","Text":"We reverse each of these and make it into a BA."},{"Start":"03:13.220 ","End":"03:15.890","Text":"I will continue on the next column."},{"Start":"03:15.890 ","End":"03:21.240","Text":"This will be BA 4 times."},{"Start":"03:23.000 ","End":"03:27.180","Text":"Now, again, I have ABABAB,"},{"Start":"03:27.180 ","End":"03:34.315","Text":"so I switch each of those and I\u0027ve got here BABABA,"},{"Start":"03:34.315 ","End":"03:37.360","Text":"and here I had an A."},{"Start":"03:38.180 ","End":"03:40.230","Text":"Now, what do I have?"},{"Start":"03:40.230 ","End":"03:43.080","Text":"AB and AB."},{"Start":"03:43.080 ","End":"03:44.970","Text":"I\u0027ve got BB."},{"Start":"03:44.970 ","End":"03:46.200","Text":"Then for set of AB,"},{"Start":"03:46.200 ","End":"03:52.080","Text":"I put BA, and again BA, and then AA."},{"Start":"03:52.080 ","End":"03:57.155","Text":"Now, I have this AB in the middle, which is BA,"},{"Start":"03:57.155 ","End":"04:04.520","Text":"that gives me BBB, and then BA, and then AAA."},{"Start":"04:04.520 ","End":"04:10.550","Text":"That is equal to B^4, A^4,"},{"Start":"04:10.550 ","End":"04:13.595","Text":"which is the right-hand side."},{"Start":"04:13.595 ","End":"04:19.920","Text":"That\u0027s done and that\u0027s it."}],"ID":10519}],"Thumbnail":null,"ID":18295},{"Name":"Matrix Inverse and its Applications","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Inverse Matrix, Intro","Duration":"8m 31s","ChapterTopicVideoID":9497,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9497.jpeg","UploadDate":"2017-07-26T08:25:37.8170000","DurationForVideoObject":"PT8M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.150","Text":"In this clip, we\u0027ll learn a possibly new concept, the inverse matrix."},{"Start":"00:06.150 ","End":"00:09.435","Text":"I\u0027ll start very informally."},{"Start":"00:09.435 ","End":"00:17.080","Text":"I can write with numbers 4 times 1/4 is equal to 1."},{"Start":"00:17.450 ","End":"00:22.080","Text":"In a certain sense, 1/4 is the inverse of 4,"},{"Start":"00:22.080 ","End":"00:24.195","Text":"because that\u0027s what it means, an inverse,"},{"Start":"00:24.195 ","End":"00:28.800","Text":"if you take it to mean a reciprocal, 1 over."},{"Start":"00:28.800 ","End":"00:36.660","Text":"We could write it as 4 times 4^minus 1 is equal to 1."},{"Start":"00:36.660 ","End":"00:39.030","Text":"Then there\u0027s a minus 1 here,"},{"Start":"00:39.030 ","End":"00:40.920","Text":"which is more than just a symbol,"},{"Start":"00:40.920 ","End":"00:43.695","Text":"it means an exponent."},{"Start":"00:43.695 ","End":"00:46.720","Text":"We\u0027ve learned that this means 1 over."},{"Start":"00:47.180 ","End":"00:50.840","Text":"In general, if I have a number a,"},{"Start":"00:50.840 ","End":"00:56.205","Text":"then a to the minus 1 is the inverse."},{"Start":"00:56.205 ","End":"00:59.705","Text":"If you multiply a number by its inverse, you get 1."},{"Start":"00:59.705 ","End":"01:02.135","Text":"By the way, what\u0027s special about 1?"},{"Start":"01:02.135 ","End":"01:07.485","Text":"1 has the property that 1 times any number,"},{"Start":"01:07.485 ","End":"01:11.580","Text":"I don\u0027t want to reuse a, let\u0027s say,"},{"Start":"01:11.580 ","End":"01:19.255","Text":"x is equal to x or the other way around x times 1 is equal to x."},{"Start":"01:19.255 ","End":"01:21.550","Text":"If you multiply anything by 1,"},{"Start":"01:21.550 ","End":"01:23.015","Text":"it leaves the same thing."},{"Start":"01:23.015 ","End":"01:24.650","Text":"That\u0027s what\u0027s special about 1."},{"Start":"01:24.650 ","End":"01:28.660","Text":"I\u0027m going to refer to that later when we get to matrices."},{"Start":"01:28.660 ","End":"01:35.700","Text":"Of course here, I should mention that a must not be 0."},{"Start":"01:35.770 ","End":"01:39.480","Text":"Let\u0027s move on."},{"Start":"01:41.070 ","End":"01:44.500","Text":"Now, in the world of matrices,"},{"Start":"01:44.500 ","End":"01:51.580","Text":"the element 1 corresponds to the identity matrix I because it"},{"Start":"01:51.580 ","End":"01:58.915","Text":"has the same properties that I times any matrix A is equal to itself."},{"Start":"01:58.915 ","End":"02:04.790","Text":"Or if you like the other way around, also holds."},{"Start":"02:05.150 ","End":"02:07.765","Text":"Just as with numbers,"},{"Start":"02:07.765 ","End":"02:10.990","Text":"if a number times some other number is 1,"},{"Start":"02:10.990 ","End":"02:15.595","Text":"then that number is the inverse of that number."},{"Start":"02:15.595 ","End":"02:18.865","Text":"Also written as to the power of minus 1."},{"Start":"02:18.865 ","End":"02:22.135","Text":"Similarly, if we have a matrix,"},{"Start":"02:22.135 ","End":"02:29.410","Text":"let\u0027s call this 1, A and I multiply it by another matrix and get the equivalent of 1,"},{"Start":"02:29.410 ","End":"02:35.905","Text":"the identity, then I want to say that this in some sense is the inverse of this."},{"Start":"02:35.905 ","End":"02:39.204","Text":"But now the minus 1 is just a symbol,"},{"Start":"02:39.204 ","End":"02:40.390","Text":"just means inverse,"},{"Start":"02:40.390 ","End":"02:43.085","Text":"it\u0027s not 1 over anymore."},{"Start":"02:43.085 ","End":"02:50.260","Text":"Another example, if this 3 by 3 matrix is called A and if I"},{"Start":"02:50.260 ","End":"02:57.190","Text":"multiply it by something else and I get the identity matrix,"},{"Start":"02:57.190 ","End":"02:58.900","Text":"we don\u0027t have to be specific."},{"Start":"02:58.900 ","End":"03:02.800","Text":"We could have said I_2 and I_3,"},{"Start":"03:02.800 ","End":"03:05.365","Text":"but just I will do."},{"Start":"03:05.365 ","End":"03:10.720","Text":"Then I would like to write this matrix as the inverse of A and"},{"Start":"03:10.720 ","End":"03:16.240","Text":"just write it with a minus 1 to remind me of the case with numbers."},{"Start":"03:16.240 ","End":"03:19.150","Text":"I didn\u0027t check the computations."},{"Start":"03:19.150 ","End":"03:24.070","Text":"I\u0027ll leave that to you or perhaps I\u0027ll just do 1 entry in each here."},{"Start":"03:24.070 ","End":"03:27.910","Text":"Let\u0027s take, for example, this with this."},{"Start":"03:27.910 ","End":"03:30.775","Text":"That will give us the last entry hopefully,"},{"Start":"03:30.775 ","End":"03:35.050","Text":"4 times minus 2 is minus 8 plus 9 is 1,"},{"Start":"03:35.050 ","End":"03:36.820","Text":"fine. What should we do here?"},{"Start":"03:36.820 ","End":"03:40.360","Text":"Let\u0027s take the middle element. Let\u0027s take this 1."},{"Start":"03:40.360 ","End":"03:44.889","Text":"Let\u0027s take third row, second column,"},{"Start":"03:44.889 ","End":"03:49.665","Text":"sorry, I meant third row, second column here."},{"Start":"03:49.665 ","End":"03:54.620","Text":"We get it from the third row here and the second column here."},{"Start":"03:54.620 ","End":"03:57.140","Text":"Let\u0027s see, 2 times 2 is 4,"},{"Start":"03:57.140 ","End":"03:59.630","Text":"1 times minus 1 is minus 1."},{"Start":"03:59.630 ","End":"04:01.310","Text":"We\u0027re down to 3,"},{"Start":"04:01.310 ","End":"04:03.140","Text":"3 times minus 1 is minus 3,"},{"Start":"04:03.140 ","End":"04:05.185","Text":"so that leaves us with 0."},{"Start":"04:05.185 ","End":"04:07.485","Text":"Anyway, check these calculations."},{"Start":"04:07.485 ","End":"04:10.610","Text":"If we have a matrix and we multiply it by another 1 and it"},{"Start":"04:10.610 ","End":"04:13.760","Text":"gives us the identity matrix of the appropriate size,"},{"Start":"04:13.760 ","End":"04:19.350","Text":"then we want to say that the other 1 is the inverse of the original 1."},{"Start":"04:20.720 ","End":"04:24.520","Text":"Continuing, now,"},{"Start":"04:24.520 ","End":"04:29.180","Text":"an important note, not every matrix has an inverse matrix."},{"Start":"04:29.180 ","End":"04:31.280","Text":"Remember in the case of numbers,"},{"Start":"04:31.280 ","End":"04:33.560","Text":"0 doesn\u0027t have an inverse."},{"Start":"04:33.560 ","End":"04:40.230","Text":"There is no number 0 times something is equal to 1."},{"Start":"04:40.810 ","End":"04:45.620","Text":"That\u0027s the only number that doesn\u0027t have an inverse in the case of numbers,"},{"Start":"04:45.620 ","End":"04:49.250","Text":"but with matrices, it\u0027s a bit more intricate."},{"Start":"04:49.250 ","End":"04:52.285","Text":"Anyway, not every matrix has an inverse matrix."},{"Start":"04:52.285 ","End":"04:58.775","Text":"If it doesn\u0027t, then it\u0027s called non-invertible or singular,"},{"Start":"04:58.775 ","End":"05:04.280","Text":"as opposed to an invertible matrix or regular matrix,"},{"Start":"05:04.280 ","End":"05:06.445","Text":"if it does have an inverse."},{"Start":"05:06.445 ","End":"05:10.070","Text":"Now, I want to give you an example of a matrix that doesn\u0027t"},{"Start":"05:10.070 ","End":"05:13.600","Text":"have an inverse and it\u0027s not all 0s,"},{"Start":"05:13.600 ","End":"05:15.634","Text":"it\u0027s not a zero matrix."},{"Start":"05:15.634 ","End":"05:17.540","Text":"Here\u0027s the example."},{"Start":"05:17.540 ","End":"05:19.500","Text":"This is my matrix,"},{"Start":"05:19.500 ","End":"05:21.090","Text":"which doesn\u0027t have an inverse,"},{"Start":"05:21.090 ","End":"05:24.030","Text":"1, 2, 0, 0."},{"Start":"05:24.030 ","End":"05:29.450","Text":"It\u0027s easy to show why there\u0027s no numbers I can put in here to make this work."},{"Start":"05:29.450 ","End":"05:32.160","Text":"Let\u0027s just look, for example,"},{"Start":"05:32.160 ","End":"05:34.935","Text":"at this 1 here,"},{"Start":"05:34.935 ","End":"05:37.170","Text":"2nd row, 2nd column."},{"Start":"05:37.170 ","End":"05:40.684","Text":"I take 2nd row with 2nd column,"},{"Start":"05:40.684 ","End":"05:43.910","Text":"0 times something plus 0 times something is 1?"},{"Start":"05:43.910 ","End":"05:48.810","Text":"No. There is no such matrix."},{"Start":"05:48.810 ","End":"05:52.535","Text":"This 1 does not have an inverse and so we"},{"Start":"05:52.535 ","End":"05:56.915","Text":"call it either non-invertible or singular and there are other names too."},{"Start":"05:56.915 ","End":"06:02.030","Text":"Let\u0027s get a bit more formal and I brought a definition."},{"Start":"06:02.030 ","End":"06:05.180","Text":"We\u0027re only talking about square matrices here."},{"Start":"06:05.180 ","End":"06:08.720","Text":"If A is a square matrix and we say that A is invertible,"},{"Start":"06:08.720 ","End":"06:18.335","Text":"if there exists another matrix and we\u0027ll call it A with a minus 1 here just as a symbol,"},{"Start":"06:18.335 ","End":"06:22.220","Text":"which satisfies that the matrix A times"},{"Start":"06:22.220 ","End":"06:28.240","Text":"this other 1 is equal to the identity matrix of the appropriate size."},{"Start":"06:28.240 ","End":"06:34.115","Text":"Then this 1 is called the inverse of this."},{"Start":"06:34.115 ","End":"06:37.490","Text":"This is the inverse of this,"},{"Start":"06:37.490 ","End":"06:41.170","Text":"A minus 1 and A."},{"Start":"06:41.170 ","End":"06:45.420","Text":"Since I pronounce this A inverse, so inverse of A."},{"Start":"06:45.420 ","End":"06:48.610","Text":"I don\u0027t usually say A to the minus 1."},{"Start":"06:48.670 ","End":"06:53.420","Text":"Turns out that the inverse as the important feature,"},{"Start":"06:53.420 ","End":"06:54.740","Text":"it\u0027s really an inverse,"},{"Start":"06:54.740 ","End":"06:59.600","Text":"that it\u0027s not just that A times A inverse is the identity."},{"Start":"06:59.600 ","End":"07:01.400","Text":"But if I do it in the other order,"},{"Start":"07:01.400 ","End":"07:07.280","Text":"A inverse times A also gives us the identity."},{"Start":"07:07.280 ","End":"07:10.490","Text":"This is important so we don\u0027t have to worry if it\u0027s an inverse,"},{"Start":"07:10.490 ","End":"07:13.070","Text":"which side we multiply it by because remember,"},{"Start":"07:13.070 ","End":"07:16.505","Text":"with matrices the order makes a difference."},{"Start":"07:16.505 ","End":"07:24.395","Text":"A variation on this is that if A times B is the identity,"},{"Start":"07:24.395 ","End":"07:29.015","Text":"then I\u0027m thinking that B is the inverse of A."},{"Start":"07:29.015 ","End":"07:35.780","Text":"Then A times B is equal to B times A and they\u0027re both equal,"},{"Start":"07:35.780 ","End":"07:38.100","Text":"of course, to the identity."},{"Start":"07:38.100 ","End":"07:42.650","Text":"This is 1 case where the order will not make a difference."},{"Start":"07:42.650 ","End":"07:44.330","Text":"If 1 is the inverse of the other,"},{"Start":"07:44.330 ","End":"07:47.340","Text":"you can multiply them in any order."},{"Start":"07:47.390 ","End":"07:55.700","Text":"Turns out there\u0027s a simple criterion for deciding if a matrix has an inverse or not."},{"Start":"07:55.700 ","End":"08:01.545","Text":"It goes as follows: a matrix A,"},{"Start":"08:01.545 ","End":"08:03.340","Text":"size n by n,"},{"Start":"08:03.340 ","End":"08:10.100","Text":"is invertible if and only if after we bring it to row echelon form,"},{"Start":"08:10.100 ","End":"08:13.565","Text":"all the n rows are non-zero."},{"Start":"08:13.565 ","End":"08:16.470","Text":"We don\u0027t have a 0 row."},{"Start":"08:17.150 ","End":"08:20.240","Text":"I hope you remember what row echelon form is."},{"Start":"08:20.240 ","End":"08:23.610","Text":"If not, we\u0027ll review it."},{"Start":"08:24.220 ","End":"08:28.610","Text":"That concludes this clip,"},{"Start":"08:28.610 ","End":"08:31.740","Text":"but more on the inverse in the following."}],"ID":9853},{"Watched":false,"Name":"Inverse Matrix, Finding","Duration":"3m 16s","ChapterTopicVideoID":9498,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9498.jpeg","UploadDate":"2017-07-26T08:26:00.8970000","DurationForVideoObject":"PT3M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this clip, we\u0027re going to show how in practice we actually"},{"Start":"00:04.560 ","End":"00:10.650","Text":"compute the inverse matrix of a matrix that has an inverse."},{"Start":"00:10.650 ","End":"00:14.625","Text":"Let\u0027s say we were given a matrix A which was 3 by 3."},{"Start":"00:14.625 ","End":"00:16.035","Text":"We then extend it,"},{"Start":"00:16.035 ","End":"00:17.985","Text":"put a vertical bar here,"},{"Start":"00:17.985 ","End":"00:22.035","Text":"and then write the identity matrix alongside it,"},{"Start":"00:22.035 ","End":"00:25.620","Text":"the 1 with 1s on the diagonal and 0 elsewhere."},{"Start":"00:25.620 ","End":"00:27.630","Text":"That\u0027s the starting point."},{"Start":"00:27.630 ","End":"00:29.730","Text":"Now we\u0027re going to do a series of"},{"Start":"00:29.730 ","End":"00:36.255","Text":"successive row operations until we get the identity matrix here."},{"Start":"00:36.255 ","End":"00:39.405","Text":"That\u0027s our goal. Here,"},{"Start":"00:39.405 ","End":"00:43.115","Text":"on the left, should be the identity matrix. Well, let\u0027s see."},{"Start":"00:43.115 ","End":"00:46.520","Text":"Let\u0027s start off by making this and this 0"},{"Start":"00:46.520 ","End":"00:51.440","Text":"by subtracting multiples of this row from each of these 2."},{"Start":"00:51.440 ","End":"00:55.060","Text":"These are the operations I\u0027m going to perform."},{"Start":"00:55.060 ","End":"00:57.080","Text":"This is what we get."},{"Start":"00:57.080 ","End":"01:00.215","Text":"I suggest you check the calculations if you want."},{"Start":"01:00.215 ","End":"01:02.690","Text":"We\u0027ve got the 0s here."},{"Start":"01:02.690 ","End":"01:06.320","Text":"Now, we want to get a 0 here,"},{"Start":"01:06.320 ","End":"01:09.870","Text":"but we also want to have a 1 here."},{"Start":"01:12.880 ","End":"01:16.610","Text":"Let\u0027s first add these 2,"},{"Start":"01:16.610 ","End":"01:18.935","Text":"put it in the last row."},{"Start":"01:18.935 ","End":"01:22.180","Text":"Add this to this, that will give us the 0."},{"Start":"01:22.180 ","End":"01:28.070","Text":"Of course, I\u0027m also doing the row operations on the right-hand part as well."},{"Start":"01:28.070 ","End":"01:33.120","Text":"For example, minus 4 plus minus 2 is minus 6, and so on."},{"Start":"01:35.120 ","End":"01:41.135","Text":"I could be multiplying this row and this row by minus 1 to get plus 1s here."},{"Start":"01:41.135 ","End":"01:42.530","Text":"But I\u0027ll hold off on that,"},{"Start":"01:42.530 ","End":"01:47.105","Text":"it suits me meanwhile to have a minus there."},{"Start":"01:47.105 ","End":"01:53.300","Text":"What I can do now is to add twice this row to this row,"},{"Start":"01:53.300 ","End":"01:55.715","Text":"and then get rid of that."},{"Start":"01:55.715 ","End":"02:00.380","Text":"Here we are. Now there really is just the matter of the sign here"},{"Start":"02:00.380 ","End":"02:04.875","Text":"because we already have a diagonal matrix on the left."},{"Start":"02:04.875 ","End":"02:08.645","Text":"Here I wrote that we\u0027ll take the negative,"},{"Start":"02:08.645 ","End":"02:12.270","Text":"multiplying by minus 1."},{"Start":"02:14.570 ","End":"02:17.415","Text":"If we do that,"},{"Start":"02:17.415 ","End":"02:19.620","Text":"we get 1, 1, 1."},{"Start":"02:19.620 ","End":"02:22.740","Text":"Here we just reverse these 2 rows."},{"Start":"02:22.740 ","End":"02:25.345","Text":"When we\u0027ve got that,"},{"Start":"02:25.345 ","End":"02:33.680","Text":"then the bit that\u0027s on the right is the inverse matrix of A."},{"Start":"02:33.680 ","End":"02:38.090","Text":"Let\u0027s do the check to see that so."},{"Start":"02:38.090 ","End":"02:40.970","Text":"Here\u0027s our original matrix A."},{"Start":"02:40.970 ","End":"02:43.835","Text":"Here\u0027s A minus 1 from here."},{"Start":"02:43.835 ","End":"02:46.085","Text":"If we multiply them,"},{"Start":"02:46.085 ","End":"02:49.565","Text":"I\u0027m not going to do the verification."},{"Start":"02:49.565 ","End":"02:53.930","Text":"Let\u0027s just check, let\u0027s say the middle term here."},{"Start":"02:53.930 ","End":"02:56.550","Text":"I\u0027ll check this 1, 2nd row,"},{"Start":"02:56.550 ","End":"02:59.390","Text":"2nd column, so I need 2nd row from here,"},{"Start":"02:59.390 ","End":"03:01.850","Text":"and 2nd column from here,"},{"Start":"03:01.850 ","End":"03:05.540","Text":"4 times 2 is 8 plus 1 is 9,"},{"Start":"03:05.540 ","End":"03:08.395","Text":"minus 8 is 1, and so on."},{"Start":"03:08.395 ","End":"03:13.220","Text":"That\u0027s the technique for finding the inverse."},{"Start":"03:13.220 ","End":"03:16.740","Text":"That\u0027s enough for this clip."}],"ID":9854},{"Watched":false,"Name":"Inverse Matrix for Solving SLE","Duration":"6m 55s","ChapterTopicVideoID":9499,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9499.jpeg","UploadDate":"2017-07-26T08:26:48.1300000","DurationForVideoObject":"PT6M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.205","Text":"In the previous clip,"},{"Start":"00:02.205 ","End":"00:06.765","Text":"we learned how to find an inverse matrix and in this clip,"},{"Start":"00:06.765 ","End":"00:11.909","Text":"I\u0027ll show you how we can use the inverse matrix to solve the system of linear equations."},{"Start":"00:11.909 ","End":"00:14.430","Text":"There\u0027s just 1 way of solving."},{"Start":"00:14.430 ","End":"00:17.070","Text":"I\u0027ll demonstrate on an example."},{"Start":"00:17.070 ","End":"00:21.135","Text":"Here\u0027s a system, 3 equations and 3 unknowns."},{"Start":"00:21.135 ","End":"00:25.350","Text":"We can write this system in matrix form."},{"Start":"00:25.350 ","End":"00:26.850","Text":"If we write it like this,"},{"Start":"00:26.850 ","End":"00:32.420","Text":"take the coefficients of the left-hand side,"},{"Start":"00:32.420 ","End":"00:34.519","Text":"put them in a 3 by 3 matrix,"},{"Start":"00:34.519 ","End":"00:40.085","Text":"put the variables in a 1 column matrix and the right-hand side."},{"Start":"00:40.085 ","End":"00:42.425","Text":"This is matrix multiplication."},{"Start":"00:42.425 ","End":"00:44.510","Text":"For example, the top element,"},{"Start":"00:44.510 ","End":"00:49.575","Text":"it\u0027s the 1st row, 1st column."},{"Start":"00:49.575 ","End":"00:55.245","Text":"We take here the 1st row and here the 1st column to get this and"},{"Start":"00:55.245 ","End":"01:01.160","Text":"what this says is that 2 times x minus 1 times y plus 1 times z is 3."},{"Start":"01:01.160 ","End":"01:02.900","Text":"That\u0027s the top row."},{"Start":"01:02.900 ","End":"01:07.505","Text":"Similarly, if I chose the 2nd row,"},{"Start":"01:07.505 ","End":"01:13.160","Text":"1st column, I\u0027d get this and if I chose this with this I\u0027d get this."},{"Start":"01:13.160 ","End":"01:16.210","Text":"This is the matrix representation."},{"Start":"01:16.210 ","End":"01:19.640","Text":"We don\u0027t stop to think about why this is true."},{"Start":"01:19.640 ","End":"01:22.700","Text":"After you\u0027ve seen this, when you have a system,"},{"Start":"01:22.700 ","End":"01:27.640","Text":"you just write the matrix with the coefficients on the left here."},{"Start":"01:27.640 ","End":"01:33.330","Text":"Then the variables and then the 3 coefficients on the right."},{"Start":"01:33.430 ","End":"01:40.795","Text":"How are we going to use the inverse matrix?"},{"Start":"01:40.795 ","End":"01:46.320","Text":"Let me give the matrix a name A and we\u0027ll give these 2 names also."},{"Start":"01:46.320 ","End":"01:50.330","Text":"Remember with the vectors which have a single column or row,"},{"Start":"01:50.330 ","End":"01:52.415","Text":"we use lower case and underscore."},{"Start":"01:52.415 ","End":"01:58.740","Text":"I\u0027ll call this x and I\u0027ll call this 1, b."},{"Start":"01:58.740 ","End":"02:03.850","Text":"It turns out that if we have this equation like Ax equals b,"},{"Start":"02:03.850 ","End":"02:05.200","Text":"which is what we have,"},{"Start":"02:05.200 ","End":"02:13.825","Text":"then x is given by the inverse of A times the vector B."},{"Start":"02:13.825 ","End":"02:16.030","Text":"You don\u0027t have to know why,"},{"Start":"02:16.030 ","End":"02:18.430","Text":"but I\u0027ll just quickly show you."},{"Start":"02:18.430 ","End":"02:23.220","Text":"If A times x is equal to b,"},{"Start":"02:23.220 ","End":"02:26.925","Text":"we multiply both sides by A minus 1,"},{"Start":"02:26.925 ","End":"02:31.710","Text":"so A minus 1 Ax is a minus 1b."},{"Start":"02:31.710 ","End":"02:34.810","Text":"But A minus 1a is"},{"Start":"02:34.810 ","End":"02:38.290","Text":"just the identity matrix"},{"Start":"02:39.620 ","End":"02:45.340","Text":"and an identity matrix times anything is itself."},{"Start":"02:45.340 ","End":"02:48.085","Text":"This just says that x equals b."},{"Start":"02:48.085 ","End":"02:49.630","Text":"You didn\u0027t need to know that,"},{"Start":"02:49.630 ","End":"02:55.620","Text":"but if you are curious. I\u0027ll erase that."},{"Start":"02:55.620 ","End":"03:00.530","Text":"Now I\u0027m just going to pull A inverse out of a hat."},{"Start":"03:00.530 ","End":"03:03.640","Text":"I will show you at the end how I got to this."},{"Start":"03:03.640 ","End":"03:07.630","Text":"We will do the process of getting the inverse of this to be this."},{"Start":"03:07.630 ","End":"03:11.620","Text":"Meanwhile, just take my word for it."},{"Start":"03:11.620 ","End":"03:14.590","Text":"According to this, we just say that x, y,"},{"Start":"03:14.590 ","End":"03:21.730","Text":"z is this matrix times this and all we have to do now is multiply out."},{"Start":"03:21.730 ","End":"03:24.020","Text":"What we get,"},{"Start":"03:24.020 ","End":"03:26.080","Text":"well, let\u0027s just see."},{"Start":"03:26.080 ","End":"03:31.390","Text":"First of all, I take this with this to get x."},{"Start":"03:31.390 ","End":"03:33.910","Text":"2 times 3 is 6,"},{"Start":"03:33.910 ","End":"03:40.155","Text":"6 minus 5 is 1 plus 0 times 11 is still 1."},{"Start":"03:40.155 ","End":"03:46.270","Text":"Next, we get y from multiplying this by this,"},{"Start":"03:46.270 ","End":"03:52.970","Text":"a twice 3 and 6 minus 15 is minus 9 plus 11 is 2."},{"Start":"03:52.970 ","End":"03:59.050","Text":"Finally z, I get from this times this minus 3,"},{"Start":"03:59.050 ","End":"04:02.800","Text":"minus 5 is minus 8 plus 11 is 3."},{"Start":"04:02.800 ","End":"04:04.495","Text":"Once we have this,"},{"Start":"04:04.495 ","End":"04:08.980","Text":"then it means that x is 1, y is 2, z is 3."},{"Start":"04:08.980 ","End":"04:13.480","Text":"We can leave the solution like this or we can explicitly say"},{"Start":"04:13.480 ","End":"04:18.250","Text":"x equals 1, y equals 2, z equals 3."},{"Start":"04:18.250 ","End":"04:20.380","Text":"But I still have a debt."},{"Start":"04:20.380 ","End":"04:25.020","Text":"I have to show you how I got to this inverse matrix."},{"Start":"04:25.020 ","End":"04:28.540","Text":"Here we are with the original matrix A"},{"Start":"04:28.540 ","End":"04:32.365","Text":"and we write it alongside the identity matrix."},{"Start":"04:32.365 ","End":"04:37.810","Text":"We want to do row operations to bring the identity on the left"},{"Start":"04:37.810 ","End":"04:42.410","Text":"and what remains on the right will be the inverse."},{"Start":"04:43.080 ","End":"04:50.170","Text":"We can get 0 here and here by adding multiples."},{"Start":"04:50.170 ","End":"04:54.890","Text":"Well, we can get 0 here if we take twice this row"},{"Start":"04:54.890 ","End":"04:59.300","Text":"minus 3 times this row because then we\u0027ll get 6 minus 6."},{"Start":"04:59.300 ","End":"05:05.885","Text":"Also, we can get 0 here by taking twice this row minus 5 times this row."},{"Start":"05:05.885 ","End":"05:08.660","Text":"If we do all that, this is what we get."},{"Start":"05:08.660 ","End":"05:10.475","Text":"I leave it up to you to check."},{"Start":"05:10.475 ","End":"05:14.790","Text":"Here, we also have changes"},{"Start":"05:14.790 ","End":"05:20.900","Text":"like twice 0 minus 3 times 1 is minus 3,"},{"Start":"05:20.900 ","End":"05:22.655","Text":"and so on."},{"Start":"05:22.655 ","End":"05:24.770","Text":"I\u0027d like to have a 0 here"},{"Start":"05:24.770 ","End":"05:28.250","Text":"so I can subtract this row from this row."},{"Start":"05:28.250 ","End":"05:31.025","Text":"If I do that,"},{"Start":"05:31.025 ","End":"05:35.885","Text":"let\u0027s see, this is what we get."},{"Start":"05:35.885 ","End":"05:40.430","Text":"I notice its the common factor 2 in the whole of the last row,"},{"Start":"05:40.430 ","End":"05:44.090","Text":"so let\u0027s have this last row."},{"Start":"05:44.090 ","End":"05:46.295","Text":"If we do that,"},{"Start":"05:46.295 ","End":"05:50.300","Text":"this is what we\u0027ll get in the last row."},{"Start":"05:50.300 ","End":"05:54.110","Text":"Next, I\u0027ll make 0 here and here by subtracting"},{"Start":"05:54.110 ","End":"05:57.890","Text":"the last row from both the 2nd row and the 1st row."},{"Start":"05:57.890 ","End":"06:06.720","Text":"Here\u0027s the notation for that and the result is this."},{"Start":"06:06.720 ","End":"06:08.255","Text":"The 0s are here."},{"Start":"06:08.255 ","End":"06:13.085","Text":"Next, I want to get rid of this element on a 0 here."},{"Start":"06:13.085 ","End":"06:17.110","Text":"I\u0027ll subtract the 2nd row from the 1st."},{"Start":"06:17.110 ","End":"06:20.270","Text":"This is what we get and now we\u0027re almost there."},{"Start":"06:20.270 ","End":"06:23.410","Text":"We\u0027re almost at the identity matrix."},{"Start":"06:23.410 ","End":"06:28.430","Text":"All I have to do is divide this top row by 2"},{"Start":"06:28.430 ","End":"06:32.450","Text":"and multiply or divide this row by minus 1."},{"Start":"06:32.450 ","End":"06:35.170","Text":"Here\u0027s the notation for that."},{"Start":"06:35.170 ","End":"06:37.205","Text":"This is what we get,"},{"Start":"06:37.205 ","End":"06:44.420","Text":"which means that this on the right is our inverse matrix of A and if you go back,"},{"Start":"06:44.420 ","End":"06:48.590","Text":"you\u0027ll see that this is what we had."},{"Start":"06:48.590 ","End":"06:55.840","Text":"This checks and we\u0027re done for this clip."}],"ID":9855},{"Watched":false,"Name":"Exercise 1","Duration":"3m 4s","ChapterTopicVideoID":9506,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9506.jpeg","UploadDate":"2017-07-26T08:29:10.6400000","DurationForVideoObject":"PT3M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"In this exercise and in the ones following it,"},{"Start":"00:03.210 ","End":"00:05.820","Text":"we\u0027re going to practice finding the inverse of"},{"Start":"00:05.820 ","End":"00:09.805","Text":"the matrix and we start off simple with a 2 by 2."},{"Start":"00:09.805 ","End":"00:13.800","Text":"What we do is we copy this matrix"},{"Start":"00:13.800 ","End":"00:17.910","Text":"into a larger matrix for the separator"},{"Start":"00:17.910 ","End":"00:21.170","Text":"where we put on the right-hand side,"},{"Start":"00:21.170 ","End":"00:24.590","Text":"the identity matrix for the appropriate size."},{"Start":"00:24.590 ","End":"00:26.855","Text":"In this case it\u0027s really I2."},{"Start":"00:26.855 ","End":"00:33.470","Text":"Then we take this larger matrix and try"},{"Start":"00:33.470 ","End":"00:41.585","Text":"to do row operations in such a way that the left-hand side becomes the identity."},{"Start":"00:41.585 ","End":"00:44.315","Text":"After this example it will be clear."},{"Start":"00:44.315 ","End":"00:48.695","Text":"Now, we do this with the row echelon form,"},{"Start":"00:48.695 ","End":"00:52.190","Text":"at least when we go beyond row echelon form,"},{"Start":"00:52.190 ","End":"00:53.750","Text":"but we start off with that."},{"Start":"00:53.750 ","End":"00:56.315","Text":"I want to make this 3 into a 0."},{"Start":"00:56.315 ","End":"01:01.415","Text":"I\u0027ll do that by subtracting 3 times the 1st row from the 2nd row."},{"Start":"01:01.415 ","End":"01:04.360","Text":"This is what it is in row notation,"},{"Start":"01:04.360 ","End":"01:07.035","Text":"and this is the result,"},{"Start":"01:07.035 ","End":"01:10.350","Text":"3 minus 3 times 1 is 0."},{"Start":"01:10.350 ","End":"01:14.800","Text":"4 minus 3 times 2 is 4 minus x is minus 2."},{"Start":"01:14.800 ","End":"01:17.840","Text":"This minus 3 times this is this."},{"Start":"01:17.840 ","End":"01:22.550","Text":"In general, I\u0027m not going to be dwelling on these computations."},{"Start":"01:22.550 ","End":"01:28.415","Text":"You\u0027re always free to pause and check them and we\u0027ll go through a bit quicker."},{"Start":"01:28.415 ","End":"01:30.995","Text":"Now from here where do we go?"},{"Start":"01:30.995 ","End":"01:36.285","Text":"Well, we don\u0027t want that minus 2 there,"},{"Start":"01:36.285 ","End":"01:38.625","Text":"so what we can do"},{"Start":"01:38.625 ","End":"01:47.540","Text":"is we\u0027re just going"},{"Start":"01:47.540 ","End":"01:52.235","Text":"to divide this row by minus 2 and get a 1 here."},{"Start":"01:52.235 ","End":"01:56.360","Text":"Or if you like, multiply by minus 1/2."},{"Start":"01:56.360 ","End":"01:59.090","Text":"That indeed gives us the 1 here,"},{"Start":"01:59.090 ","End":"02:00.995","Text":"but it also changes here,"},{"Start":"02:00.995 ","End":"02:05.055","Text":"this 2 plus 1.5 and this 2 minus 1/2."},{"Start":"02:05.055 ","End":"02:07.490","Text":"We\u0027re very close to an identity matrix,"},{"Start":"02:07.490 ","End":"02:09.485","Text":"we just have to get a 0 here."},{"Start":"02:09.485 ","End":"02:17.510","Text":"What I suggest is subtracting twice the 2nd row from the 1st row."},{"Start":"02:17.510 ","End":"02:20.355","Text":"This is it in row notation,"},{"Start":"02:20.355 ","End":"02:23.129","Text":"and this is the result."},{"Start":"02:23.129 ","End":"02:25.080","Text":"Now, what do we do with this?"},{"Start":"02:25.080 ","End":"02:31.430","Text":"Well, the part on the right is simply the inverse matrix."},{"Start":"02:31.430 ","End":"02:33.275","Text":"That\u0027s this part here."},{"Start":"02:33.275 ","End":"02:36.674","Text":"But usually, it\u0027s a good idea to check."},{"Start":"02:36.674 ","End":"02:40.910","Text":"The way to check would be to multiply"},{"Start":"02:40.910 ","End":"02:46.825","Text":"this inverse matrix by the original matrix A and see if we get the identity."},{"Start":"02:46.825 ","End":"02:50.370","Text":"Here\u0027s our inverse matrix,"},{"Start":"02:50.370 ","End":"02:53.960","Text":"and I just copied the original matrix is easy to remember,"},{"Start":"02:53.960 ","End":"02:56.030","Text":"it was 1, 2, 3, 4."},{"Start":"02:56.030 ","End":"02:58.880","Text":"I\u0027ll leave you to do the computations and you can"},{"Start":"02:58.880 ","End":"03:01.805","Text":"check that this gives us really the identity matrix."},{"Start":"03:01.805 ","End":"03:05.500","Text":"This really is the inverse and we\u0027re done."}],"ID":9856},{"Watched":false,"Name":"Exercise 2","Duration":"2m 16s","ChapterTopicVideoID":9507,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9507.jpeg","UploadDate":"2017-07-26T08:29:17.6700000","DurationForVideoObject":"PT2M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.165","Text":"Hey, we have another 1 of those 2-by-2 matrices that we have to find the inverse of."},{"Start":"00:06.165 ","End":"00:11.100","Text":"We\u0027ve done 1 before at least."},{"Start":"00:11.100 ","End":"00:12.900","Text":"I\u0027ll repeat the technique."},{"Start":"00:12.900 ","End":"00:20.510","Text":"We take the matrix A and then put a dividing line,"},{"Start":"00:20.510 ","End":"00:22.340","Text":"and then the identity matrix,"},{"Start":"00:22.340 ","End":"00:25.615","Text":"we make a larger augmented matrix."},{"Start":"00:25.615 ","End":"00:34.270","Text":"We do a row operations on this to get the identity matrix on the left."},{"Start":"00:34.270 ","End":"00:36.885","Text":"I\u0027d like to have a 0 here."},{"Start":"00:36.885 ","End":"00:41.330","Text":"The way I can do that is if I take this times 5 minus"},{"Start":"00:41.330 ","End":"00:47.545","Text":"this times 7 or the other way around and put it into the 2nd row,"},{"Start":"00:47.545 ","End":"00:54.380","Text":"then we really do get a 0 here and also 5 times 3 minus 7 times 2."},{"Start":"00:54.380 ","End":"00:55.880","Text":"15 minus 14 is 1."},{"Start":"00:55.880 ","End":"00:58.410","Text":"I\u0027ll leave you to check the others."},{"Start":"00:58.720 ","End":"01:00.530","Text":"That\u0027s good."},{"Start":"01:00.530 ","End":"01:04.070","Text":"At this point, there\u0027s more than 1 way I could go."},{"Start":"01:04.070 ","End":"01:07.730","Text":"I could divide the top row by 5 and get a 1 here."},{"Start":"01:07.730 ","End":"01:10.910","Text":"But it seems to be easier to get the 0 here first by"},{"Start":"01:10.910 ","End":"01:15.095","Text":"subtracting twice the last row from the 1st row,"},{"Start":"01:15.095 ","End":"01:19.820","Text":"which I\u0027ve just written in a row notation."},{"Start":"01:19.820 ","End":"01:23.720","Text":"After this, we get this, for example,"},{"Start":"01:23.720 ","End":"01:29.610","Text":"1 minus twice minus 7 is 15, and so on."},{"Start":"01:29.950 ","End":"01:35.000","Text":"Now all we have to do is to divide this 1 by 5,"},{"Start":"01:35.000 ","End":"01:37.865","Text":"of course, and that will give us the 1 here."},{"Start":"01:37.865 ","End":"01:40.340","Text":"Sure enough we do."},{"Start":"01:40.340 ","End":"01:43.435","Text":"The right-hand side,"},{"Start":"01:43.435 ","End":"01:47.560","Text":"this is our inverse matrix."},{"Start":"01:47.560 ","End":"01:50.885","Text":"A inverse A to the minus 1, it\u0027s written."},{"Start":"01:50.885 ","End":"01:54.560","Text":"But as I said previously,"},{"Start":"01:54.560 ","End":"01:57.740","Text":"the best thing to do is to check that this answer is"},{"Start":"01:57.740 ","End":"02:01.810","Text":"correct by multiplying this by the original A."},{"Start":"02:01.810 ","End":"02:07.190","Text":"We take the A minus 1 and this was the original A."},{"Start":"02:07.190 ","End":"02:09.350","Text":"I\u0027ll leave you to do the matrix multiplication."},{"Start":"02:09.350 ","End":"02:11.700","Text":"The result really is the identity matrix,"},{"Start":"02:11.700 ","End":"02:14.540","Text":"and that proves that this really is the inverse."},{"Start":"02:14.540 ","End":"02:16.590","Text":"We\u0027re done."}],"ID":9857},{"Watched":false,"Name":"Exercise 3","Duration":"3m 6s","ChapterTopicVideoID":9508,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9508.jpeg","UploadDate":"2017-07-26T08:29:25.7570000","DurationForVideoObject":"PT3M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"In this exercise, we have to find the inverse of this matrix,"},{"Start":"00:04.350 ","End":"00:06.495","Text":"just 2 by 2."},{"Start":"00:06.495 ","End":"00:11.490","Text":"The usual technique, we make a larger matrix with"},{"Start":"00:11.490 ","End":"00:17.400","Text":"a separator and we copy the entries into here,"},{"Start":"00:17.400 ","End":"00:21.810","Text":"and on the other side of the separator we put the identity matrix and then we do row"},{"Start":"00:21.810 ","End":"00:26.850","Text":"operations to try and get the identity to be on the left."},{"Start":"00:26.850 ","End":"00:29.220","Text":"Now, if we were going routinely about it,"},{"Start":"00:29.220 ","End":"00:33.795","Text":"we\u0027d probably take a 1/2 of the top row and subtract it"},{"Start":"00:33.795 ","End":"00:38.605","Text":"from the last row and get a 0 down here."},{"Start":"00:38.605 ","End":"00:41.990","Text":"But that would be a lot of messing with fractions."},{"Start":"00:41.990 ","End":"00:43.565","Text":"I mean, 1/2 the top row,"},{"Start":"00:43.565 ","End":"00:46.100","Text":"that if 1 and a 1/2, I\u0027ll get 3/4."},{"Start":"00:46.100 ","End":"00:47.915","Text":"I don\u0027t like working with fractions."},{"Start":"00:47.915 ","End":"00:54.275","Text":"I\u0027m going to show you an alternative approach which gets rid of the fractions."},{"Start":"00:54.275 ","End":"01:00.625","Text":"What I\u0027m suggesting is we multiply the top row by 2."},{"Start":"01:00.625 ","End":"01:04.970","Text":"But if we do that, we\u0027ll get an 8 here."},{"Start":"01:04.970 ","End":"01:06.110","Text":"Yeah, we get rid of the fraction,"},{"Start":"01:06.110 ","End":"01:07.390","Text":"but there\u0027s also an 8 here."},{"Start":"01:07.390 ","End":"01:09.200","Text":"Why don\u0027t I have the same opportunity,"},{"Start":"01:09.200 ","End":"01:12.230","Text":"multiply this by 4 and then I\u0027ll get 8 and 8."},{"Start":"01:12.230 ","End":"01:14.630","Text":"That\u0027s what I wrote."},{"Start":"01:14.630 ","End":"01:16.100","Text":"This 1 by 4,"},{"Start":"01:16.100 ","End":"01:17.875","Text":"this 1 by 2,"},{"Start":"01:17.875 ","End":"01:21.075","Text":"and indeed we get 8 in both places."},{"Start":"01:21.075 ","End":"01:25.010","Text":"We\u0027re not going to be working with fractions except at the very end, possibly,"},{"Start":"01:25.010 ","End":"01:30.424","Text":"in order to get ones we might have to do some division."},{"Start":"01:30.424 ","End":"01:32.510","Text":"There might be a fraction in the inverse,"},{"Start":"01:32.510 ","End":"01:34.100","Text":"but during the whole way,"},{"Start":"01:34.100 ","End":"01:36.120","Text":"we won\u0027t have fractions,"},{"Start":"01:36.120 ","End":"01:37.470","Text":"we\u0027ll just have whole numbers."},{"Start":"01:37.470 ","End":"01:39.380","Text":"Now let\u0027s do the subtraction."},{"Start":"01:39.380 ","End":"01:41.810","Text":"Subtract row 1 from row 2,"},{"Start":"01:41.810 ","End":"01:44.360","Text":"and as a result, I will get 0 down here"},{"Start":"01:44.360 ","End":"01:46.190","Text":"and 4 minus 3 is 1,"},{"Start":"01:46.190 ","End":"01:48.815","Text":"0 minus 2 is minus 2, and so on."},{"Start":"01:48.815 ","End":"01:54.300","Text":"At this point I have the 1 there."},{"Start":"01:54.300 ","End":"01:57.700","Text":"Now, how do I proceed?"},{"Start":"01:57.700 ","End":"02:01.160","Text":"I don\u0027t want to divide by 8 and get fractions,"},{"Start":"02:01.160 ","End":"02:02.495","Text":"that misses the whole point."},{"Start":"02:02.495 ","End":"02:06.770","Text":"What I\u0027ll do, I\u0027ll get the 0 here by subtracting"},{"Start":"02:06.770 ","End":"02:12.975","Text":"the last row multiplied by 3 from the 1st row."},{"Start":"02:12.975 ","End":"02:15.255","Text":"That\u0027s it in row notation."},{"Start":"02:15.255 ","End":"02:18.810","Text":"After the subtraction of thrice this from this,"},{"Start":"02:18.810 ","End":"02:23.410","Text":"I get 0 here, and so on."},{"Start":"02:24.290 ","End":"02:29.460","Text":"2 minus 3 times minus 2 is 8, and so on."},{"Start":"02:29.460 ","End":"02:31.730","Text":"We\u0027re very close."},{"Start":"02:31.730 ","End":"02:35.660","Text":"All we have to do now to get an identity matrix here is divide this by 8,"},{"Start":"02:35.660 ","End":"02:38.560","Text":"and this is the only point at which fractions will come in."},{"Start":"02:38.560 ","End":"02:41.480","Text":"Here\u0027s the notation, 1/8 of the 1st row,"},{"Start":"02:41.480 ","End":"02:44.630","Text":"and that gives us the identity matrix here."},{"Start":"02:44.630 ","End":"02:46.745","Text":"This is the inverse."},{"Start":"02:46.745 ","End":"02:49.190","Text":"Although we could just stop here,"},{"Start":"02:49.190 ","End":"02:54.280","Text":"it\u0027s recommended to check your result by multiplying with the original matrix."},{"Start":"02:54.280 ","End":"02:59.210","Text":"Check that if you multiply the inverse we got,"},{"Start":"02:59.210 ","End":"03:01.570","Text":"by the matrix we started out with,"},{"Start":"03:01.570 ","End":"03:03.890","Text":"that we really do get the identity matrix."},{"Start":"03:03.890 ","End":"03:07.290","Text":"I\u0027ll leave that to you to check. We\u0027re done."}],"ID":9858},{"Watched":false,"Name":"Exercise 4","Duration":"3m 47s","ChapterTopicVideoID":9509,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9509.jpeg","UploadDate":"2017-07-26T08:29:41.5030000","DurationForVideoObject":"PT3M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, we have to find the inverse of this matrix,"},{"Start":"00:03.720 ","End":"00:05.495","Text":"which is 3 by 3,"},{"Start":"00:05.495 ","End":"00:10.110","Text":"and we use the method we\u0027ve been using so far,"},{"Start":"00:10.110 ","End":"00:11.820","Text":"the method in this section,"},{"Start":"00:11.820 ","End":"00:19.110","Text":"which is the copy the matrix or rather extend it to a larger matrix"},{"Start":"00:19.110 ","End":"00:23.160","Text":"where we have the original matrix on the left side of"},{"Start":"00:23.160 ","End":"00:27.540","Text":"a partition and we have the identity matrix on the right side of a partition."},{"Start":"00:27.540 ","End":"00:29.265","Text":"In this case that of a 3 by 3,"},{"Start":"00:29.265 ","End":"00:30.900","Text":"we have a 3 by 6,"},{"Start":"00:30.900 ","End":"00:34.400","Text":"and now we\u0027re going to do row operations on"},{"Start":"00:34.400 ","End":"00:40.850","Text":"this combined larger matrix until we get the identity here,"},{"Start":"00:40.850 ","End":"00:46.805","Text":"and at the end when we have that on the right-hand side will be the inverse of A."},{"Start":"00:46.805 ","End":"00:54.680","Text":"Let\u0027s get started and see how we can get the identity matrix here."},{"Start":"00:54.680 ","End":"00:57.710","Text":"We do the most obvious thing here,"},{"Start":"00:57.710 ","End":"01:04.340","Text":"which is to get 0s below the 1 by subtracting 4 times this whole top"},{"Start":"01:04.340 ","End":"01:08.060","Text":"row from the 2nd row and twice the top row"},{"Start":"01:08.060 ","End":"01:12.885","Text":"from the 3rd row as I\u0027ve written that here."},{"Start":"01:12.885 ","End":"01:16.775","Text":"When we\u0027ve done that, we do get 2 0s here."},{"Start":"01:16.775 ","End":"01:20.615","Text":"Don\u0027t forget to also do on the right side."},{"Start":"01:20.615 ","End":"01:26.270","Text":"For example, 0 minus 4 times 1 is minus 4 and 0 minus twice,"},{"Start":"01:26.270 ","End":"01:28.325","Text":"1 is minus 2 and so on."},{"Start":"01:28.325 ","End":"01:31.430","Text":"We\u0027re working on both sides, just stressing it."},{"Start":"01:31.430 ","End":"01:33.730","Text":"Sometimes people forget."},{"Start":"01:33.770 ","End":"01:36.500","Text":"Let\u0027s take a look where we are now,"},{"Start":"01:36.500 ","End":"01:38.659","Text":"what we can do next."},{"Start":"01:38.659 ","End":"01:41.060","Text":"I suggest taking an easy shot,"},{"Start":"01:41.060 ","End":"01:43.535","Text":"just adding these 2 together,"},{"Start":"01:43.535 ","End":"01:45.335","Text":"putting it in the last row,"},{"Start":"01:45.335 ","End":"01:47.320","Text":"that would give us a 0 here,"},{"Start":"01:47.320 ","End":"01:49.230","Text":"is what I said in row notation,"},{"Start":"01:49.230 ","End":"01:54.410","Text":"and when we add this to this we do get 0 here and check that the rest of them,"},{"Start":"01:54.410 ","End":"01:58.475","Text":"for example, minus 4 and minus 2 is minus 6 and so on."},{"Start":"01:58.475 ","End":"02:00.625","Text":"We\u0027re getting closer."},{"Start":"02:00.625 ","End":"02:05.210","Text":"Next thing to do would be I would say,"},{"Start":"02:05.210 ","End":"02:08.510","Text":"get rid of the 2 here, make that as 0."},{"Start":"02:08.510 ","End":"02:12.785","Text":"Then we\u0027ll get a diagonal matrix which is very close to the identity."},{"Start":"02:12.785 ","End":"02:15.985","Text":"Add twice the last row to this row."},{"Start":"02:15.985 ","End":"02:18.510","Text":"Here it is in row notation,"},{"Start":"02:18.510 ","End":"02:20.460","Text":"and what we actually get is,"},{"Start":"02:20.460 ","End":"02:23.030","Text":"here we get the same thing with the 0 here,"},{"Start":"02:23.030 ","End":"02:29.910","Text":"and we also have twice this 1 plus this 0 here,"},{"Start":"02:29.910 ","End":"02:33.020","Text":"twice minus 6 plus 1 is minus 11 and so on."},{"Start":"02:33.020 ","End":"02:35.310","Text":"So this is what we get."},{"Start":"02:36.170 ","End":"02:42.770","Text":"At this point I think it\u0027s fairly clear that all we have to do is take these 2 rows,"},{"Start":"02:42.770 ","End":"02:45.470","Text":"multiply them, or divide them by minus"},{"Start":"02:45.470 ","End":"02:49.450","Text":"1 makes no difference and we\u0027ll get just the identity."},{"Start":"02:49.450 ","End":"02:52.875","Text":"After we make reverse all this row,"},{"Start":"02:52.875 ","End":"02:54.530","Text":"here we get 0, 1, 0."},{"Start":"02:54.530 ","End":"02:57.800","Text":"But notice, don\u0027t forget to do it on the right-hand side also,"},{"Start":"02:57.800 ","End":"02:59.090","Text":"these have changed signs,"},{"Start":"02:59.090 ","End":"03:00.920","Text":"and similarly for the last row,"},{"Start":"03:00.920 ","End":"03:02.990","Text":"everything\u0027s changed sign,"},{"Start":"03:02.990 ","End":"03:06.500","Text":"and now we do have the identity matrix here."},{"Start":"03:06.500 ","End":"03:08.285","Text":"When we get that,"},{"Start":"03:08.285 ","End":"03:14.225","Text":"then on the right-hand side is the inverse of the original matrix A,"},{"Start":"03:14.225 ","End":"03:18.590","Text":"and I usually recommend not to stop here,"},{"Start":"03:18.590 ","End":"03:21.935","Text":"but to check that this times the original matrix,"},{"Start":"03:21.935 ","End":"03:24.799","Text":"like A minus 1 here."},{"Start":"03:24.799 ","End":"03:29.235","Text":"It doesn\u0027t matter if you put the A on the left or the right,"},{"Start":"03:29.235 ","End":"03:31.985","Text":"either will do to check for inverse,"},{"Start":"03:31.985 ","End":"03:33.770","Text":"multiply them together,"},{"Start":"03:33.770 ","End":"03:35.735","Text":"and you should get this."},{"Start":"03:35.735 ","End":"03:38.570","Text":"I\u0027m not going to do the calculation for you,"},{"Start":"03:38.570 ","End":"03:44.240","Text":"you can always pause the clip and check the computations."},{"Start":"03:44.240 ","End":"03:46.920","Text":"So we\u0027re done."}],"ID":9859},{"Watched":false,"Name":"Exercise 5","Duration":"3m 48s","ChapterTopicVideoID":9510,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9510.jpeg","UploadDate":"2017-07-26T08:29:57.6800000","DurationForVideoObject":"PT3M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"Here we have yet another inverse matrix problem which"},{"Start":"00:04.440 ","End":"00:09.280","Text":"we\u0027re going to do with the method of extending the matrix."},{"Start":"00:09.440 ","End":"00:14.655","Text":"Like so by sticking an identity matrix alongside it."},{"Start":"00:14.655 ","End":"00:18.570","Text":"Then we do row operations on the whole thing,"},{"Start":"00:18.570 ","End":"00:24.435","Text":"the 3 by 6 matrix until we get the identity matrix on the left of the bar."},{"Start":"00:24.435 ","End":"00:28.305","Text":"For starters, I could get a 0 here"},{"Start":"00:28.305 ","End":"00:35.535","Text":"by adding 5 times this row to twice this row and putting it here."},{"Start":"00:35.535 ","End":"00:38.370","Text":"I don\u0027t want to mess with fractions by getting a 1 here,"},{"Start":"00:38.370 ","End":"00:46.640","Text":"so 5 times this plus twice this here will give me this, the 0 here."},{"Start":"00:46.640 ","End":"00:50.975","Text":"This is the row notation and you can check the calculations."},{"Start":"00:50.975 ","End":"00:53.195","Text":"Let\u0027s see what should we do now?"},{"Start":"00:53.195 ","End":"00:57.830","Text":"Well, I could make a 0 here."},{"Start":"00:57.830 ","End":"01:01.685","Text":"If I take this plus twice this,"},{"Start":"01:01.685 ","End":"01:04.715","Text":"but stick it in the bottom row."},{"Start":"01:04.715 ","End":"01:09.200","Text":"In other words, we take twice row 3 plus row 2,"},{"Start":"01:09.200 ","End":"01:11.535","Text":"put it into row 3."},{"Start":"01:11.535 ","End":"01:19.190","Text":"Then we will get the following and check a couple of numbers."},{"Start":"01:19.190 ","End":"01:23.180","Text":"Twice 1 and minus 1 is 1, twice minus 5,"},{"Start":"01:23.180 ","End":"01:30.090","Text":"and 0 is 10 and so on. What next?"},{"Start":"01:31.010 ","End":"01:36.935","Text":"I\u0027m going to go for getting these 2 to be 0 and I can do this"},{"Start":"01:36.935 ","End":"01:43.270","Text":"by adding the last row to the 2nd but subtracting it from the 1st."},{"Start":"01:43.270 ","End":"01:44.850","Text":"In row notation,"},{"Start":"01:44.850 ","End":"01:46.380","Text":"this is what I mean."},{"Start":"01:46.380 ","End":"01:49.150","Text":"The result of doing that,"},{"Start":"01:49.150 ","End":"01:53.510","Text":"I\u0027ll leave you to check is this with the 2 0s here."},{"Start":"01:53.510 ","End":"01:57.350","Text":"Now I want to get rid of this 1 here."},{"Start":"01:57.350 ","End":"01:59.650","Text":"I want to have just diagonal."},{"Start":"01:59.650 ","End":"02:02.085","Text":"There\u0027s more than 1 way to go about it."},{"Start":"02:02.085 ","End":"02:07.740","Text":"I could if I didn\u0027t want fractions to take twice"},{"Start":"02:07.740 ","End":"02:15.210","Text":"this and subtract this but this row is all even."},{"Start":"02:15.210 ","End":"02:17.130","Text":"Why don\u0027t we just divide it by 2,"},{"Start":"02:17.130 ","End":"02:19.570","Text":"when you see something divides by something"},{"Start":"02:19.570 ","End":"02:22.445","Text":"then just go ahead and do it simplifies the number."},{"Start":"02:22.445 ","End":"02:23.780","Text":"Here we get a 1,"},{"Start":"02:23.780 ","End":"02:26.300","Text":"but we also have the whole of this,"},{"Start":"02:26.300 ","End":"02:29.825","Text":"and we\u0027ve got minus 5, 1 and 2 instead of this."},{"Start":"02:29.825 ","End":"02:35.360","Text":"Now, the obvious thing to do is"},{"Start":"02:35.360 ","End":"02:41.700","Text":"to subtract the 2nd row from the top row."},{"Start":"02:42.080 ","End":"02:47.150","Text":"If we do that, we almost got the identity matrix here."},{"Start":"02:47.150 ","End":"02:49.370","Text":"We\u0027ve got a diagonal matrix, we even have 2 1s."},{"Start":"02:49.370 ","End":"02:55.640","Text":"All we have to do obviously now is divide the top row by 2 here and look,"},{"Start":"02:55.640 ","End":"02:58.640","Text":"we didn\u0027t even get any fractions that was lucky."},{"Start":"02:58.640 ","End":"03:01.010","Text":"Now at this point,"},{"Start":"03:01.010 ","End":"03:04.340","Text":"we have the identity on the left of the separator."},{"Start":"03:04.340 ","End":"03:10.015","Text":"We have the inverse of a on the right side of it."},{"Start":"03:10.015 ","End":"03:17.435","Text":"We could just enclose these in brackets and say this is the inverse but I always think,"},{"Start":"03:17.435 ","End":"03:21.785","Text":"well, if you have the time, it\u0027s good to check that this times the original matrix."},{"Start":"03:21.785 ","End":"03:26.855","Text":"In this case, it doesn\u0027t really matter if you put a times a minus 1, a minus 1 times a."},{"Start":"03:26.855 ","End":"03:29.695","Text":"Check that this product gives you 1."},{"Start":"03:29.695 ","End":"03:32.755","Text":"I\u0027ll just check 1 of the entries for you."},{"Start":"03:32.755 ","End":"03:37.820","Text":"Let\u0027s try this with this and see if we get this."},{"Start":"03:37.820 ","End":"03:40.370","Text":"Let\u0027s see 2 times 8 is 16,"},{"Start":"03:40.370 ","End":"03:42.124","Text":"minus 5 is 11,"},{"Start":"03:42.124 ","End":"03:43.595","Text":"minus 10 is 1,"},{"Start":"03:43.595 ","End":"03:45.320","Text":"and so on for the 8 others,"},{"Start":"03:45.320 ","End":"03:46.849","Text":"these 9 entries altogether."},{"Start":"03:46.849 ","End":"03:49.270","Text":"We\u0027re done."}],"ID":9860},{"Watched":false,"Name":"Exercise 6","Duration":"3m 14s","ChapterTopicVideoID":9500,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9500.jpeg","UploadDate":"2017-07-26T08:27:11.7930000","DurationForVideoObject":"PT3M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"Continuing with finding inverse of"},{"Start":"00:02.340 ","End":"00:04.920","Text":"matrix using the method we\u0027ve been using in this section,"},{"Start":"00:04.920 ","End":"00:07.980","Text":"I\u0027m going to go a little bit quicker this time."},{"Start":"00:07.980 ","End":"00:10.680","Text":"We copy the matrix here,"},{"Start":"00:10.680 ","End":"00:11.940","Text":"we extend it,"},{"Start":"00:11.940 ","End":"00:16.560","Text":"and put identity matrix to the right of the separator."},{"Start":"00:16.560 ","End":"00:19.620","Text":"Then we start to do row operations with the aim"},{"Start":"00:19.620 ","End":"00:22.830","Text":"of getting the identity on the left of the separator."},{"Start":"00:22.830 ","End":"00:26.720","Text":"What I\u0027ll do first is to get 0 here and here,"},{"Start":"00:26.720 ","End":"00:28.610","Text":"but without using fractions."},{"Start":"00:28.610 ","End":"00:32.155","Text":"If I do twice this minus 3 times this,"},{"Start":"00:32.155 ","End":"00:34.545","Text":"like 6 minus 6 is 0."},{"Start":"00:34.545 ","End":"00:39.500","Text":"Also, I can take twice this minus 5 times this and here I\u0027ll get 10 minus 10 is"},{"Start":"00:39.500 ","End":"00:45.889","Text":"0 and if I apply these 2 operations to the whole of the extended matrix,"},{"Start":"00:45.889 ","End":"00:47.160","Text":"the 3 by 6,"},{"Start":"00:47.160 ","End":"00:49.785","Text":"then we get this."},{"Start":"00:49.785 ","End":"00:52.865","Text":"That\u0027s the 1st column looking good."},{"Start":"00:52.865 ","End":"00:56.335","Text":"Now how about getting 0 here and here?"},{"Start":"00:56.335 ","End":"00:58.440","Text":"Sorry, I don\u0027t need 0 here,"},{"Start":"00:58.440 ","End":"01:01.215","Text":"I could but I just need the 0 here."},{"Start":"01:01.215 ","End":"01:05.165","Text":"First of all, I\u0027ll get it into echelon form if I want."},{"Start":"01:05.165 ","End":"01:11.510","Text":"I\u0027m just subtracting the second row from the third row and this is what we get."},{"Start":"01:11.510 ","End":"01:13.330","Text":"Note that the last row,"},{"Start":"01:13.330 ","End":"01:17.935","Text":"they\u0027re all divisible by 2 and I recommended that when you have something gets divisible,"},{"Start":"01:17.935 ","End":"01:19.865","Text":"it\u0027s better to divide."},{"Start":"01:19.865 ","End":"01:24.490","Text":"We have the last row and we get just 1s"},{"Start":"01:24.490 ","End":"01:29.260","Text":"instead of the 2s with these minuses are appropriate."},{"Start":"01:29.260 ","End":"01:33.250","Text":"Now, let\u0027s just subtract"},{"Start":"01:33.250 ","End":"01:39.290","Text":"the last row from each of the 1st 2 rows and that should give us 0s here."},{"Start":"01:39.290 ","End":"01:41.380","Text":"Here\u0027s in row notation,"},{"Start":"01:41.380 ","End":"01:44.665","Text":"and here\u0027s the result focusing on the 0s here."},{"Start":"01:44.665 ","End":"01:51.755","Text":"Don\u0027t forget to do it to the right-hand side of the extended matrix also."},{"Start":"01:51.755 ","End":"01:54.640","Text":"For example,"},{"Start":"01:59.750 ","End":"02:03.545","Text":"I meant 1 takeaway minus 1 is 2,"},{"Start":"02:03.545 ","End":"02:05.570","Text":"0 takeaway minus 1 is 1,"},{"Start":"02:05.570 ","End":"02:09.660","Text":"0 takeaway 1 is 1 so on and so on."},{"Start":"02:10.250 ","End":"02:19.935","Text":"Now I\u0027d want to make a 0 here so just subtract the 2nd row from the 1st."},{"Start":"02:19.935 ","End":"02:21.740","Text":"If I do that subtraction,"},{"Start":"02:21.740 ","End":"02:25.160","Text":"then I will just get the 0 here."},{"Start":"02:25.160 ","End":"02:30.445","Text":"Now I have a diagonal but not an identity matrix."},{"Start":"02:30.445 ","End":"02:32.680","Text":"I mean the 1 here is fine,"},{"Start":"02:32.680 ","End":"02:35.920","Text":"but here we have to divide or multiply by minus 1,"},{"Start":"02:35.920 ","End":"02:38.050","Text":"and here we have to half it."},{"Start":"02:38.050 ","End":"02:42.660","Text":"We do those 2 operations and this is what we get."},{"Start":"02:42.660 ","End":"02:48.220","Text":"Notice that we have on the left of the partition the identity."},{"Start":"02:48.220 ","End":"02:50.440","Text":"You could write it as I_3 if you want,"},{"Start":"02:50.440 ","End":"02:52.105","Text":"we\u0027ll just write I."},{"Start":"02:52.105 ","End":"02:54.460","Text":"This must be the inverse."},{"Start":"02:54.460 ","End":"02:56.980","Text":"This part here is the inverse of A"},{"Start":"02:56.980 ","End":"03:02.260","Text":"and we\u0027re done except that I recommend checking by"},{"Start":"03:02.260 ","End":"03:06.605","Text":"multiplying this with the original A"},{"Start":"03:06.605 ","End":"03:08.560","Text":"on the right or on the left, it doesn\u0027t matter,"},{"Start":"03:08.560 ","End":"03:10.300","Text":"to show that we get the identity"},{"Start":"03:10.300 ","End":"03:11.170","Text":"and when we\u0027ve done that,"},{"Start":"03:11.170 ","End":"03:13.430","Text":"then we\u0027re really done."}],"ID":9861},{"Watched":false,"Name":"Exercise 7","Duration":"2m 18s","ChapterTopicVideoID":9501,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9501.jpeg","UploadDate":"2017-07-26T08:27:24.7670000","DurationForVideoObject":"PT2M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.189","Text":"Continuing with the previous series of exercises of finding the inverse."},{"Start":"00:05.189 ","End":"00:08.580","Text":"The same method, but we\u0027re a bit more ambitious this time and we"},{"Start":"00:08.580 ","End":"00:13.815","Text":"have a 4 by 4 matrix so same technique."},{"Start":"00:13.815 ","End":"00:18.520","Text":"We make a larger matrix made up of 2 halves on the left,"},{"Start":"00:18.520 ","End":"00:22.055","Text":"the original, and on the right, the identity."},{"Start":"00:22.055 ","End":"00:24.400","Text":"This is actually I_4,"},{"Start":"00:24.400 ","End":"00:27.915","Text":"identity for 4 by 4 matrices."},{"Start":"00:27.915 ","End":"00:31.910","Text":"We\u0027re going to be doing row operations on the combined,"},{"Start":"00:31.910 ","End":"00:35.030","Text":"until we get the identity on the left."},{"Start":"00:35.030 ","End":"00:38.600","Text":"The 1st thing we can do to 0 out the rest of"},{"Start":"00:38.600 ","End":"00:42.030","Text":"the 1st column is to subtract the 1st row from each of the 2nd,"},{"Start":"00:42.030 ","End":"00:45.090","Text":"3rd, 4th, as I\u0027ve written here."},{"Start":"00:45.090 ","End":"00:49.700","Text":"That should give us 0s here and indeed it does."},{"Start":"00:49.700 ","End":"00:54.065","Text":"Don\u0027t forget to do the operations on both halves of the matrix."},{"Start":"00:54.065 ","End":"01:01.305","Text":"Now, we want to get 0 here and here so what I suggest we do"},{"Start":"01:01.305 ","End":"01:05.840","Text":"is to subtract this 2nd row from the 3rd and 4th and that should zero"},{"Start":"01:05.840 ","End":"01:10.060","Text":"out these elements and indeed it does."},{"Start":"01:10.060 ","End":"01:14.210","Text":"Now, we can get it to be in"},{"Start":"01:14.210 ","End":"01:20.305","Text":"echelon form if we subtract the 3rd row from the 4th row."},{"Start":"01:20.305 ","End":"01:23.930","Text":"Here we are. Look, we\u0027re really lucky."},{"Start":"01:23.930 ","End":"01:28.384","Text":"Not only is it in echelon form,"},{"Start":"01:28.384 ","End":"01:32.720","Text":"but it\u0027s actually a diagonal on the left to the separated."},{"Start":"01:32.720 ","End":"01:33.740","Text":"Once it\u0027s diagonal,"},{"Start":"01:33.740 ","End":"01:36.110","Text":"all we have to do is divide when necessary."},{"Start":"01:36.110 ","End":"01:37.295","Text":"Well, this one\u0027s okay."},{"Start":"01:37.295 ","End":"01:38.870","Text":"Divide this by 2, this by 3,"},{"Start":"01:38.870 ","End":"01:40.735","Text":"and this by 4."},{"Start":"01:40.735 ","End":"01:43.830","Text":"If we do that in row notation,"},{"Start":"01:43.830 ","End":"01:45.615","Text":"this is what I mean."},{"Start":"01:45.615 ","End":"01:47.970","Text":"Then this is what we get."},{"Start":"01:47.970 ","End":"01:50.360","Text":"Yeah, there\u0027s some fractions on here,"},{"Start":"01:50.360 ","End":"01:55.115","Text":"but we now know that we have the inverse of a A the right."},{"Start":"01:55.115 ","End":"01:57.530","Text":"You know me, I like to do a check"},{"Start":"01:57.530 ","End":"02:03.835","Text":"so we compute A times A inverse or the alleged inverse."},{"Start":"02:03.835 ","End":"02:06.050","Text":"We\u0027re checking it and see that we get"},{"Start":"02:06.050 ","End":"02:08.600","Text":"the identity or you could do it the other way around,"},{"Start":"02:08.600 ","End":"02:10.400","Text":"doesn\u0027t matter, A minus 1 times A,"},{"Start":"02:10.400 ","End":"02:12.675","Text":"you just have to do 1 of them, whichever."},{"Start":"02:12.675 ","End":"02:14.840","Text":"We do, in fact, get identity."},{"Start":"02:14.840 ","End":"02:18.630","Text":"I\u0027ll leave you to check the computations and so we\u0027re done."}],"ID":9862},{"Watched":false,"Name":"Exercise 8","Duration":"4m 56s","ChapterTopicVideoID":9502,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9502.jpeg","UploadDate":"2017-07-26T08:27:47.7930000","DurationForVideoObject":"PT4M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.665","Text":"In this exercise, we\u0027re given a 3 by 3 matrix,"},{"Start":"00:04.665 ","End":"00:07.890","Text":"but it has a parameter k."},{"Start":"00:07.890 ","End":"00:13.860","Text":"We can\u0027t say find its inverse."},{"Start":"00:13.860 ","End":"00:16.710","Text":"The question more suitable for this would"},{"Start":"00:16.710 ","End":"00:21.195","Text":"be find the values of k for which it\u0027s even invertible."},{"Start":"00:21.195 ","End":"00:24.870","Text":"Now, it goes without saying,"},{"Start":"00:24.870 ","End":"00:30.990","Text":"or maybe it needs to be said that for a matrix to be invertible,"},{"Start":"00:30.990 ","End":"00:32.475","Text":"it has to be square."},{"Start":"00:32.475 ","End":"00:37.620","Text":"This is indeed a 3 by 3 square matrix."},{"Start":"00:37.620 ","End":"00:39.194","Text":"That\u0027s a good start."},{"Start":"00:39.194 ","End":"00:49.050","Text":"Now, there is a theorem or a proposition or a statement that one way to do this,"},{"Start":"00:53.270 ","End":"00:55.680","Text":"is to, first of all,"},{"Start":"00:55.680 ","End":"01:01.690","Text":"bring it to row-echelon form."},{"Start":"01:02.600 ","End":"01:08.800","Text":"Then if the diagonal,"},{"Start":"01:09.230 ","End":"01:11.580","Text":"when I say the diagonal,"},{"Start":"01:11.580 ","End":"01:17.250","Text":"I mean the main diagonal always is nonzero."},{"Start":"01:17.250 ","End":"01:18.600","Text":"When I say nonzero,"},{"Start":"01:18.600 ","End":"01:21.030","Text":"I mean everything is nonzero."},{"Start":"01:21.030 ","End":"01:31.395","Text":"Then the matrix to say it is invertible,"},{"Start":"01:31.395 ","End":"01:32.640","Text":"and the other way around."},{"Start":"01:32.640 ","End":"01:40.215","Text":"If it\u0027s invertible and spelling is not the greatest I see, it\u0027s invertible."},{"Start":"01:40.215 ","End":"01:45.610","Text":"That\u0027s what we\u0027re going to do, bring it to row-echelon form and put"},{"Start":"01:45.610 ","End":"01:51.395","Text":"a condition on the diagonal that the n elements are all nonzero,"},{"Start":"01:51.395 ","End":"01:54.790","Text":"which would involve k somehow and we get n condition on k."},{"Start":"01:54.790 ","End":"02:01.015","Text":"Let\u0027s start doing row operations."},{"Start":"02:01.015 ","End":"02:04.240","Text":"Let me say bring it to row-echelon form using row operations,"},{"Start":"02:04.240 ","End":"02:08.230","Text":"so let\u0027s take 5 times this from this."},{"Start":"02:08.230 ","End":"02:14.315","Text":"What I say is basically row 2 minus 5,"},{"Start":"02:14.315 ","End":"02:18.365","Text":"row 1, put that into row 2."},{"Start":"02:18.365 ","End":"02:19.955","Text":"Similarly, with the 3,"},{"Start":"02:19.955 ","End":"02:24.795","Text":"I take row 3 and subtract thrice,"},{"Start":"02:24.795 ","End":"02:28.050","Text":"3 times row 1,"},{"Start":"02:28.050 ","End":"02:32.055","Text":"and put that into row 3,"},{"Start":"02:32.055 ","End":"02:37.220","Text":"and then what we get is these 2 zeros here."},{"Start":"02:37.220 ","End":"02:41.570","Text":"The other calculations, I\u0027ll leave you to do,"},{"Start":"02:41.570 ","End":"02:43.190","Text":"I just give you one example."},{"Start":"02:43.190 ","End":"02:45.245","Text":"Row 2 minus 5, row 1,"},{"Start":"02:45.245 ","End":"02:47.015","Text":"so minus 7,"},{"Start":"02:47.015 ","End":"02:50.760","Text":"minus 5 times row 1."},{"Start":"02:50.760 ","End":"02:53.880","Text":"Minus 7 plus 5 is minus 2,"},{"Start":"02:53.880 ","End":"02:57.160","Text":"so that entry checks and check the rest."},{"Start":"02:57.260 ","End":"03:00.050","Text":"We\u0027re getting close. We got 2 zeros here."},{"Start":"03:00.050 ","End":"03:01.610","Text":"I\u0027d like this to be 0."},{"Start":"03:01.610 ","End":"03:05.930","Text":"How about I just add this row to this row?"},{"Start":"03:05.930 ","End":"03:11.210","Text":"What I\u0027m going to propose is we take row 3 and"},{"Start":"03:11.210 ","End":"03:17.120","Text":"add row 2 to it and put the result in row 3."},{"Start":"03:17.120 ","End":"03:20.360","Text":"That certainly gives us a 0 here."},{"Start":"03:20.360 ","End":"03:23.585","Text":"Meanwhile, this is getting a bit more complicated."},{"Start":"03:23.585 ","End":"03:26.660","Text":"To do the calculations,"},{"Start":"03:26.660 ","End":"03:29.150","Text":"like here is not that difficult."},{"Start":"03:29.150 ","End":"03:32.840","Text":"This plus this and just reorder it, we get this."},{"Start":"03:32.840 ","End":"03:37.670","Text":"Now already it is in echelon form."},{"Start":"03:37.670 ","End":"03:40.945","Text":"You can see the staircase,"},{"Start":"03:40.945 ","End":"03:43.830","Text":"and it has a diagonal."},{"Start":"03:43.830 ","End":"03:46.855","Text":"This is what I call a diagonal."},{"Start":"03:46.855 ","End":"03:50.360","Text":"We want all the diagonal to be nonzero."},{"Start":"03:50.360 ","End":"03:52.920","Text":"Well, 1 and minus 2,"},{"Start":"03:52.920 ","End":"03:57.720","Text":"are not nonzero but the problem might be with"},{"Start":"03:57.720 ","End":"04:03.770","Text":"this entry because it\u0027s got a parameter k. It\u0027s not easy to see if it\u0027s 0 or not,"},{"Start":"04:03.770 ","End":"04:07.510","Text":"or in fact, we can actually give conditions."},{"Start":"04:07.510 ","End":"04:10.125","Text":"I need to find the 2 roots,"},{"Start":"04:10.125 ","End":"04:16.150","Text":"and it turns out that the roots are 1 and minus 2."},{"Start":"04:16.150 ","End":"04:21.990","Text":"These roots would be what you\u0027d get if you solve this equals 0."},{"Start":"04:23.090 ","End":"04:27.660","Text":"Again, I\u0027m not doing the computation we can just check 1 squared plus"},{"Start":"04:27.660 ","End":"04:33.210","Text":"1 minus 2 is 0 and 4 minus 2 minus 2 is 0."},{"Start":"04:33.210 ","End":"04:37.205","Text":"This is going to be 0 when k is one of these,"},{"Start":"04:37.205 ","End":"04:41.790","Text":"so we have to have nonzero,"},{"Start":"04:41.790 ","End":"04:45.060","Text":"meaning that k is neither minus 2,"},{"Start":"04:45.060 ","End":"04:49.300","Text":"nor 1, it can\u0027t be either of these is not this and it\u0027s not this."},{"Start":"04:49.300 ","End":"04:56.060","Text":"That\u0027s the condition for the matrix to be invertible. We\u0027re done."}],"ID":9863},{"Watched":false,"Name":"Exercise 9","Duration":"6m 53s","ChapterTopicVideoID":9503,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9503.jpeg","UploadDate":"2017-07-26T08:28:13.3170000","DurationForVideoObject":"PT6M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:03.810","Text":"Here we have a real exercise. What do we have?"},{"Start":"00:03.810 ","End":"00:09.900","Text":"We have a 5 by 5 matrix with the parameter k,"},{"Start":"00:09.900 ","End":"00:15.600","Text":"and we have to find out for which values of k this matrix is invertible."},{"Start":"00:15.600 ","End":"00:19.500","Text":"I want to just give you the highlights of how we do this."},{"Start":"00:19.500 ","End":"00:29.500","Text":"We bring it to row echelon form using elementary row operations."},{"Start":"00:30.860 ","End":"00:36.705","Text":"It\u0027s a square, this is 5 by 5."},{"Start":"00:36.705 ","End":"00:40.030","Text":"Then we look at the diagonal,"},{"Start":"00:40.760 ","End":"00:46.965","Text":"and if the diagonal is all not 0,"},{"Start":"00:46.965 ","End":"00:52.210","Text":"then bingo, it\u0027s invertible."},{"Start":"00:52.790 ","End":"00:59.445","Text":"Otherwise, otw is otherwise, not."},{"Start":"00:59.445 ","End":"01:06.550","Text":"The main thing is getting it into row echelon form."},{"Start":"01:08.750 ","End":"01:12.945","Text":"We just better get started,"},{"Start":"01:12.945 ","End":"01:15.695","Text":"5 by 5 we can handle."},{"Start":"01:15.695 ","End":"01:17.890","Text":"Look at the 1st column,"},{"Start":"01:17.890 ","End":"01:19.285","Text":"we got a 1 here,"},{"Start":"01:19.285 ","End":"01:21.910","Text":"why shouldn\u0027t we just make the rest 0?"},{"Start":"01:21.910 ","End":"01:27.075","Text":"Just 0 them out by subtracting multiples of this by the appropriate number,"},{"Start":"01:27.075 ","End":"01:28.970","Text":"subtracting directly from this, this,"},{"Start":"01:28.970 ","End":"01:32.000","Text":"and this and k times from the last 1."},{"Start":"01:32.000 ","End":"01:34.720","Text":"Here\u0027s what I just said in compact form,"},{"Start":"01:34.720 ","End":"01:37.550","Text":"and here\u0027s what we get after all that work,"},{"Start":"01:37.550 ","End":"01:39.110","Text":"which I\u0027ll leave you to check,"},{"Start":"01:39.110 ","End":"01:43.610","Text":"but of course we have to see that there are 4 0s here."},{"Start":"01:43.610 ","End":"01:46.460","Text":"The rest of this column is being zeroed out."},{"Start":"01:46.460 ","End":"01:49.565","Text":"We\u0027re getting closer to echelon form."},{"Start":"01:49.565 ","End":"01:51.475","Text":"Now what would we do here?"},{"Start":"01:51.475 ","End":"01:54.470","Text":"The fact that I have a 0 here is a bit of"},{"Start":"01:54.470 ","End":"02:00.260","Text":"a problem because now it\u0027s going to be hard to 0 out the rest of the column,"},{"Start":"02:00.260 ","End":"02:03.140","Text":"but there is a standard trick we can use."},{"Start":"02:03.140 ","End":"02:05.105","Text":"We can swap rows."},{"Start":"02:05.105 ","End":"02:07.220","Text":"I don\u0027t like this 0 either."},{"Start":"02:07.220 ","End":"02:09.440","Text":"Let\u0027s swap these 2 with these 2,"},{"Start":"02:09.440 ","End":"02:13.835","Text":"something like row 2 swaps with row 5"},{"Start":"02:13.835 ","End":"02:20.090","Text":"and row 3 swaps with row 4,"},{"Start":"02:20.090 ","End":"02:24.350","Text":"and that gives us this."},{"Start":"02:24.350 ","End":"02:26.015","Text":"We just put this row here."},{"Start":"02:26.015 ","End":"02:27.530","Text":"You\u0027ll see, anyway."},{"Start":"02:27.530 ","End":"02:34.680","Text":"Now we don\u0027t have the 0s there that are going to put a stop to things."},{"Start":"02:35.270 ","End":"02:38.875","Text":"We have 1 minus k here."},{"Start":"02:38.875 ","End":"02:43.250","Text":"Notice that k minus 1 is the inverse,"},{"Start":"02:43.250 ","End":"02:45.110","Text":"the negative of 1 minus k."},{"Start":"02:45.110 ","End":"02:50.420","Text":"I\u0027ve used that as an opportunity to get rid of this element"},{"Start":"02:50.420 ","End":"02:53.060","Text":"here by adding the second row to"},{"Start":"02:53.060 ","End":"02:58.290","Text":"the third row as indicated and that brings us up to here."},{"Start":"02:58.290 ","End":"03:01.275","Text":"0 is here, good, 0 here, good."},{"Start":"03:01.275 ","End":"03:04.620","Text":"Now, what about this and this?"},{"Start":"03:04.620 ","End":"03:09.910","Text":"These 2 won\u0027t bother me now."},{"Start":"03:10.060 ","End":"03:15.830","Text":"Notice that this is k minus 1 and this is 1 minus"},{"Start":"03:15.830 ","End":"03:22.170","Text":"k. 1 minus k plus k minus 1 would be 0."},{"Start":"03:22.170 ","End":"03:30.140","Text":"Actually, I\u0027m going to add row 3 to row 4 and now we have another 0."},{"Start":"03:30.140 ","End":"03:34.610","Text":"Now the only thing that prevents us from being an echelon form is here."},{"Start":"03:34.610 ","End":"03:35.960","Text":"But the same trick again,"},{"Start":"03:35.960 ","End":"03:39.380","Text":"if I add this to this, in other words,"},{"Start":"03:39.380 ","End":"03:43.605","Text":"I want to add row 4 to row 5,"},{"Start":"03:43.605 ","End":"03:45.315","Text":"that\u0027s what it says here."},{"Start":"03:45.315 ","End":"03:48.030","Text":"Now we finally get this 0,"},{"Start":"03:48.030 ","End":"03:54.185","Text":"and we are indeed in echelon form,"},{"Start":"03:54.185 ","End":"03:57.100","Text":"and we have a diagonal."},{"Start":"03:57.100 ","End":"03:59.580","Text":"Diagonals this, this,"},{"Start":"03:59.580 ","End":"04:02.955","Text":"this, this, and this."},{"Start":"04:02.955 ","End":"04:05.910","Text":"I want them all to be non 0."},{"Start":"04:05.910 ","End":"04:11.115","Text":"I want 1 to be not equal to 0, got that."},{"Start":"04:11.115 ","End":"04:13.455","Text":"1 minus k shouldn\u0027t be 0."},{"Start":"04:13.455 ","End":"04:15.145","Text":"K not equal to 1."},{"Start":"04:15.145 ","End":"04:18.140","Text":"For here I need k to be not equal to 1."},{"Start":"04:18.140 ","End":"04:21.460","Text":"From here I need k to be not equal to 1."},{"Start":"04:21.460 ","End":"04:25.820","Text":"From here, if I take the roots of this polynomial,"},{"Start":"04:25.820 ","End":"04:27.290","Text":"this happens to be,"},{"Start":"04:27.290 ","End":"04:29.300","Text":"I can also give you the factorization."},{"Start":"04:29.300 ","End":"04:38.140","Text":"It\u0027s k minus 4"},{"Start":"04:38.140 ","End":"04:42.110","Text":"times k plus 1,"},{"Start":"04:42.110 ","End":"04:45.850","Text":"which means that the roots are 4 and minus 1."},{"Start":"04:45.850 ","End":"04:50.150","Text":"We must have the k is not equal to 4,"},{"Start":"04:50.150 ","End":"04:55.310","Text":"and also the k is not equal to minus 1."},{"Start":"04:55.310 ","End":"04:58.810","Text":"If I combine all these facts together,"},{"Start":"04:58.810 ","End":"05:02.410","Text":"I just have 3 values that k needs to avoid."},{"Start":"05:02.410 ","End":"05:04.690","Text":"It needs to avoid being 1,"},{"Start":"05:04.690 ","End":"05:07.015","Text":"4, or minus 1."},{"Start":"05:07.015 ","End":"05:10.060","Text":"Whoops, I got the signs backwards."},{"Start":"05:10.060 ","End":"05:13.320","Text":"That\u0027s a minus, that\u0027s a plus,"},{"Start":"05:13.320 ","End":"05:16.865","Text":"k cannot equal minus 4,"},{"Start":"05:16.865 ","End":"05:20.430","Text":"and k cannot equal to 1."},{"Start":"05:20.430 ","End":"05:26.360","Text":"Really it all boils down to this is not a condition on k at all."},{"Start":"05:26.360 ","End":"05:29.270","Text":"We have k not equal to 1,"},{"Start":"05:29.270 ","End":"05:31.340","Text":"1, 1, 1 or minus 4."},{"Start":"05:31.340 ","End":"05:32.780","Text":"Well, the 1s redundant,"},{"Start":"05:32.780 ","End":"05:43.200","Text":"so k is just not equal to 1 or minus 4."},{"Start":"05:43.200 ","End":"05:48.320","Text":"Mathia was working here on the cases where A is not invertible,"},{"Start":"05:48.320 ","End":"05:49.625","Text":"I forget what they are."},{"Start":"05:49.625 ","End":"05:52.175","Text":"But for not invertible,"},{"Start":"05:52.175 ","End":"05:56.710","Text":"then we have to have that 1 minus k is not 0,"},{"Start":"05:56.710 ","End":"06:04.055","Text":"and 4 minus 3 minus k squared is not equal to 0."},{"Start":"06:04.055 ","End":"06:08.435","Text":"Which gives us the k is not equal to 1,"},{"Start":"06:08.435 ","End":"06:11.645","Text":"and k is not equal to minus 4,"},{"Start":"06:11.645 ","End":"06:15.300","Text":"and then it is invertible."},{"Start":"06:18.250 ","End":"06:20.300","Text":"I\u0027m actually going to go back."},{"Start":"06:20.300 ","End":"06:25.740","Text":"I\u0027m just curious, we want to know the values invertible or not invertible."},{"Start":"06:27.620 ","End":"06:30.510","Text":"It was invertible,"},{"Start":"06:30.510 ","End":"06:36.310","Text":"not equal to 1 or minus 4."},{"Start":"06:37.070 ","End":"06:42.530","Text":"This is the answer for the values of k. All"},{"Start":"06:42.530 ","End":"06:47.640","Text":"the values on the number line with 2 exceptions,"},{"Start":"06:47.640 ","End":"06:54.490","Text":"only these 2 bad ones spoil it from being invertible. We\u0027re done."}],"ID":9864},{"Watched":false,"Name":"Exercise 10","Duration":"3m 55s","ChapterTopicVideoID":9504,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9504.jpeg","UploadDate":"2017-07-26T08:28:30.6000000","DurationForVideoObject":"PT3M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"Here we have this system of linear equations,"},{"Start":"00:04.020 ","End":"00:05.880","Text":"3 equations and 3 unknowns,"},{"Start":"00:05.880 ","End":"00:09.810","Text":"and we\u0027re going to solve it using the inverse matrix method."},{"Start":"00:09.810 ","End":"00:13.140","Text":"The first step is to write this in matrix form."},{"Start":"00:13.140 ","End":"00:15.540","Text":"We\u0027ve seen this before, it\u0027s a review."},{"Start":"00:15.540 ","End":"00:18.090","Text":"We just take the coefficients in order,"},{"Start":"00:18.090 ","End":"00:20.670","Text":"and if there\u0027s any missing, you put zeros and so on."},{"Start":"00:20.670 ","End":"00:22.830","Text":"Here\u0027s the matrix of coefficients."},{"Start":"00:22.830 ","End":"00:26.700","Text":"Here is the vector of the variables x,"},{"Start":"00:26.700 ","End":"00:29.160","Text":"y, z, and here is the vector of constants."},{"Start":"00:29.160 ","End":"00:31.740","Text":"Let\u0027s just check 1 of the entries, say the top 1."},{"Start":"00:31.740 ","End":"00:36.100","Text":"We take this, multiply it element-wise with this,"},{"Start":"00:36.100 ","End":"00:44.435","Text":"and add 2 times x minus 1 times y plus 1 times z equals 3."},{"Start":"00:44.435 ","End":"00:46.070","Text":"That\u0027s exactly what\u0027s written here."},{"Start":"00:46.070 ","End":"00:48.110","Text":"Similarly for the other 2."},{"Start":"00:48.110 ","End":"00:50.300","Text":"Let\u0027s give them names."},{"Start":"00:50.300 ","End":"00:54.815","Text":"This matrix will be a,"},{"Start":"00:54.815 ","End":"01:00.740","Text":"this vector which is just a 1 column matrix is x."},{"Start":"01:00.740 ","End":"01:03.710","Text":"Sometimes we use boldface,"},{"Start":"01:03.710 ","End":"01:06.625","Text":"sometimes x with an underscore."},{"Start":"01:06.625 ","End":"01:08.990","Text":"Here\u0027s b, which is the constants."},{"Start":"01:08.990 ","End":"01:14.230","Text":"Again, you could have written it this way or often it\u0027s done in boldface."},{"Start":"01:14.270 ","End":"01:21.140","Text":"The theory of such matrix form equations is"},{"Start":"01:21.140 ","End":"01:27.950","Text":"that we can get x almost directly if we know the inverse of A."},{"Start":"01:27.950 ","End":"01:33.470","Text":"What we do once we have it in this form is say that the solution x, which is xyz,"},{"Start":"01:33.470 ","End":"01:39.790","Text":"is just the inverse matrix of a times column matrix or vector, b."},{"Start":"01:39.790 ","End":"01:42.635","Text":"Now how do we find this inverse?"},{"Start":"01:42.635 ","End":"01:44.510","Text":"I\u0027m just going to give it to you."},{"Start":"01:44.510 ","End":"01:48.205","Text":"We\u0027ve actually done it in a previous exercise."},{"Start":"01:48.205 ","End":"01:54.450","Text":"This is A minus 1 and this was in exercise,"},{"Start":"01:54.450 ","End":"01:56.720","Text":"well, the numbering might change,"},{"Start":"01:56.720 ","End":"01:58.340","Text":"but at the time I wrote this,"},{"Start":"01:58.340 ","End":"02:01.170","Text":"this was the exercise number."},{"Start":"02:02.000 ","End":"02:10.300","Text":"Continuing from here, all we have to do is multiply out the matrix,"},{"Start":"02:10.300 ","End":"02:13.305","Text":"which is a inverse with the column vector."},{"Start":"02:13.305 ","End":"02:17.830","Text":"Oh, yeah, before that I just wanted to label them so you can see what\u0027s going on here."},{"Start":"02:17.830 ","End":"02:22.030","Text":"This is the right-hand side of this arrow, this implication."},{"Start":"02:22.030 ","End":"02:25.395","Text":"Here is the x, here is the A inverse and here\u0027s the b."},{"Start":"02:25.395 ","End":"02:27.880","Text":"Now we do the multiplication."},{"Start":"02:27.880 ","End":"02:35.085","Text":"We take this, times this and that gives us the 1st entry,"},{"Start":"02:35.085 ","End":"02:36.990","Text":"we should come out to this."},{"Start":"02:36.990 ","End":"02:38.415","Text":"Let\u0027s just check 1 of them."},{"Start":"02:38.415 ","End":"02:40.830","Text":"2 times 3 is 6,"},{"Start":"02:40.830 ","End":"02:51.285","Text":"minus 5 is 1 plus 0 times 11 is still 1."},{"Start":"02:51.285 ","End":"02:52.620","Text":"I\u0027ll go through them all."},{"Start":"02:52.620 ","End":"02:57.375","Text":"Next, this time this."},{"Start":"02:57.375 ","End":"03:00.945","Text":"2 times 3 is 6 minus 15 plus"},{"Start":"03:00.945 ","End":"03:09.120","Text":"11 is 17 minus 15 is 2, we\u0027re okay."},{"Start":"03:09.120 ","End":"03:12.885","Text":"The last 1, minus 3,"},{"Start":"03:12.885 ","End":"03:18.165","Text":"minus 5, that\u0027s down to minus 8 plus 11 gives us the 3."},{"Start":"03:18.165 ","End":"03:20.910","Text":"Now we have x, y, and z,"},{"Start":"03:20.910 ","End":"03:27.230","Text":"and you might want to write it more explicitly and maybe even put a nice box around it."},{"Start":"03:27.300 ","End":"03:35.405","Text":"Now, in case you missed the exercise where we did the computation of A inverse,"},{"Start":"03:35.405 ","End":"03:41.975","Text":"I rewrote it in condensed form using this system,"},{"Start":"03:41.975 ","End":"03:43.160","Text":"these are all the steps."},{"Start":"03:43.160 ","End":"03:49.640","Text":"It goes down here and from here it continues up to here and then ends up here."},{"Start":"03:49.640 ","End":"03:51.230","Text":"But as I said, we\u0027ve done it before,"},{"Start":"03:51.230 ","End":"03:53.870","Text":"so I\u0027ll just leave this with you for reference."},{"Start":"03:53.870 ","End":"03:56.220","Text":"We\u0027re done."}],"ID":9865},{"Watched":false,"Name":"Exercise 11","Duration":"3m 50s","ChapterTopicVideoID":9505,"CourseChapterTopicPlaylistID":7283,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9505.jpeg","UploadDate":"2017-07-26T08:28:58.4570000","DurationForVideoObject":"PT3M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.700","Text":"In this exercise, we have here a system of linear equations."},{"Start":"00:05.700 ","End":"00:08.790","Text":"I guess could have put curly braces here to"},{"Start":"00:08.790 ","End":"00:11.970","Text":"emphasize the system as a whole, 4 equations,"},{"Start":"00:11.970 ","End":"00:16.890","Text":"4 unknowns, x, y, z, and t."},{"Start":"00:16.890 ","End":"00:20.400","Text":"We want to solve that using the inverse matrix method."},{"Start":"00:20.400 ","End":"00:24.540","Text":"First of all, let\u0027s write it in matrix form."},{"Start":"00:24.540 ","End":"00:29.190","Text":"We just take the coefficients making sure that the variables are all in order."},{"Start":"00:29.190 ","End":"00:31.890","Text":"Then any missing variable gets a 0,"},{"Start":"00:31.890 ","End":"00:38.190","Text":"like here we had a missing t and here we have a missing x."},{"Start":"00:38.190 ","End":"00:43.110","Text":"Then the column matrix which is a vector,"},{"Start":"00:43.110 ","End":"00:44.785","Text":"x, y, z, t,"},{"Start":"00:44.785 ","End":"00:48.360","Text":"the variables and then the constants 1, 0, 1, 0,"},{"Start":"00:48.950 ","End":"00:54.435","Text":"label these matrix A, vector x, vector b."},{"Start":"00:54.435 ","End":"00:58.855","Text":"The theory says that when you have it in this form,"},{"Start":"00:58.855 ","End":"01:01.655","Text":"that is the form A x equals b,"},{"Start":"01:01.655 ","End":"01:08.310","Text":"then the solution x is given by the inverse of A times b."},{"Start":"01:09.620 ","End":"01:13.790","Text":"We do actually have the inverse of A because we did this in"},{"Start":"01:13.790 ","End":"01:19.765","Text":"a previous exercise where we found that the inverse value was this,"},{"Start":"01:19.765 ","End":"01:22.090","Text":"just put the labels on these."},{"Start":"01:22.090 ","End":"01:26.260","Text":"The way we got to this,"},{"Start":"01:26.470 ","End":"01:31.520","Text":"the exercise number at the time that I\u0027m recording this,"},{"Start":"01:31.520 ","End":"01:35.105","Text":"it may have changed since is this."},{"Start":"01:35.105 ","End":"01:37.310","Text":"So go and check,"},{"Start":"01:37.310 ","End":"01:41.300","Text":"and that\u0027s what we got for A inverse."},{"Start":"01:41.300 ","End":"01:50.800","Text":"Okay. All that remains to do is to multiply out this by this."},{"Start":"01:50.800 ","End":"01:53.390","Text":"Here\u0027s the result of the calculation."},{"Start":"01:53.390 ","End":"01:55.070","Text":"I\u0027ll go through it with you."},{"Start":"01:55.070 ","End":"02:03.650","Text":"The first entry here should be equal to this row times this column."},{"Start":"02:03.650 ","End":"02:05.360","Text":"When I say row times a column, you know what I mean."},{"Start":"02:05.360 ","End":"02:07.595","Text":"Pairwise product and add,"},{"Start":"02:07.595 ","End":"02:11.195","Text":"7 times 1 is 7,"},{"Start":"02:11.195 ","End":"02:17.370","Text":"minus nothing, minus 20 plus nothing,"},{"Start":"02:17.370 ","End":"02:19.980","Text":"7 minus 20 is minus 13."},{"Start":"02:19.980 ","End":"02:22.015","Text":"That one\u0027s okay."},{"Start":"02:22.015 ","End":"02:32.130","Text":"Next 1, minus 2, 0, 6, 0, minus 2 and 6 is 4."},{"Start":"02:32.130 ","End":"02:33.150","Text":"That\u0027s okay."},{"Start":"02:33.150 ","End":"02:44.955","Text":"Next 1, 3, 0, minus 8, 0, 3 minus 8, is minus 5."},{"Start":"02:44.955 ","End":"02:55.470","Text":"The last, minus 1, 0, 3, 0 altogether 2."},{"Start":"02:55.470 ","End":"02:59.900","Text":"It might be nicer to write explicitly what x is,"},{"Start":"02:59.900 ","End":"03:04.370","Text":"y is and z and t are and put it in a nice box."},{"Start":"03:04.370 ","End":"03:09.065","Text":"In case you somehow are missing this exercise,"},{"Start":"03:09.065 ","End":"03:13.205","Text":"I\u0027ll give you a summary of what it looks like."},{"Start":"03:13.205 ","End":"03:19.490","Text":"Here we are, here\u0027s a summary of these steps to get from A to A inverse."},{"Start":"03:19.490 ","End":"03:21.530","Text":"Yes, I know it doesn\u0027t quite all fit in."},{"Start":"03:21.530 ","End":"03:23.480","Text":"I\u0027ll scroll down in the moment."},{"Start":"03:23.480 ","End":"03:28.225","Text":"It starts from here and then continues up to here,"},{"Start":"03:28.225 ","End":"03:36.270","Text":"and then from here to here,"},{"Start":"03:36.270 ","End":"03:37.860","Text":"and then down here."},{"Start":"03:37.860 ","End":"03:44.140","Text":"Now I\u0027m going to just scroll a bit so you can have the last bit."},{"Start":"03:44.140 ","End":"03:46.760","Text":"As I said, we did do it in a previous exercise."},{"Start":"03:46.760 ","End":"03:50.940","Text":"This is just in case you missed it. We\u0027re done."}],"ID":9866}],"Thumbnail":null,"ID":7283},{"Name":"Properties of the Matrix Inverse","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Inverse Matrix, Rules","Duration":"3m 6s","ChapterTopicVideoID":13533,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13533.jpeg","UploadDate":"2018-09-06T09:54:35.4100000","DurationForVideoObject":"PT3M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"Now we come to some rules"},{"Start":"00:02.280 ","End":"00:05.595","Text":"or properties of the inverse matrix."},{"Start":"00:05.595 ","End":"00:08.325","Text":"This is really part of the definition,"},{"Start":"00:08.325 ","End":"00:11.475","Text":"but we can make it as a rule of property."},{"Start":"00:11.475 ","End":"00:16.150","Text":"A matrix times its inverse is the identity matrix."},{"Start":"00:16.150 ","End":"00:18.470","Text":"If we take the inverse matrix"},{"Start":"00:18.470 ","End":"00:20.585","Text":"and then take the inverse of that,"},{"Start":"00:20.585 ","End":"00:23.840","Text":"then we\u0027re back to the original matrix."},{"Start":"00:23.840 ","End":"00:26.180","Text":"Before I comment on this,"},{"Start":"00:26.180 ","End":"00:29.255","Text":"let me just say that for a matrix A,"},{"Start":"00:29.255 ","End":"00:31.700","Text":"we can raise to a power."},{"Start":"00:31.700 ","End":"00:34.730","Text":"If I write something like A^3,"},{"Start":"00:34.730 ","End":"00:39.185","Text":"I just mean the product A times A times A."},{"Start":"00:39.185 ","End":"00:45.800","Text":"We don\u0027t have a meaning for A to a negative number,"},{"Start":"00:45.800 ","End":"00:50.270","Text":"but what we can define it as"},{"Start":"00:50.270 ","End":"00:57.200","Text":"is either the inverse of A^n or A^n,"},{"Start":"00:57.200 ","End":"00:58.640","Text":"and then the inverse of that."},{"Start":"00:58.640 ","End":"01:00.080","Text":"It turns out it doesn\u0027t matter"},{"Start":"01:00.080 ","End":"01:04.880","Text":"if you first take the inverse and then raise to a power,"},{"Start":"01:04.880 ","End":"01:07.655","Text":"and take an inverse in both cases it\u0027s the same thing,"},{"Start":"01:07.655 ","End":"01:11.815","Text":"and we call it A^minus n."},{"Start":"01:11.815 ","End":"01:14.810","Text":"This rule reminds me of the transpose"},{"Start":"01:14.810 ","End":"01:17.900","Text":"because there also we had an inversion."},{"Start":"01:17.900 ","End":"01:21.125","Text":"Now if you take the inverse of AB,"},{"Start":"01:21.125 ","End":"01:22.640","Text":"it\u0027s not the inverse of A,"},{"Start":"01:22.640 ","End":"01:25.070","Text":"inverse of B, you have to reverse the order,"},{"Start":"01:25.070 ","End":"01:27.980","Text":"the inverse of B times the inverse of A."},{"Start":"01:27.980 ","End":"01:29.240","Text":"Very important."},{"Start":"01:29.240 ","End":"01:32.965","Text":"Take note, do not get confused."},{"Start":"01:32.965 ","End":"01:38.345","Text":"If I multiply a matrix by a scalar K,"},{"Start":"01:38.345 ","End":"01:41.640","Text":"well it should not be 0,"},{"Start":"01:42.020 ","End":"01:44.430","Text":"and I\u0027m going to take the inverse,"},{"Start":"01:44.430 ","End":"01:52.014","Text":"it\u0027s the same as taking the inverse and then dividing by 1 over K times."},{"Start":"01:52.014 ","End":"01:56.750","Text":"Let\u0027s see a few more rules."},{"Start":"01:57.530 ","End":"02:01.330","Text":"Well not exactly rules relating to the inverse,"},{"Start":"02:01.330 ","End":"02:02.995","Text":"but while we\u0027re at it,"},{"Start":"02:02.995 ","End":"02:06.190","Text":"a couple of more exponent rules."},{"Start":"02:06.190 ","End":"02:10.670","Text":"Just like with numbers, A^n, A^m,"},{"Start":"02:10.670 ","End":"02:16.705","Text":"we add the exponents and also a power of a power then we multiply the exponents."},{"Start":"02:16.705 ","End":"02:19.095","Text":"That\u0027s just like with numbers."},{"Start":"02:19.095 ","End":"02:23.625","Text":"I want to end with some non rule."},{"Start":"02:23.625 ","End":"02:25.515","Text":"This is not a rule."},{"Start":"02:25.515 ","End":"02:32.160","Text":"I mentioned it because often people get confused and make such a rule."},{"Start":"02:32.530 ","End":"02:36.985","Text":"AB^n, you cannot say it\u0027s A^n, B^n."},{"Start":"02:36.985 ","End":"02:43.670","Text":"It doesn\u0027t even work with 2 because AB squared is AB,"},{"Start":"02:43.670 ","End":"02:47.570","Text":"AB, but that\u0027s not the same."},{"Start":"02:47.570 ","End":"02:49.310","Text":"They could be the same,"},{"Start":"02:49.310 ","End":"02:52.460","Text":"but not necessarily the same as AA,"},{"Start":"02:52.460 ","End":"02:56.390","Text":"BB which is A squared, B squared."},{"Start":"02:56.390 ","End":"03:01.200","Text":"Because the order makes a difference, you cannot conclude."},{"Start":"03:02.330 ","End":"03:06.340","Text":"I\u0027m going to end it at that."}],"ID":14186},{"Watched":false,"Name":"Exercise 1 Parts 1-3","Duration":"5m 56s","ChapterTopicVideoID":13534,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13534.jpeg","UploadDate":"2018-09-06T09:55:28.6870000","DurationForVideoObject":"PT5M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"This exercise is in 6 parts."},{"Start":"00:04.350 ","End":"00:06.195","Text":"In each part we\u0027ll assume that"},{"Start":"00:06.195 ","End":"00:11.490","Text":"all the matrices that are mentioned here are all invertible with"},{"Start":"00:11.490 ","End":"00:18.765","Text":"the same order N by N really maybe we could say,"},{"Start":"00:18.765 ","End":"00:22.545","Text":"and we have to extract the x means solve for x,"},{"Start":"00:22.545 ","End":"00:24.480","Text":"say what x equals."},{"Start":"00:24.480 ","End":"00:26.970","Text":"We\u0027ll start with the first 1."},{"Start":"00:26.970 ","End":"00:32.040","Text":"We have that AX times C equals D."},{"Start":"00:32.040 ","End":"00:35.190","Text":"We can do 1 of 2 things."},{"Start":"00:35.190 ","End":"00:37.290","Text":"We can get rid of the A or get rid of the C."},{"Start":"00:37.290 ","End":"00:38.420","Text":"Let\u0027s get rid of the A."},{"Start":"00:38.420 ","End":"00:40.000","Text":"How do we do that?"},{"Start":"00:40.000 ","End":"00:42.680","Text":"If we multiply A by its inverse,"},{"Start":"00:42.680 ","End":"00:44.360","Text":"we get just the identity."},{"Start":"00:44.360 ","End":"00:50.905","Text":"So what we do is multiply both sides by A inverse from the left."},{"Start":"00:50.905 ","End":"00:57.770","Text":"It\u0027s important that you have to multiply both sides of the equation on the same side."},{"Start":"00:57.770 ","End":"00:59.615","Text":"If I multiply here on the left,"},{"Start":"00:59.615 ","End":"01:03.410","Text":"I have to put here A minus 1 to the left of D."},{"Start":"01:03.410 ","End":"01:11.150","Text":"Now, this is the identity matrix, strictly speaking,"},{"Start":"01:11.150 ","End":"01:13.860","Text":"I sub n, so order n."},{"Start":"01:13.860 ","End":"01:19.670","Text":"Anyway, identity multiplied by anything is just itself."},{"Start":"01:19.670 ","End":"01:26.660","Text":"So we\u0027re left with just x times C is equal to A inverse D."},{"Start":"01:26.660 ","End":"01:28.790","Text":"Now we\u0027re going to get rid of the C"},{"Start":"01:28.790 ","End":"01:31.760","Text":"and we\u0027ll do that similar to how we got rid of a,"},{"Start":"01:31.760 ","End":"01:33.620","Text":"will multiply by C inverse,"},{"Start":"01:33.620 ","End":"01:35.080","Text":"but on the right."},{"Start":"01:35.080 ","End":"01:41.540","Text":"We\u0027ve got XC multiplied by C inverse and do the same thing to the right-hand side,"},{"Start":"01:41.540 ","End":"01:44.945","Text":"multiply it on the right by C-inverse."},{"Start":"01:44.945 ","End":"01:51.840","Text":"Once again, C times C inverse is I."},{"Start":"01:51.840 ","End":"01:54.680","Text":"Anything times I is itself."},{"Start":"01:54.680 ","End":"01:58.489","Text":"We get that x is equal to A inverse,"},{"Start":"01:58.489 ","End":"02:03.755","Text":"DC inverse and that\u0027s the answer."},{"Start":"02:03.755 ","End":"02:06.220","Text":"On to part 2."},{"Start":"02:06.220 ","End":"02:08.020","Text":"Now part 2,"},{"Start":"02:08.020 ","End":"02:11.860","Text":"which is somewhat similar to part 1,"},{"Start":"02:11.860 ","End":"02:18.920","Text":"just a little bit more involved, more letters."},{"Start":"02:19.260 ","End":"02:23.410","Text":"We want to isolate or extract X."},{"Start":"02:23.410 ","End":"02:26.950","Text":"I want to get rid of the A inverse and the C."},{"Start":"02:26.950 ","End":"02:28.870","Text":"Now in exercise 1,"},{"Start":"02:28.870 ","End":"02:31.794","Text":"we got rid of A by multiplying by A inverse."},{"Start":"02:31.794 ","End":"02:35.890","Text":"Similarly, we can get rid of an A inverse by multiplying by A."},{"Start":"02:35.890 ","End":"02:43.280","Text":"If I multiply both on the left by A I should still get an equation."},{"Start":"02:43.280 ","End":"02:48.795","Text":"I have here also A times A minus 1,"},{"Start":"02:48.795 ","End":"02:56.850","Text":"DC, AA minus 1 is the identity matrix."},{"Start":"02:56.850 ","End":"02:59.930","Text":"When you see this, you can just basically throw it out"},{"Start":"02:59.930 ","End":"03:03.505","Text":"because identity times anything is itself."},{"Start":"03:03.505 ","End":"03:09.965","Text":"Here and here we can throw them out and we\u0027ve got XC equals DC."},{"Start":"03:09.965 ","End":"03:13.010","Text":"Now we want to get rid of this C because we just want X so you"},{"Start":"03:13.010 ","End":"03:16.305","Text":"multiply on the right by C inverse."},{"Start":"03:16.305 ","End":"03:21.695","Text":"So XCC inverse is DCC inverse,"},{"Start":"03:21.695 ","End":"03:24.470","Text":"CC inverse is the identity matrix."},{"Start":"03:24.470 ","End":"03:25.985","Text":"Just throw it out."},{"Start":"03:25.985 ","End":"03:27.860","Text":"X is equal to D."},{"Start":"03:27.860 ","End":"03:35.800","Text":"That solves part 2 on to the next part."},{"Start":"03:37.010 ","End":"03:41.570","Text":"This 1 has a transpose in it,"},{"Start":"03:41.570 ","End":"03:45.830","Text":"but it\u0027s not going to be very difficult at all."},{"Start":"03:45.830 ","End":"03:52.850","Text":"You\u0027ll see this is what we have and we want to isolate X."},{"Start":"03:52.850 ","End":"03:55.340","Text":"Let\u0027s first of all go for X transpose,"},{"Start":"03:55.340 ","End":"04:01.160","Text":"we can get rid of this P inverse by multiplying on the left by its inverse,"},{"Start":"04:01.160 ","End":"04:02.330","Text":"which is just P."},{"Start":"04:02.330 ","End":"04:05.060","Text":"We\u0027ve got PP inverse,"},{"Start":"04:05.060 ","End":"04:06.890","Text":"X transpose P."},{"Start":"04:06.890 ","End":"04:13.225","Text":"Here also multiply by P. But on the left, this disappears."},{"Start":"04:13.225 ","End":"04:17.595","Text":"X transpose P is PA."},{"Start":"04:17.595 ","End":"04:19.110","Text":"Now I want to get rid of this P."},{"Start":"04:19.110 ","End":"04:22.495","Text":"I multiply on the right by P inverse."},{"Start":"04:22.495 ","End":"04:33.990","Text":"X transpose PP inverse is PAP inverse."},{"Start":"04:33.990 ","End":"04:36.199","Text":"PP inverse is the identity,"},{"Start":"04:36.199 ","End":"04:37.835","Text":"we just throw it out."},{"Start":"04:37.835 ","End":"04:44.885","Text":"We\u0027ve got that X transpose is PAP inverse."},{"Start":"04:44.885 ","End":"04:49.465","Text":"But we don\u0027t want X transpose, we want X."},{"Start":"04:49.465 ","End":"04:54.320","Text":"The thing to do here is to take the transpose of both sides if 2 things are equal,"},{"Start":"04:54.320 ","End":"04:57.500","Text":"their transposes are equal continue over here."},{"Start":"04:57.500 ","End":"05:01.750","Text":"I\u0027ve got x transpose, transpose."},{"Start":"05:01.750 ","End":"05:06.445","Text":"You know why, that\u0027s good for me because X transpose will be just X"},{"Start":"05:06.445 ","End":"05:11.420","Text":"has got to equal PAP inverse transpose."},{"Start":"05:11.420 ","End":"05:14.195","Text":"Like I said, this leaves me with just X."},{"Start":"05:14.195 ","End":"05:22.865","Text":"Here I have to use the rule that when you take a transpose of a product of matrices,"},{"Start":"05:22.865 ","End":"05:25.430","Text":"we can reverse the order."},{"Start":"05:25.430 ","End":"05:28.820","Text":"We take the transpose of each 1 but in reverse order."},{"Start":"05:28.820 ","End":"05:33.905","Text":"So we get P inverse transpose and then A transpose,"},{"Start":"05:33.905 ","End":"05:37.050","Text":"and then P transpose."},{"Start":"05:37.460 ","End":"05:41.595","Text":"I guess that\u0027s the answer."},{"Start":"05:41.595 ","End":"05:45.650","Text":"If you prefer, you could write it as P transpose inverse."},{"Start":"05:45.650 ","End":"05:46.910","Text":"It makes no difference."},{"Start":"05:46.910 ","End":"05:56.880","Text":"We can leave it like this I think we\u0027ll continue with Part 4 in the next clip."}],"ID":14187},{"Watched":false,"Name":"Exercise 1 Parts 4-5","Duration":"5m 56s","ChapterTopicVideoID":13535,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13535.jpeg","UploadDate":"2018-09-06T09:56:23.0500000","DurationForVideoObject":"PT5M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"We\u0027re continuing from the previous clip"},{"Start":"00:03.690 ","End":"00:07.560","Text":"and we\u0027re now up to Part 4,"},{"Start":"00:07.560 ","End":"00:12.300","Text":"which is this, and we want to extract X."},{"Start":"00:12.300 ","End":"00:15.570","Text":"Looks like we want to get rid of this C inverse"},{"Start":"00:15.570 ","End":"00:21.750","Text":"by multiplying both sides on the left by C."},{"Start":"00:21.750 ","End":"00:26.955","Text":"This is what we get here and here also, C times I."},{"Start":"00:26.955 ","End":"00:31.445","Text":"Now, this times this is just identity matrix and we can throw it out."},{"Start":"00:31.445 ","End":"00:37.440","Text":"What we\u0027re left with is A plus X D to the minus 2."},{"Start":"00:37.440 ","End":"00:39.420","Text":"C times I is just C."},{"Start":"00:39.420 ","End":"00:44.810","Text":"Now we multiply both of them on the right by,"},{"Start":"00:44.810 ","End":"00:48.960","Text":"you probably can guess, by D squared,"},{"Start":"00:48.960 ","End":"00:53.100","Text":"and this is equal to CD squared."},{"Start":"00:53.100 ","End":"00:57.515","Text":"Actually D to the minus 2 is defined to be the inverse of D squared,"},{"Start":"00:57.515 ","End":"00:59.495","Text":"it\u0027s not like numbers, but anyway,"},{"Start":"00:59.495 ","End":"01:01.970","Text":"it\u0027s defined to be the inverse of this."},{"Start":"01:01.970 ","End":"01:04.130","Text":"When we multiply these 2,"},{"Start":"01:04.130 ","End":"01:09.640","Text":"we get the identity."},{"Start":"01:09.640 ","End":"01:11.310","Text":"I\u0027ll continue over here,"},{"Start":"01:11.310 ","End":"01:14.670","Text":"so we\u0027ve got A plus X."},{"Start":"01:14.670 ","End":"01:17.580","Text":"No, I thought I had forgotten the D squared on the right."},{"Start":"01:17.580 ","End":"01:19.365","Text":"No everything\u0027s okay,"},{"Start":"01:19.365 ","End":"01:24.930","Text":"A plus X equals CD squared."},{"Start":"01:24.930 ","End":"01:29.090","Text":"All I have to do now is subtract A from both sides."},{"Start":"01:29.090 ","End":"01:30.560","Text":"With addition and subtraction,"},{"Start":"01:30.560 ","End":"01:33.545","Text":"there\u0027s no on the left or on the right makes no difference,"},{"Start":"01:33.545 ","End":"01:38.250","Text":"so I could just as well put it subtracted on the right."},{"Start":"01:38.250 ","End":"01:42.520","Text":"CD squared minus A,"},{"Start":"01:42.590 ","End":"01:46.510","Text":"that\u0027s the answer to Part 4."},{"Start":"01:46.610 ","End":"01:48.860","Text":"Onto the next part,"},{"Start":"01:48.860 ","End":"01:50.390","Text":"which is Part 5,"},{"Start":"01:50.390 ","End":"01:53.135","Text":"which I\u0027ll just copy."},{"Start":"01:53.135 ","End":"02:00.125","Text":"This one is a bit trickier because we have X on both sides."},{"Start":"02:00.125 ","End":"02:02.305","Text":"What we\u0027re going to do here,"},{"Start":"02:02.305 ","End":"02:04.430","Text":"I\u0027m going to take the inverse of both sides."},{"Start":"02:04.430 ","End":"02:07.235","Text":"If 2 things are equal, their inverses are equal."},{"Start":"02:07.235 ","End":"02:11.320","Text":"I don\u0027t want this as it is with the inverse sign,"},{"Start":"02:11.320 ","End":"02:13.820","Text":"because if I get rid of this,"},{"Start":"02:13.820 ","End":"02:16.280","Text":"then I can do additions and subtractions."},{"Start":"02:16.280 ","End":"02:17.660","Text":"I can break it up as it is."},{"Start":"02:17.660 ","End":"02:19.145","Text":"I can\u0027t do anything with it."},{"Start":"02:19.145 ","End":"02:21.560","Text":"But here I don\u0027t have any pluses or minuses,"},{"Start":"02:21.560 ","End":"02:24.835","Text":"so I prefer the inverse to be on the right."},{"Start":"02:24.835 ","End":"02:31.385","Text":"Basically what I\u0027m saying is I\u0027ll take the inverse of this and the inverse of this,"},{"Start":"02:31.385 ","End":"02:33.380","Text":"and that should also be equal."},{"Start":"02:33.380 ","End":"02:38.010","Text":"On the left, we\u0027re just left with A minus AX,"},{"Start":"02:38.010 ","End":"02:42.545","Text":"the inverse of the inverse is it itself and the inverse here,"},{"Start":"02:42.545 ","End":"02:44.660","Text":"when we take the inverse of a product,"},{"Start":"02:44.660 ","End":"02:47.165","Text":"we have to take it in reverse order."},{"Start":"02:47.165 ","End":"02:54.000","Text":"It\u0027s C inverse and then X inverse, inverse."},{"Start":"02:54.770 ","End":"02:58.770","Text":"Now on the left I want to take A out of the brackets."},{"Start":"02:58.770 ","End":"03:03.345","Text":"You just got to think of A as like A times I."},{"Start":"03:03.345 ","End":"03:05.960","Text":"If we take a out the brackets,"},{"Start":"03:05.960 ","End":"03:10.640","Text":"we have A times I minus X."},{"Start":"03:10.640 ","End":"03:19.240","Text":"X inverse, inverse is X, we have C inverse X."},{"Start":"03:20.660 ","End":"03:26.765","Text":"Here, what I\u0027d like to do is get rid of this inverse,"},{"Start":"03:26.765 ","End":"03:31.280","Text":"because it doesn\u0027t suit me to have an inverse when I have a plus or a minus."},{"Start":"03:31.280 ","End":"03:35.210","Text":"What I\u0027ll do is just take the inverse of both sides"},{"Start":"03:35.210 ","End":"03:39.160","Text":"because the inverse of the inverse will be the thing itself."},{"Start":"03:39.160 ","End":"03:44.000","Text":"Here we\u0027re left with just A minus AX and on the right,"},{"Start":"03:44.000 ","End":"03:46.114","Text":"when we take an inverse of a product,"},{"Start":"03:46.114 ","End":"03:47.780","Text":"we take the inverse of each one,"},{"Start":"03:47.780 ","End":"03:49.895","Text":"bur we have to reverse the order."},{"Start":"03:49.895 ","End":"03:55.380","Text":"It\u0027s C inverse and then well,"},{"Start":"03:55.380 ","End":"04:01.635","Text":"just X because really X inverse inverse is just X."},{"Start":"04:01.635 ","End":"04:04.550","Text":"Now, I want to try and extract X,"},{"Start":"04:04.550 ","End":"04:07.295","Text":"but I have it on left and the right-hand side."},{"Start":"04:07.295 ","End":"04:11.400","Text":"What I\u0027ll do is, I\u0027ll rewrite this."},{"Start":"04:11.400 ","End":"04:15.660","Text":"I\u0027ll put the AX on the right-hand side,"},{"Start":"04:15.660 ","End":"04:18.390","Text":"so I\u0027ve got A is equal to,"},{"Start":"04:18.390 ","End":"04:19.590","Text":"I can add it,"},{"Start":"04:19.590 ","End":"04:21.530","Text":"on the left or on the right, doesn\u0027t matter."},{"Start":"04:21.530 ","End":"04:26.760","Text":"I\u0027ll write it as AX plus C inverse X."},{"Start":"04:26.760 ","End":"04:29.640","Text":"Now we can take X out the brackets"},{"Start":"04:29.640 ","End":"04:39.170","Text":"and say that A is equal to A plus C inverse times X."},{"Start":"04:39.170 ","End":"04:41.930","Text":"Notice that the X has to stay on the right."},{"Start":"04:41.930 ","End":"04:44.240","Text":"In each of these, it has to be on the same side."},{"Start":"04:44.240 ","End":"04:47.215","Text":"If it\u0027s on the right, then here it\u0027s on the right."},{"Start":"04:47.215 ","End":"04:51.560","Text":"Now, what I have to do is multiply both sides."},{"Start":"04:51.560 ","End":"04:55.085","Text":"Let me just write it in the other order,"},{"Start":"04:55.085 ","End":"04:58.770","Text":"A plus C inverse X equals A."},{"Start":"04:58.770 ","End":"05:00.470","Text":"I prefer to have the X on the left."},{"Start":"05:00.470 ","End":"05:05.015","Text":"Now if I multiply both sides by the inverse of this,"},{"Start":"05:05.015 ","End":"05:08.160","Text":"I\u0027ll continue over here."},{"Start":"05:08.660 ","End":"05:13.670","Text":"I\u0027ll get A plus C inverse,"},{"Start":"05:13.670 ","End":"05:20.585","Text":"inverse times A plus C inverse times X."},{"Start":"05:20.585 ","End":"05:25.700","Text":"The same thing here, A plus C inverse,"},{"Start":"05:25.700 ","End":"05:30.860","Text":"inverse after multiply that by the A."},{"Start":"05:30.860 ","End":"05:39.585","Text":"Of course, these 2 cancel each other out because their I times X is X."},{"Start":"05:39.585 ","End":"05:45.530","Text":"We get the answer that X equals A plus C inverse,"},{"Start":"05:45.530 ","End":"05:50.510","Text":"inverse A, and there\u0027s not really any way to simplify this."},{"Start":"05:50.510 ","End":"05:55.260","Text":"This is our answer and we\u0027re done."}],"ID":14188},{"Watched":false,"Name":"Exercise 1 Part 6","Duration":"5m 11s","ChapterTopicVideoID":13536,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13536.jpeg","UploadDate":"2018-09-06T09:57:11.9900000","DurationForVideoObject":"PT5M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:03.750","Text":"Now the 6 and last part."},{"Start":"00:03.750 ","End":"00:07.290","Text":"This looks like a lot of letters and there are."},{"Start":"00:07.290 ","End":"00:11.350","Text":"We\u0027re trying to isolate X, which is here."},{"Start":"00:12.320 ","End":"00:15.930","Text":"There is a slow way and a slightly quicker way."},{"Start":"00:15.930 ","End":"00:21.690","Text":"The slow way is to peel like layers of an onion."},{"Start":"00:21.690 ","End":"00:30.750","Text":"I could multiply both on the left by A inverse here and here."},{"Start":"00:30.750 ","End":"00:32.970","Text":"I might even do 2 steps in 1"},{"Start":"00:32.970 ","End":"00:37.125","Text":"and also multiply on the right by C inverse."},{"Start":"00:37.125 ","End":"00:39.420","Text":"That will get rid of these 2 and these 2,"},{"Start":"00:39.420 ","End":"00:42.450","Text":"then I peel off another layer and so on."},{"Start":"00:42.450 ","End":"00:47.390","Text":"It\u0027s either 3 steps or 6 steps depending on"},{"Start":"00:47.390 ","End":"00:48.965","Text":"whether you do them in pairs."},{"Start":"00:48.965 ","End":"00:54.215","Text":"But we can actually do something a bit quicker."},{"Start":"00:54.215 ","End":"00:56.780","Text":"Because I could consider,"},{"Start":"00:56.780 ","End":"00:59.795","Text":"this is 1 thing in brackets,"},{"Start":"00:59.795 ","End":"01:01.900","Text":"and this also,"},{"Start":"01:01.900 ","End":"01:06.245","Text":"and then if I multiply on the left"},{"Start":"01:06.245 ","End":"01:10.415","Text":"by the inverse of this and on the right by the inverse of this,"},{"Start":"01:10.415 ","End":"01:15.470","Text":"then I can do 3 steps in 1 or might even be considered 6 steps,"},{"Start":"01:15.470 ","End":"01:18.965","Text":"so I\u0027m working on the left and the right simultaneously."},{"Start":"01:18.965 ","End":"01:25.860","Text":"We get that ABC transpose inverse,"},{"Start":"01:30.170 ","End":"01:35.760","Text":"then the ABC transpose,"},{"Start":"01:35.760 ","End":"01:37.830","Text":"now the X inverse,"},{"Start":"01:37.830 ","End":"01:43.025","Text":"now this BA transpose C,"},{"Start":"01:43.025 ","End":"01:50.665","Text":"and now the inverse of this BA transpose C inverse."},{"Start":"01:50.665 ","End":"01:55.775","Text":"On the right, I have to multiply by this and by this."},{"Start":"01:55.775 ","End":"02:04.400","Text":"I\u0027ve got ABC transpose inverse, then AB transpose."},{"Start":"02:04.400 ","End":"02:12.545","Text":"Then this, which is BA transpose C inverse."},{"Start":"02:12.545 ","End":"02:18.690","Text":"Now this with its inverse is going to give me the identity."},{"Start":"02:19.390 ","End":"02:26.030","Text":"Now this with its inverse is also going to give me the identity matrix."},{"Start":"02:26.030 ","End":"02:33.370","Text":"All I\u0027m left with on the left is X inverse."},{"Start":"02:33.370 ","End":"02:36.124","Text":"Let\u0027s see what I have on the right."},{"Start":"02:36.124 ","End":"02:40.685","Text":"Now, remember that when you take the inverse of a product,"},{"Start":"02:40.685 ","End":"02:44.440","Text":"take the inverse of each factor,"},{"Start":"02:44.440 ","End":"02:47.000","Text":"but you also have to reverse the order."},{"Start":"02:47.000 ","End":"02:50.735","Text":"I\u0027ll just write this to remind us that we\u0027re reversing the order."},{"Start":"02:50.735 ","End":"02:56.315","Text":"I, first of all, invert the C transpose here,"},{"Start":"02:56.315 ","End":"03:00.570","Text":"then the B, and then the A."},{"Start":"03:00.710 ","End":"03:08.430","Text":"Then I have what I had before, AB transpose."},{"Start":"03:08.430 ","End":"03:14.940","Text":"Now I have to invert each of these starting from the C."},{"Start":"03:14.940 ","End":"03:17.145","Text":"It\u0027s C inverse,"},{"Start":"03:17.145 ","End":"03:24.045","Text":"A transpose inverse, B inverse."},{"Start":"03:24.045 ","End":"03:26.100","Text":"Now to get X,"},{"Start":"03:26.100 ","End":"03:31.675","Text":"I just take the inverse of both sides."},{"Start":"03:31.675 ","End":"03:35.805","Text":"That gives me X inverse, inverse is X."},{"Start":"03:35.805 ","End":"03:37.340","Text":"Here, once again,"},{"Start":"03:37.340 ","End":"03:42.914","Text":"we\u0027re going to have to read it from right to left."},{"Start":"03:42.914 ","End":"03:44.700","Text":"I\u0027m starting with the B."},{"Start":"03:44.700 ","End":"03:46.310","Text":"We take the inverse of each piece."},{"Start":"03:46.310 ","End":"03:50.420","Text":"B inverse inverse is B."},{"Start":"03:50.420 ","End":"03:55.255","Text":"This inverse, the inverse just drops out."},{"Start":"03:55.255 ","End":"04:00.005","Text":"Here, I have C transpose inverse."},{"Start":"04:00.005 ","End":"04:05.730","Text":"Then B transpose inverse."},{"Start":"04:06.160 ","End":"04:09.340","Text":"With something canceled out,"},{"Start":"04:09.340 ","End":"04:14.880","Text":"I just noticed that the A minus 1 with the A is identity."},{"Start":"04:14.880 ","End":"04:18.720","Text":"That disappears, so I don\u0027t need anything for that."},{"Start":"04:18.720 ","End":"04:22.555","Text":"Then B inverse inverse is B."},{"Start":"04:22.555 ","End":"04:29.250","Text":"Then the inverse here just throughout the minus 1, C transpose."},{"Start":"04:29.350 ","End":"04:36.590","Text":"I just noticed I made a small typo when I took here the inverse,"},{"Start":"04:36.590 ","End":"04:40.325","Text":"should be C inverse, not C transpose."},{"Start":"04:40.325 ","End":"04:44.700","Text":"This is C inverse."},{"Start":"04:48.130 ","End":"04:54.615","Text":"This is not a t, it\u0027s an inverse."},{"Start":"04:54.615 ","End":"04:59.550","Text":"C inverse inverse is just C."},{"Start":"04:59.550 ","End":"05:00.990","Text":"I\u0027ve got a bit of a gap here,"},{"Start":"05:00.990 ","End":"05:03.160","Text":"I\u0027ll put a dot there."},{"Start":"05:05.210 ","End":"05:08.505","Text":"This is our answer for X."},{"Start":"05:08.505 ","End":"05:10.840","Text":"We\u0027re done."}],"ID":14189},{"Watched":false,"Name":"Exercise 2","Duration":"8m 8s","ChapterTopicVideoID":13537,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13537.jpeg","UploadDate":"2018-09-06T09:58:21.0430000","DurationForVideoObject":"PT8M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this exercise, we\u0027re given a matrix B as follows,"},{"Start":"00:05.070 ","End":"00:13.845","Text":"and we\u0027re also given the following equation holds as written."},{"Start":"00:13.845 ","End":"00:17.910","Text":"Notice that there are various matrices here."},{"Start":"00:17.910 ","End":"00:21.000","Text":"There\u0027s B, there\u0027s X, there\u0027s I."},{"Start":"00:21.000 ","End":"00:26.980","Text":"We assume that they\u0027re all 2 by 2 and our task is to find X."},{"Start":"00:27.350 ","End":"00:30.465","Text":"Everything else is given, I is given,"},{"Start":"00:30.465 ","End":"00:32.910","Text":"we know what it is and B is this."},{"Start":"00:32.910 ","End":"00:35.190","Text":"Let\u0027s see what we can do."},{"Start":"00:35.190 ","End":"00:38.865","Text":"We want to isolate, extract X,"},{"Start":"00:38.865 ","End":"00:43.130","Text":"so if I multiply on the left and then multiply on the right by inverses."},{"Start":"00:43.130 ","End":"00:45.160","Text":"I should be able to get it."},{"Start":"00:45.160 ","End":"00:48.165","Text":"Anyway, let me just start by copying it."},{"Start":"00:48.165 ","End":"00:55.365","Text":"Now we\u0027re at the point where we can do 2 things in once I can multiply by from the left,"},{"Start":"00:55.365 ","End":"00:57.345","Text":"and multiply from the right."},{"Start":"00:57.345 ","End":"01:06.780","Text":"From the left, I\u0027m going to take the inverse of B squared and then B squared and then X,"},{"Start":"01:06.780 ","End":"01:12.330","Text":"then 2B inverse and the inverse of"},{"Start":"01:12.330 ","End":"01:18.410","Text":"this is 2B because inverse,"},{"Start":"01:18.410 ","End":"01:20.735","Text":"inverse is the thing itself."},{"Start":"01:20.735 ","End":"01:26.930","Text":"Now, after you do the same thing to the right-hand side also this bit I put on the left,"},{"Start":"01:26.930 ","End":"01:29.090","Text":"this bit I put on the right,"},{"Start":"01:29.090 ","End":"01:30.545","Text":"you have to do that here too."},{"Start":"01:30.545 ","End":"01:37.010","Text":"I start off with B squared inverse and then"},{"Start":"01:37.010 ","End":"01:45.480","Text":"B plus I, and then 2B."},{"Start":"01:46.420 ","End":"01:49.190","Text":"Now on the left-hand side,"},{"Start":"01:49.190 ","End":"01:56.750","Text":"these 2 combine their inverses to give me the identity so I don\u0027t have to write anything."},{"Start":"01:56.750 ","End":"01:58.340","Text":"Can these 2 combine?"},{"Start":"01:58.340 ","End":"02:05.450","Text":"That\u0027s why I did this multiplication and we\u0027re left with just X on the right."},{"Start":"02:05.450 ","End":"02:10.385","Text":"What I can do is I can certainly take the 2 out front,"},{"Start":"02:10.385 ","End":"02:18.840","Text":"so we can write this as 2 I\u0027ll write this as B to the minus 2."},{"Start":"02:18.840 ","End":"02:23.280","Text":"Then I have B plus I times B."},{"Start":"02:23.280 ","End":"02:25.620","Text":"I\u0027m going to take the B"},{"Start":"02:25.620 ","End":"02:30.185","Text":"and multiply by the B plus I."},{"Start":"02:30.185 ","End":"02:33.785","Text":"If I do that, I get B squared."},{"Start":"02:33.785 ","End":"02:40.200","Text":"I get B times B plus I times B. I times B is just B."},{"Start":"02:40.760 ","End":"02:49.390","Text":"Now I can multiply on the left by B to the minus 2 by using the distributive law."},{"Start":"02:49.390 ","End":"02:55.335","Text":"I\u0027ve got B to the minus 2B squared is just I."},{"Start":"02:55.335 ","End":"03:02.715","Text":"This with this and this with this B to the minus 2B is just B to the minus 1,"},{"Start":"03:02.715 ","End":"03:05.350","Text":"minus 2 plus 1."},{"Start":"03:07.340 ","End":"03:09.675","Text":"That\u0027s what X is."},{"Start":"03:09.675 ","End":"03:14.995","Text":"Now, what we\u0027re missing is B minus 1."},{"Start":"03:14.995 ","End":"03:17.375","Text":"This is what I need."},{"Start":"03:17.375 ","End":"03:23.360","Text":"What I\u0027m given is B. I have to get its inverse,"},{"Start":"03:23.360 ","End":"03:28.155","Text":"so now begins the story of finding the inverse."},{"Start":"03:28.155 ","End":"03:31.135","Text":"Let\u0027s do that."},{"Start":"03:31.135 ","End":"03:32.735","Text":"Remember how we do it."},{"Start":"03:32.735 ","End":"03:40.950","Text":"We start with a matrix which is here I put 1, 2,"},{"Start":"03:40.950 ","End":"03:43.680","Text":"4, 9 as here,"},{"Start":"03:43.680 ","End":"03:52.530","Text":"a separator and then the identity matrix of size 2 is 1, 0, 0, 1."},{"Start":"03:52.750 ","End":"04:02.140","Text":"Now we\u0027ll do a series of row operations until I get the identity in the left part,"},{"Start":"04:02.140 ","End":"04:04.160","Text":"see I just want to keep,"},{"Start":"04:04.160 ","End":"04:09.900","Text":"make sure I can still see this when I get B minus 1."},{"Start":"04:10.120 ","End":"04:14.270","Text":"I\u0027m not going to write every thing in row notation."},{"Start":"04:14.270 ","End":"04:15.680","Text":"I\u0027ll just write this symbol,"},{"Start":"04:15.680 ","End":"04:19.220","Text":"meaning after several row, after row operations,"},{"Start":"04:19.220 ","End":"04:26.700","Text":"what I can get is I\u0027m going to subtract 4 times this row from this row,"},{"Start":"04:26.700 ","End":"04:28.980","Text":"I\u0027m not writing it down in row notation,"},{"Start":"04:28.980 ","End":"04:33.780","Text":"so 4 minus 4 1s is 0, 9 minus 4,"},{"Start":"04:33.780 ","End":"04:39.720","Text":"2s is 1 and so I have this also,"},{"Start":"04:39.720 ","End":"04:43.890","Text":"0 minus 4 1s is minus 4,"},{"Start":"04:43.890 ","End":"04:46.570","Text":"1 minus 4 0s is still 1."},{"Start":"04:46.570 ","End":"04:49.025","Text":"Top row is unchanged."},{"Start":"04:49.025 ","End":"04:57.365","Text":"Now, what I want to do is subtract twice the second row from the first row,"},{"Start":"04:57.365 ","End":"05:02.135","Text":"so I\u0027ll get, let\u0027s see."},{"Start":"05:02.135 ","End":"05:05.425","Text":"Second row will be unchanged."},{"Start":"05:05.425 ","End":"05:11.010","Text":"Now subtract twice 0 from 1 is still 1,"},{"Start":"05:11.010 ","End":"05:14.835","Text":"twice 1 from 2 is 0."},{"Start":"05:14.835 ","End":"05:19.530","Text":"Subtract minus 8 from 1,"},{"Start":"05:19.530 ","End":"05:26.500","Text":"and that gives me 9 and subtract 2 from 0 is minus 2."},{"Start":"05:28.940 ","End":"05:33.830","Text":"If this in the beginning was A and I,"},{"Start":"05:33.830 ","End":"05:35.825","Text":"at the very end, we get,"},{"Start":"05:35.825 ","End":"05:39.625","Text":"I here, then it\u0027s A minus 1 here."},{"Start":"05:39.625 ","End":"05:41.700","Text":"Now, I don\u0027t always do this,"},{"Start":"05:41.700 ","End":"05:46.025","Text":"but I think it\u0027s a good idea to check when you have an inverse,"},{"Start":"05:46.025 ","End":"05:49.280","Text":"and let\u0027s just check that say A times A minus 1."},{"Start":"05:49.280 ","End":"05:52.460","Text":"If I take 1, 2, 4, 9,"},{"Start":"05:52.460 ","End":"05:56.855","Text":"and I multiply it by 9 minus 2,"},{"Start":"05:56.855 ","End":"05:58.640","Text":"minus 4, 1."},{"Start":"05:58.640 ","End":"06:00.110","Text":"Let\u0027s see what we get."},{"Start":"06:00.110 ","End":"06:03.445","Text":"I\u0027m hoping to get the identity matrix."},{"Start":"06:03.445 ","End":"06:10.670","Text":"The first element is this times this 1 times 9 is 9,"},{"Start":"06:10.670 ","End":"06:12.485","Text":"2 times minus 4 is minus 8,"},{"Start":"06:12.485 ","End":"06:16.645","Text":"9 minus 8 is 1."},{"Start":"06:16.645 ","End":"06:20.230","Text":"Next was mentally this with this,"},{"Start":"06:20.230 ","End":"06:23.164","Text":"minus 2 plus 2 is 0."},{"Start":"06:23.164 ","End":"06:29.840","Text":"Now this with this 36 minus 36 is 0 and then 4,"},{"Start":"06:29.840 ","End":"06:31.340","Text":"9 with minus 2,"},{"Start":"06:31.340 ","End":"06:33.560","Text":"1 is minus 8 plus 9,"},{"Start":"06:33.560 ","End":"06:35.015","Text":"it is 1,"},{"Start":"06:35.015 ","End":"06:38.315","Text":"so that\u0027s the inverse matrix."},{"Start":"06:38.315 ","End":"06:43.655","Text":"The only thing is I called it A and I should have called it B the same calculations,"},{"Start":"06:43.655 ","End":"06:47.195","Text":"so erase this, erase this,"},{"Start":"06:47.195 ","End":"06:51.785","Text":"B and B, no harm done."},{"Start":"06:51.785 ","End":"06:58.085","Text":"Now that we have B minus 1 B inverse, which is this."},{"Start":"06:58.085 ","End":"07:01.675","Text":"Now we can do the computation of X."},{"Start":"07:01.675 ","End":"07:11.300","Text":"Continuing over here, we get that X is equal to twice."},{"Start":"07:11.300 ","End":"07:15.680","Text":"Now I is 1, 0, 0, 1"},{"Start":"07:15.680 ","End":"07:21.035","Text":"and B inverse is what we have here,"},{"Start":"07:21.035 ","End":"07:25.115","Text":"is 9 minus 2 minus 4,"},{"Start":"07:25.115 ","End":"07:31.475","Text":"1 and this is equal to twice."},{"Start":"07:31.475 ","End":"07:33.620","Text":"Now if I add them,"},{"Start":"07:33.620 ","End":"07:36.920","Text":"just have to add 1s to the diagonal here."},{"Start":"07:36.920 ","End":"07:39.440","Text":"This minus 2 and this minus 4 stay,"},{"Start":"07:39.440 ","End":"07:41.120","Text":"this 9 becomes a 10,"},{"Start":"07:41.120 ","End":"07:46.230","Text":"the 1 becomes a 2 and then finally,"},{"Start":"07:46.540 ","End":"07:48.770","Text":"if I do this,"},{"Start":"07:48.770 ","End":"07:54.780","Text":"the answer comes out as 20."},{"Start":"07:54.780 ","End":"07:57.120","Text":"Double this is minus 4,"},{"Start":"07:57.120 ","End":"07:59.430","Text":"double this minus 8,"},{"Start":"07:59.430 ","End":"08:01.890","Text":"double this is 4"},{"Start":"08:01.890 ","End":"08:09.760","Text":"and that\u0027s our answer for what X is and we\u0027re done."}],"ID":14190},{"Watched":false,"Name":"Exercise 3","Duration":"11m 15s","ChapterTopicVideoID":13538,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13538.jpeg","UploadDate":"2018-09-06T10:00:03.9570000","DurationForVideoObject":"PT11M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.050","Text":"In this exercise, we\u0027re given that B inverse is this,"},{"Start":"00:07.050 ","End":"00:08.760","Text":"it\u0027s not B, it\u0027s the inverse,"},{"Start":"00:08.760 ","End":"00:10.815","Text":"so there must be a B somewhere."},{"Start":"00:10.815 ","End":"00:14.490","Text":"We\u0027re also given that this equation holds B,"},{"Start":"00:14.490 ","End":"00:18.240","Text":"why B transpose is B inverse plus B,"},{"Start":"00:18.240 ","End":"00:23.650","Text":"we have to find Y means extract or solve for Y from this equation."},{"Start":"00:24.650 ","End":"00:29.820","Text":"That\u0027s why we say all the matrices are square 3 by 3,"},{"Start":"00:29.820 ","End":"00:31.485","Text":"when I say all well,"},{"Start":"00:31.485 ","End":"00:32.940","Text":"B inverse already is,"},{"Start":"00:32.940 ","End":"00:35.110","Text":"so B must be."},{"Start":"00:35.270 ","End":"00:39.930","Text":"I guess it just means that Y is also a 3 by 3."},{"Start":"00:39.930 ","End":"00:44.055","Text":"Anyway, maybe it doesn\u0027t hurt to say."},{"Start":"00:44.055 ","End":"00:48.165","Text":"Let\u0027s start with this part,"},{"Start":"00:48.165 ","End":"00:51.095","Text":"and I just copied it over here."},{"Start":"00:51.095 ","End":"00:53.570","Text":"Now I want to extract Y."},{"Start":"00:53.570 ","End":"00:56.900","Text":"We\u0027re going to multiply on the left by the inverse of"},{"Start":"00:56.900 ","End":"01:00.710","Text":"B on the right by the inverse of B transpose."},{"Start":"01:00.710 ","End":"01:04.425","Text":"B inverse B,"},{"Start":"01:04.425 ","End":"01:06.435","Text":"then the Y,"},{"Start":"01:06.435 ","End":"01:08.565","Text":"then the B transpose,"},{"Start":"01:08.565 ","End":"01:16.525","Text":"and then B transpose inverse has got a equal."},{"Start":"01:16.525 ","End":"01:25.640","Text":"Well, I added this on the left and this on the right of the expression,"},{"Start":"01:25.640 ","End":"01:27.940","Text":"so on the other side of the equation,"},{"Start":"01:27.940 ","End":"01:30.210","Text":"I better do the same thing."},{"Start":"01:30.210 ","End":"01:39.240","Text":"I put a B minus 1 here and a B transpose to the minus 1 here."},{"Start":"01:39.590 ","End":"01:43.940","Text":"Just for consistency, I\u0027ll highlight it again."},{"Start":"01:43.940 ","End":"01:46.265","Text":"In the middle I put this."},{"Start":"01:46.265 ","End":"01:49.070","Text":"But I have to protect it with brackets,"},{"Start":"01:49.070 ","End":"01:53.395","Text":"and B minus 1 plus B."},{"Start":"01:53.395 ","End":"01:56.840","Text":"Now, we know what we\u0027re going to get on the left-hand side,"},{"Start":"01:56.840 ","End":"01:58.295","Text":"we\u0027re going to get just Y."},{"Start":"01:58.295 ","End":"02:01.100","Text":"That\u0027s how we chose all these things."},{"Start":"02:01.100 ","End":"02:03.290","Text":"This cancels with this and this with this,"},{"Start":"02:03.290 ","End":"02:06.470","Text":"at least the product is the identity and so on, so on."},{"Start":"02:06.470 ","End":"02:09.530","Text":"We just have to compute the left-hand side."},{"Start":"02:09.530 ","End":"02:16.325","Text":"First, let me multiply this with this using the distributive law,"},{"Start":"02:16.325 ","End":"02:24.110","Text":"B minus 1 B minus 1 is B to the minus 2 or I could just leave it as B minus 1 squared."},{"Start":"02:24.110 ","End":"02:25.609","Text":"That would be okay."},{"Start":"02:25.609 ","End":"02:28.700","Text":"Reason I\u0027m doing that is that I have B minus 1,"},{"Start":"02:28.700 ","End":"02:30.575","Text":"so that suits me."},{"Start":"02:30.575 ","End":"02:36.685","Text":"B minus 1 times B is just the identity matrix."},{"Start":"02:36.685 ","End":"02:38.740","Text":"All this now,"},{"Start":"02:38.740 ","End":"02:41.930","Text":"I still have to multiply by this."},{"Start":"02:41.930 ","End":"02:48.305","Text":"Now, you can change the places of the transpose on the inverse is one of the rules,"},{"Start":"02:48.305 ","End":"02:53.315","Text":"so we have B inverse transpose."},{"Start":"02:53.315 ","End":"02:58.450","Text":"The reason I did that again is because I have B inverse over here."},{"Start":"02:58.450 ","End":"03:06.060","Text":"Something to notice is that these computations enable me to work just with B minus 1,"},{"Start":"03:06.060 ","End":"03:14.490","Text":"I don\u0027t actually have to find B. I have B inverse B minus 1."},{"Start":"03:14.490 ","End":"03:17.330","Text":"I don\u0027t need to do any matrix inversions,"},{"Start":"03:17.330 ","End":"03:19.715","Text":"just do some computations."},{"Start":"03:19.715 ","End":"03:21.905","Text":"I\u0027d like to do one of them with the side,"},{"Start":"03:21.905 ","End":"03:25.805","Text":"this B inverse squared I\u0027d like to do over here,"},{"Start":"03:25.805 ","End":"03:32.580","Text":"it\u0027s just B inverse is 1, 0,"},{"Start":"03:32.580 ","End":"03:35.805","Text":"2, 4, minus 1,"},{"Start":"03:35.805 ","End":"03:40.650","Text":"8, 2, 1, 3."},{"Start":"03:40.650 ","End":"03:43.855","Text":"I copy pasted another one,"},{"Start":"03:43.855 ","End":"03:45.350","Text":"it made a bit of room."},{"Start":"03:45.350 ","End":"03:52.440","Text":"Let\u0027s see, we have to fit in another 3 by 3 matrix."},{"Start":"03:52.440 ","End":"03:57.015","Text":"Let\u0027s say, let\u0027s go for the top row."},{"Start":"03:57.015 ","End":"04:03.950","Text":"Now this one with this one will give"},{"Start":"04:03.950 ","End":"04:11.285","Text":"me 1 plus 0 plus 4,"},{"Start":"04:11.285 ","End":"04:17.740","Text":"that will be 5."},{"Start":"04:17.740 ","End":"04:22.335","Text":"Next, I\u0027ll do this row with this column,"},{"Start":"04:22.335 ","End":"04:26.370","Text":"and let\u0027s see we get, 1 times 0,"},{"Start":"04:26.370 ","End":"04:31.720","Text":"0 minus 1, 2 times 1 is 2."},{"Start":"04:31.810 ","End":"04:35.005","Text":"Then with the third column,"},{"Start":"04:35.005 ","End":"04:38.545","Text":"we get 1 times 2 plus nothing,"},{"Start":"04:38.545 ","End":"04:40.930","Text":"plus 2 times 3,"},{"Start":"04:40.930 ","End":"04:46.075","Text":"which is 2 plus 6, which is 8."},{"Start":"04:46.075 ","End":"04:49.885","Text":"The next row with the first column,"},{"Start":"04:49.885 ","End":"04:55.655","Text":"we\u0027ll get 4 minus 4 plus 16."},{"Start":"04:55.655 ","End":"04:58.995","Text":"That would be 16."},{"Start":"04:58.995 ","End":"05:05.910","Text":"Then this with this is 0 plus 1 plus 8 is 9,"},{"Start":"05:05.910 ","End":"05:08.440","Text":"and then 4 times 2 is 8,"},{"Start":"05:08.440 ","End":"05:14.060","Text":"minus 8 is 0 plus 8 times 3 is 24."},{"Start":"05:14.060 ","End":"05:19.170","Text":"Last row. This with this is 2 plus"},{"Start":"05:19.170 ","End":"05:26.775","Text":"4 plus 6 is 12,"},{"Start":"05:26.775 ","End":"05:36.840","Text":"and then 0 minus 1 plus 3 is 2,"},{"Start":"05:36.840 ","End":"05:40.740","Text":"and finally, 4 and 8 is 12,"},{"Start":"05:40.740 ","End":"05:44.350","Text":"and 9 is 21."},{"Start":"05:45.020 ","End":"05:50.770","Text":"That is B inverse squared."},{"Start":"05:53.030 ","End":"06:01.625","Text":"This is B inverse B inverse B inverse squared."},{"Start":"06:01.625 ","End":"06:07.680","Text":"The other thing we want is B inverse transpose."},{"Start":"06:08.720 ","End":"06:14.315","Text":"Well, we can just write that B inverse transpose would"},{"Start":"06:14.315 ","End":"06:21.965","Text":"be just taking B inverse say from here or from here,"},{"Start":"06:21.965 ","End":"06:26.715","Text":"and just writing the rows as columns."},{"Start":"06:26.715 ","End":"06:28.525","Text":"Like 1, 0, 2,"},{"Start":"06:28.525 ","End":"06:30.170","Text":"1, 0, 2."},{"Start":"06:30.170 ","End":"06:31.400","Text":"4 minus 1,"},{"Start":"06:31.400 ","End":"06:36.859","Text":"8 here, and then 2, 1, 3."},{"Start":"06:36.859 ","End":"06:39.410","Text":"Now we have all the ingredients,"},{"Start":"06:39.410 ","End":"06:42.485","Text":"so we can start making the computation."},{"Start":"06:42.485 ","End":"06:49.100","Text":"What we have is B minus 1 squared from here is the 3 by 3,"},{"Start":"06:49.100 ","End":"06:51.950","Text":"5, 2,"},{"Start":"06:51.950 ","End":"06:56.000","Text":"8, 16, 9,"},{"Start":"06:56.000 ","End":"07:00.785","Text":"24, 12, 2,"},{"Start":"07:00.785 ","End":"07:07.960","Text":"21 plus I. I is just 1,"},{"Start":"07:07.960 ","End":"07:11.600","Text":"1, 1 and everything else is 0."},{"Start":"07:14.190 ","End":"07:19.475","Text":"Then B inverse transpose or just copy from here,"},{"Start":"07:19.475 ","End":"07:22.015","Text":"just copy pasted it."},{"Start":"07:22.015 ","End":"07:27.050","Text":"Now this addition is fairly straight forward,"},{"Start":"07:27.120 ","End":"07:31.850","Text":"what we get is,"},{"Start":"07:31.850 ","End":"07:35.845","Text":"it\u0027s pretty much the same as this except we add 1 to the diagonal,"},{"Start":"07:35.845 ","End":"07:37.420","Text":"so instead of 5,"},{"Start":"07:37.420 ","End":"07:38.500","Text":"I have 6,"},{"Start":"07:38.500 ","End":"07:40.360","Text":"instead of 9, I get 10,"},{"Start":"07:40.360 ","End":"07:42.685","Text":"instead of 21 I get 22,"},{"Start":"07:42.685 ","End":"07:49.290","Text":"but all the rest of it stays the same, just copying."},{"Start":"07:49.290 ","End":"07:54.750","Text":"This I have to multiply by this one,"},{"Start":"07:54.750 ","End":"07:57.570","Text":"I copy that here,"},{"Start":"07:57.570 ","End":"08:00.200","Text":"sorry and now we have"},{"Start":"08:00.200 ","End":"08:06.430","Text":"one more matrix multiplication to do in order to figure out what Y is."},{"Start":"08:06.430 ","End":"08:11.615","Text":"Let\u0027s see. Y is going to be also a 3 by 3."},{"Start":"08:11.615 ","End":"08:14.660","Text":"Let\u0027s see if we can do this quickly."},{"Start":"08:14.660 ","End":"08:17.960","Text":"I\u0027m going to work on the first row."},{"Start":"08:17.960 ","End":"08:20.670","Text":"For the first column,"},{"Start":"08:21.080 ","End":"08:27.030","Text":"I get 6 times 1,"},{"Start":"08:27.030 ","End":"08:29.835","Text":"2 times 0, 8 times 2,"},{"Start":"08:29.835 ","End":"08:33.845","Text":"6 to the 16 is 22."},{"Start":"08:33.845 ","End":"08:38.330","Text":"Next, the second column,"},{"Start":"08:38.330 ","End":"08:40.805","Text":"6 times 4,"},{"Start":"08:40.805 ","End":"08:51.110","Text":"minus 2 plus 64."},{"Start":"08:51.110 ","End":"08:53.985","Text":"24 minus 2 is 22,"},{"Start":"08:53.985 ","End":"08:58.980","Text":"plus 64 is 86."},{"Start":"08:58.980 ","End":"09:01.679","Text":"Our first row with the third column,"},{"Start":"09:01.679 ","End":"09:11.500","Text":"12 and 2 and 24 is 38."},{"Start":"09:12.090 ","End":"09:18.490","Text":"Now on to the second row with the first column,16 and"},{"Start":"09:18.490 ","End":"09:28.540","Text":"0 and 48 is 64."},{"Start":"09:28.540 ","End":"09:37.960","Text":"16 and 4, 64 minus 10 plus,"},{"Start":"09:37.960 ","End":"09:42.050","Text":"let\u0027s see, 24 times 8 is 198,"},{"Start":"09:42.780 ","End":"09:47.230","Text":"I make it 246."},{"Start":"09:47.230 ","End":"09:53.800","Text":"Next second row with third column 32"},{"Start":"09:53.800 ","End":"10:02.330","Text":"and 10 and 72 is 114."},{"Start":"10:02.990 ","End":"10:14.205","Text":"This row with this column 12 and 0 and 44 is 56,"},{"Start":"10:14.205 ","End":"10:18.289","Text":"and then with this column, this last row,"},{"Start":"10:18.289 ","End":"10:24.750","Text":"48 minus 2 plus,"},{"Start":"10:25.840 ","End":"10:28.625","Text":"26 of 8 176,"},{"Start":"10:28.625 ","End":"10:34.670","Text":"and I make the total 222 because it\u0027s"},{"Start":"10:34.670 ","End":"10:39.565","Text":"48 minus 2 is"},{"Start":"10:39.565 ","End":"10:46.590","Text":"46 and 176 is 222."},{"Start":"10:46.590 ","End":"10:54.810","Text":"Finally, last row with last column is 24 and 2 and 66,"},{"Start":"10:54.810 ","End":"11:06.790","Text":"is 26 and 66, which is 92."},{"Start":"11:07.940 ","End":"11:12.455","Text":"I believe we finally got to the answer."},{"Start":"11:12.455 ","End":"11:16.440","Text":"We found y and we\u0027re done."}],"ID":14191},{"Watched":false,"Name":"Exercise 4","Duration":"13m 9s","ChapterTopicVideoID":13539,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13539.jpeg","UploadDate":"2018-09-06T10:02:07.7570000","DurationForVideoObject":"PT13M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.055","Text":"In this exercise we\u0027re given a matrix equation"},{"Start":"00:05.055 ","End":"00:12.580","Text":"here and we\u0027re also given that the inverse of matrix A is this,"},{"Start":"00:12.580 ","End":"00:16.980","Text":"and we have to solve for B and assuming everything\u0027s 2 by 2."},{"Start":"00:16.980 ","End":"00:20.925","Text":"This I is a 2 by 2 matrix and so is B,"},{"Start":"00:20.925 ","End":"00:22.995","Text":"A of course is."},{"Start":"00:22.995 ","End":"00:25.860","Text":"Let me just copy that."},{"Start":"00:25.860 ","End":"00:29.610","Text":"Now what I want to do is first of all take these constants out,"},{"Start":"00:29.610 ","End":"00:32.940","Text":"but you have to be careful."},{"Start":"00:32.940 ","End":"00:39.515","Text":"If I have a constant times a matrix it could be A in general,"},{"Start":"00:39.515 ","End":"00:45.515","Text":"and I take the inverse there\u0027s a rule that that\u0027s 1/k times the inverse of A."},{"Start":"00:45.515 ","End":"00:49.945","Text":"If I had kA^minus 2,"},{"Start":"00:49.945 ","End":"00:53.685","Text":"that would just mean this thing squared it would be"},{"Start":"00:53.685 ","End":"01:00.800","Text":"1/k A inverse times 1/k A inverse."},{"Start":"01:00.800 ","End":"01:06.805","Text":"It would mean 1/k squared A^minus 2."},{"Start":"01:06.805 ","End":"01:10.910","Text":"In fact in general for positive or negative powers,"},{"Start":"01:10.910 ","End":"01:20.130","Text":"kA^n is k^n A^n."},{"Start":"01:20.130 ","End":"01:26.365","Text":"There\u0027s different interpretations of a number to the power and the matrix to the power."},{"Start":"01:26.365 ","End":"01:32.150","Text":"Matrix to a positive power multiplied by itself and if it\u0027s a negative power,"},{"Start":"01:32.150 ","End":"01:35.255","Text":"it\u0027s the inverse multiplied by itself,"},{"Start":"01:35.255 ","End":"01:37.820","Text":"twice n times whatever."},{"Start":"01:37.820 ","End":"01:40.870","Text":"In any event, what we get here."},{"Start":"01:40.870 ","End":"01:47.890","Text":"On the right-hand side, we have 1/7 squared,"},{"Start":"01:48.200 ","End":"01:52.065","Text":"and here we have A^minus 2."},{"Start":"01:52.065 ","End":"01:57.195","Text":"I\u0027m also going to bring the 5 over so I get 1/5,"},{"Start":"01:57.195 ","End":"02:01.590","Text":"and I\u0027m left on the left with"},{"Start":"02:01.590 ","End":"02:06.840","Text":"A transpose B and then"},{"Start":"02:06.840 ","End":"02:13.710","Text":"I plus 2A^minus 2."},{"Start":"02:13.710 ","End":"02:18.990","Text":"Continuing the numbers, 49 times 5 is 245."},{"Start":"02:18.990 ","End":"02:22.150","Text":"That\u0027s the number part."},{"Start":"02:23.810 ","End":"02:29.640","Text":"Then A^minus 2 we could have just left it as A^minus 1 squared"},{"Start":"02:29.640 ","End":"02:33.765","Text":"because I see I have A^minus 1 A inverse,"},{"Start":"02:33.765 ","End":"02:37.520","Text":"and I\u0027ll be able to compute its square in a moment."},{"Start":"02:40.730 ","End":"02:45.400","Text":"The same thing on the left,"},{"Start":"02:46.570 ","End":"02:50.060","Text":"I just copied it."},{"Start":"02:50.060 ","End":"02:52.730","Text":"Now we want to isolate just B,"},{"Start":"02:52.730 ","End":"02:55.520","Text":"so we can multiply on the left by the inverse"},{"Start":"02:55.520 ","End":"02:59.185","Text":"of this and on the right by the inverse of this."},{"Start":"02:59.185 ","End":"03:07.955","Text":"What we\u0027ll get is A transpose inverse and then all of this,"},{"Start":"03:07.955 ","End":"03:12.005","Text":"this part I just copied from here."},{"Start":"03:12.005 ","End":"03:19.950","Text":"Then I want to put the inverse of this which is I plus 2A."},{"Start":"03:19.950 ","End":"03:21.500","Text":"Instead of minus 2,"},{"Start":"03:21.500 ","End":"03:24.840","Text":"I put 2 and that will be its inverse."},{"Start":"03:25.880 ","End":"03:29.540","Text":"I multiply it on the left by this and on the right by this,"},{"Start":"03:29.540 ","End":"03:32.670","Text":"so I have to do it on the other side also."},{"Start":"03:34.580 ","End":"03:38.220","Text":"Well, the constant can come upfront in any event,"},{"Start":"03:38.220 ","End":"03:41.730","Text":"constants don\u0027t matter,"},{"Start":"03:41.730 ","End":"03:47.295","Text":"so I\u0027ll just copy these 2 bits and I wrap them around"},{"Start":"03:47.295 ","End":"03:52.845","Text":"the A^minus 1 or A inverse squared."},{"Start":"03:52.845 ","End":"03:56.560","Text":"The other number comes upfront, you can always do that."},{"Start":"03:57.010 ","End":"03:59.450","Text":"Now on the left-hand side,"},{"Start":"03:59.450 ","End":"04:02.449","Text":"we deliberately multiplied by inverses."},{"Start":"04:02.449 ","End":"04:06.865","Text":"This with this will give me the identity matrix,"},{"Start":"04:06.865 ","End":"04:09.590","Text":"and this with this will also give me"},{"Start":"04:09.590 ","End":"04:14.000","Text":"the identity matrix I because it\u0027s something with its inverse."},{"Start":"04:14.000 ","End":"04:22.690","Text":"All I\u0027m left with on the left-hand side is B which is equal to 1/245."},{"Start":"04:22.690 ","End":"04:28.305","Text":"Now, this I\u0027m rewriting as A inverse transpose."},{"Start":"04:28.305 ","End":"04:33.590","Text":"It\u0027s 1 of the rules and I want to do it because A inverse I have,"},{"Start":"04:33.590 ","End":"04:40.415","Text":"and then I have A inverse squared."},{"Start":"04:40.415 ","End":"04:42.850","Text":"Then the last bit,"},{"Start":"04:42.850 ","End":"04:53.785","Text":"if I write this as I plus 2A times I plus 2A instead of squared"},{"Start":"04:53.785 ","End":"05:00.590","Text":"then what we get is I times I is I"},{"Start":"05:00.590 ","End":"05:06.320","Text":"and then I times 2A or 2A times I,"},{"Start":"05:06.320 ","End":"05:07.190","Text":"I is the identity."},{"Start":"05:07.190 ","End":"05:08.370","Text":"In each case it gives me 2A,"},{"Start":"05:08.370 ","End":"05:12.130","Text":"so 2A plus 2A is 4A."},{"Start":"05:12.230 ","End":"05:18.530","Text":"Then I also get 2A times 2A"},{"Start":"05:18.530 ","End":"05:24.600","Text":"and I\u0027m going to take the 2 times 2 first and it\u0027s A squared."},{"Start":"05:24.910 ","End":"05:30.220","Text":"2A times 2A is just 2 times 2 is 4,"},{"Start":"05:30.220 ","End":"05:32.890","Text":"and A times A is A squared."},{"Start":"05:32.890 ","End":"05:35.510","Text":"Now we want to proceed carefully here,"},{"Start":"05:35.510 ","End":"05:38.460","Text":"if not we\u0027re going to end up doing too much work."},{"Start":"05:38.460 ","End":"05:40.730","Text":"This should be easy to compute."},{"Start":"05:40.730 ","End":"05:41.990","Text":"We have A inverse."},{"Start":"05:41.990 ","End":"05:44.540","Text":"To compute a transpose is easy."},{"Start":"05:44.540 ","End":"05:48.305","Text":"To compute this thing squared multiplying a 2 by 2;"},{"Start":"05:48.305 ","End":"05:50.200","Text":"multiply by itself,"},{"Start":"05:50.200 ","End":"05:52.980","Text":"they\u0027re not too bad, but A,"},{"Start":"05:52.980 ","End":"05:56.330","Text":"we\u0027d have to compute the inverse of the inverse and then we"},{"Start":"05:56.330 ","End":"06:01.190","Text":"also have to square it so I suggest we do some more algebra."},{"Start":"06:01.190 ","End":"06:05.110","Text":"Take this A^minus 2 and multiply it throughout."},{"Start":"06:05.110 ","End":"06:06.300","Text":"We can avoid,"},{"Start":"06:06.300 ","End":"06:10.215","Text":"I\u0027m claiming actually having to find A."},{"Start":"06:10.215 ","End":"06:13.320","Text":"What I\u0027m going to do is as follows,"},{"Start":"06:13.320 ","End":"06:15.465","Text":"this part I keep,"},{"Start":"06:15.465 ","End":"06:20.175","Text":"A^minus 1 transpose I keep."},{"Start":"06:20.175 ","End":"06:25.080","Text":"This which is really A^minus 2."},{"Start":"06:25.080 ","End":"06:27.705","Text":"We have our rules of exponents,"},{"Start":"06:27.705 ","End":"06:35.010","Text":"so if I multiply the brackets this becomes A^minus 2 or you know what?"},{"Start":"06:35.010 ","End":"06:42.285","Text":"I\u0027ll keep it as A^minus 1 squared because they have a inverse."},{"Start":"06:42.285 ","End":"06:46.800","Text":"Then multiplying by the second,"},{"Start":"06:46.800 ","End":"06:51.560","Text":"A^minus 2 times A is just A^minus 1."},{"Start":"06:51.560 ","End":"06:56.670","Text":"The rules of exponents work and the 4 stays,"},{"Start":"06:56.670 ","End":"07:04.410","Text":"so I got 4A inverse and then A^minus 2 with A squared gives me the identity"},{"Start":"07:04.410 ","End":"07:13.080","Text":"so I\u0027ve just got 4 times the identity matrix."},{"Start":"07:13.080 ","End":"07:16.350","Text":"Now I don\u0027t have just plain A."},{"Start":"07:16.350 ","End":"07:24.040","Text":"It always appears as A inverse which I have and I should have recorded it,"},{"Start":"07:24.040 ","End":"07:25.910","Text":"and so I\u0027ll write it here."},{"Start":"07:25.910 ","End":"07:30.970","Text":"A inverse was a 2 by 2 matrix,"},{"Start":"07:30.970 ","End":"07:37.140","Text":"and it was 2, 3, 4, 7. We\u0027ll need that."},{"Start":"07:38.840 ","End":"07:45.245","Text":"There\u0027s really just 2 main matrix computation to have to perform."},{"Start":"07:45.245 ","End":"07:51.560","Text":"I\u0027ll need the transpose of A inverse and I\u0027ll need the square of A inverse."},{"Start":"07:51.560 ","End":"07:55.755","Text":"I\u0027ll change the colors."},{"Start":"07:55.755 ","End":"08:05.820","Text":"Let\u0027s go with the first 1 which is A inverse transpose."},{"Start":"08:05.820 ","End":"08:08.620","Text":"I just look at this A inverse and transpose it,"},{"Start":"08:08.620 ","End":"08:13.165","Text":"it means flip it along the diagonal or write columns as rows."},{"Start":"08:13.165 ","End":"08:17.125","Text":"Top row; 2, 3 becomes the first column,"},{"Start":"08:17.125 ","End":"08:22.170","Text":"the bottom row; 4, 7, becomes the last column"},{"Start":"08:22.170 ","End":"08:24.405","Text":"so that\u0027s that 1."},{"Start":"08:24.405 ","End":"08:27.585","Text":"As for the square,"},{"Start":"08:27.585 ","End":"08:39.420","Text":"A inverse squared is just this 2, 3, 4, 7"},{"Start":"08:39.420 ","End":"08:43.110","Text":"and then I have to multiply it by itself"},{"Start":"08:43.110 ","End":"08:46.205","Text":"and let\u0027s see what we get."},{"Start":"08:46.205 ","End":"08:48.605","Text":"Let\u0027s do this quickly."},{"Start":"08:48.605 ","End":"08:53.095","Text":"First row with 1st column,"},{"Start":"08:53.095 ","End":"08:56.410","Text":"4 plus 12 is 16."},{"Start":"08:56.410 ","End":"08:59.950","Text":"First row with 2nd column, 2, 3"},{"Start":"08:59.950 ","End":"09:06.375","Text":"and 3,7 is 6 and 21 is 27."},{"Start":"09:06.375 ","End":"09:10.480","Text":"Then the 2nd row, 1st column;"},{"Start":"09:11.210 ","End":"09:13.350","Text":"4 times 2 is 8,"},{"Start":"09:13.350 ","End":"09:15.390","Text":"7 times 4 is 28."},{"Start":"09:15.390 ","End":"09:17.385","Text":"That\u0027s 36."},{"Start":"09:17.385 ","End":"09:19.350","Text":"Then this row with this column;"},{"Start":"09:19.350 ","End":"09:20.835","Text":"4 times 3 is 12,"},{"Start":"09:20.835 ","End":"09:25.660","Text":"plus 49 is 61."},{"Start":"09:26.930 ","End":"09:31.745","Text":"I think we\u0027re about ready for the final attack on B,"},{"Start":"09:31.745 ","End":"09:37.820","Text":"so we get that B is equal to 1/245."},{"Start":"09:37.820 ","End":"09:44.135","Text":"Now, this is what we computed here which is this."},{"Start":"09:44.135 ","End":"09:48.060","Text":"That\u0027s the 2, 4, 3, 7,"},{"Start":"09:48.060 ","End":"09:49.830","Text":"that\u0027s A^minus 1 transpose."},{"Start":"09:49.830 ","End":"09:55.815","Text":"This 1, we computed over here but it\u0027s in the brackets,"},{"Start":"09:55.815 ","End":"10:04.400","Text":"so it\u0027s 16, 27, 36, 61"},{"Start":"10:04.400 ","End":"10:09.525","Text":"plus 4 times A inverse."},{"Start":"10:09.525 ","End":"10:14.375","Text":"4 times a matrix, we can do and I just multiply all the entries by 4."},{"Start":"10:14.375 ","End":"10:16.745","Text":"So 2 times 4 is 8,"},{"Start":"10:16.745 ","End":"10:18.260","Text":"3 times 4 is 12,"},{"Start":"10:18.260 ","End":"10:19.670","Text":"4 times 4 is 16,"},{"Start":"10:19.670 ","End":"10:21.875","Text":"7 times 4 is 28."},{"Start":"10:21.875 ","End":"10:25.955","Text":"That\u0027s the 4 times this, A inverse."},{"Start":"10:25.955 ","End":"10:30.780","Text":"Then for I is 4 times this 1."},{"Start":"10:30.780 ","End":"10:31.940","Text":"1 on the diagonal,"},{"Start":"10:31.940 ","End":"10:32.990","Text":"so it becomes 4, 0,"},{"Start":"10:32.990 ","End":"10:37.140","Text":"4, and the 0s stays 0s."},{"Start":"10:37.430 ","End":"10:41.270","Text":"I have now an addition of 3 things"},{"Start":"10:41.270 ","End":"10:44.000","Text":"and then a multiplication and multiplication by a constant,"},{"Start":"10:44.000 ","End":"10:48.755","Text":"we\u0027re getting there, 1/245,"},{"Start":"10:48.755 ","End":"10:52.280","Text":"2, 4, 3, 7."},{"Start":"10:52.280 ","End":"10:56.055","Text":"Now we can add 3 at a time."},{"Start":"10:56.055 ","End":"10:57.510","Text":"16, and 8,"},{"Start":"10:57.510 ","End":"11:01.860","Text":"and 4 is 28,"},{"Start":"11:01.860 ","End":"11:04.050","Text":"27 and 12,"},{"Start":"11:04.050 ","End":"11:08.235","Text":"and 0 is 39,"},{"Start":"11:08.235 ","End":"11:10.710","Text":"36 and 16,"},{"Start":"11:10.710 ","End":"11:15.075","Text":"and 0 is 52,"},{"Start":"11:15.075 ","End":"11:17.565","Text":"61 and 28,"},{"Start":"11:17.565 ","End":"11:21.210","Text":"and 4 is 61,"},{"Start":"11:21.210 ","End":"11:27.730","Text":"and 32 is 93."},{"Start":"11:31.340 ","End":"11:35.460","Text":"Now we have another multiplication."},{"Start":"11:35.460 ","End":"11:40.230","Text":"We\u0027re really practicing our matrices here."},{"Start":"11:41.450 ","End":"11:45.345","Text":"This with this."},{"Start":"11:45.345 ","End":"11:53.210","Text":"56 and 208,"},{"Start":"11:53.210 ","End":"11:56.015","Text":"I make it 264."},{"Start":"11:56.015 ","End":"11:59.400","Text":"Let\u0027s put some brackets here."},{"Start":"11:59.810 ","End":"12:08.445","Text":"Then I want 2 times 39 is 78 plus 4 times 93,"},{"Start":"12:08.445 ","End":"12:11.760","Text":"I make it 450."},{"Start":"12:11.760 ","End":"12:18.840","Text":"Next, 3 times 28 plus 7 times"},{"Start":"12:18.840 ","End":"12:28.140","Text":"52 is 84 plus 364."},{"Start":"12:28.140 ","End":"12:31.770","Text":"I make it 448."},{"Start":"12:31.770 ","End":"12:37.780","Text":"Then last row with last column."},{"Start":"12:38.570 ","End":"12:41.130","Text":"Let\u0027s do it on the calculator."},{"Start":"12:41.130 ","End":"12:53.845","Text":"3 times 39 plus 7 times 93, I get 768."},{"Start":"12:53.845 ","End":"12:58.130","Text":"I don\u0027t think there\u0027s any point in putting these over the fractions,"},{"Start":"12:58.130 ","End":"13:05.310","Text":"I would leave it like this and say lo and behold we finally got our answer,"},{"Start":"13:05.310 ","End":"13:07.020","Text":"we worked hard,"},{"Start":"13:07.020 ","End":"13:10.030","Text":"and now we\u0027re done."}],"ID":14192},{"Watched":false,"Name":"Exercise 5","Duration":"5m 49s","ChapterTopicVideoID":13540,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13540.jpeg","UploadDate":"2018-09-06T10:03:01.2230000","DurationForVideoObject":"PT5M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"This is a 2-part question,"},{"Start":"00:02.970 ","End":"00:05.475","Text":"but they\u0027re both similar."},{"Start":"00:05.475 ","End":"00:12.450","Text":"In each case, we\u0027re given an equation that A satisfies here or here."},{"Start":"00:12.450 ","End":"00:18.360","Text":"In each case, we have to show that A is an invertible matrix."},{"Start":"00:18.360 ","End":"00:22.865","Text":"We actually have to find its inverse in terms of A and I,"},{"Start":"00:22.865 ","End":"00:27.500","Text":"I being the identity matrix for whatever size it is."},{"Start":"00:27.500 ","End":"00:30.390","Text":"They\u0027re all square of the same size."},{"Start":"00:30.670 ","End":"00:33.425","Text":"Starting with A,"},{"Start":"00:33.425 ","End":"00:35.960","Text":"but they both have the general idea."},{"Start":"00:35.960 ","End":"00:37.685","Text":"I\u0027ll tell you the strategy."},{"Start":"00:37.685 ","End":"00:40.820","Text":"The strategy is to find some other matrix."},{"Start":"00:40.820 ","End":"00:43.550","Text":"Let\u0027s just call it smiley,"},{"Start":"00:43.550 ","End":"00:48.185","Text":"such that A times smiley equals the identity,"},{"Start":"00:48.185 ","End":"00:49.370","Text":"or the other way around."},{"Start":"00:49.370 ","End":"00:51.665","Text":"If you find smiley times A is the identity,"},{"Start":"00:51.665 ","End":"00:59.945","Text":"then that means that the matrix that you found is A inverse."},{"Start":"00:59.945 ","End":"01:06.990","Text":"I\u0027m using this as a placeholder instead of using a plain box or something."},{"Start":"01:06.990 ","End":"01:08.960","Text":"In the first 1,"},{"Start":"01:08.960 ","End":"01:10.580","Text":"what I can do is as follows."},{"Start":"01:10.580 ","End":"01:13.370","Text":"I start off with what I\u0027m given, just copying it."},{"Start":"01:13.370 ","End":"01:18.300","Text":"A squared minus 5A minus 2I equals 0."},{"Start":"01:18.300 ","End":"01:23.405","Text":"Now I\u0027m going to take this with the I to the other side."},{"Start":"01:23.405 ","End":"01:28.140","Text":"I squared minus 5A equals 2I."},{"Start":"01:28.160 ","End":"01:32.040","Text":"I\u0027m trying to get it to A times something equals I."},{"Start":"01:32.040 ","End":"01:33.620","Text":"I have a 2 here."},{"Start":"01:33.620 ","End":"01:37.810","Text":"That doesn\u0027t really bother me, it\u0027s a constant."},{"Start":"01:37.810 ","End":"01:42.840","Text":"But here I can take A out of the brackets,"},{"Start":"01:42.840 ","End":"01:45.780","Text":"doesn\u0027t matter on the left or the right."},{"Start":"01:45.780 ","End":"01:48.210","Text":"Normally, I would say that this is,"},{"Start":"01:48.210 ","End":"01:50.550","Text":"I take it out on the right."},{"Start":"01:50.550 ","End":"01:56.300","Text":"But it\u0027s not A minus 5 times A because remember,"},{"Start":"01:56.300 ","End":"01:58.865","Text":"we don\u0027t just put a number on its own with a matrix,"},{"Start":"01:58.865 ","End":"02:01.130","Text":"we have to throw in this I"},{"Start":"02:01.130 ","End":"02:07.875","Text":"because I could have written it I times A is equal to 2I,"},{"Start":"02:07.875 ","End":"02:15.630","Text":"but what I can do is just put I here and bring the 2 to the other side as 1.5."},{"Start":"02:15.630 ","End":"02:18.974","Text":"Now if I look at this,"},{"Start":"02:18.974 ","End":"02:22.990","Text":"I\u0027m claiming that this is my,"},{"Start":"02:23.120 ","End":"02:26.335","Text":"what I called smiley."},{"Start":"02:26.335 ","End":"02:28.070","Text":"Could be the other way around."},{"Start":"02:28.070 ","End":"02:30.200","Text":"Like I said, it could be that this is on the left,"},{"Start":"02:30.200 ","End":"02:32.220","Text":"is on the right, doesn\u0027t matter."},{"Start":"02:32.220 ","End":"02:35.185","Text":"That\u0027s my A inverse."},{"Start":"02:35.185 ","End":"02:40.200","Text":"From here, I can conclude that A inverse is"},{"Start":"02:40.200 ","End":"02:46.660","Text":"1/2 of A minus 5I."},{"Start":"02:47.780 ","End":"02:51.240","Text":"That\u0027s Part A. Now,"},{"Start":"02:51.240 ","End":"02:54.190","Text":"let\u0027s move on to Part B."},{"Start":"02:54.500 ","End":"03:00.335","Text":"Let me write that\u0027s Part A and now Part B."},{"Start":"03:00.335 ","End":"03:02.240","Text":"Here we have a different equation."},{"Start":"03:02.240 ","End":"03:05.880","Text":"We have A minus 3I,"},{"Start":"03:06.370 ","End":"03:12.425","Text":"A plus 2I equals 0."},{"Start":"03:12.425 ","End":"03:15.890","Text":"I want to multiply this out,"},{"Start":"03:15.890 ","End":"03:21.335","Text":"so A times A is A squared."},{"Start":"03:21.335 ","End":"03:23.660","Text":"Now, I times anything is itself,"},{"Start":"03:23.660 ","End":"03:26.690","Text":"so minus and the constants just stay."},{"Start":"03:26.690 ","End":"03:30.055","Text":"I get minus 3A."},{"Start":"03:30.055 ","End":"03:33.225","Text":"I\u0027ll write it as minus 3A,"},{"Start":"03:33.225 ","End":"03:37.070","Text":"and then from here and here I get plus 2A,"},{"Start":"03:37.070 ","End":"03:41.510","Text":"and then I get minus 6I squared,"},{"Start":"03:41.510 ","End":"03:43.670","Text":"but I times anything is itself,"},{"Start":"03:43.670 ","End":"03:48.510","Text":"so it\u0027s just minus 6I is 0."},{"Start":"03:50.500 ","End":"03:57.270","Text":"A squared minus 3A plus 2A is minus A."},{"Start":"03:57.270 ","End":"04:01.960","Text":"I can bring the 6I to the other side."},{"Start":"04:01.960 ","End":"04:04.210","Text":"I\u0027ll continue over here."},{"Start":"04:04.210 ","End":"04:07.355","Text":"Now I can take A out the brackets."},{"Start":"04:07.355 ","End":"04:10.560","Text":"I\u0027ve got A times,"},{"Start":"04:10.560 ","End":"04:13.120","Text":"doesn\u0027t matter if I take it out on the left or the right."},{"Start":"04:13.120 ","End":"04:17.380","Text":"This is AI or IA, whichever you prefer."},{"Start":"04:17.380 ","End":"04:23.870","Text":"It\u0027s A, A minus I equals 6I."},{"Start":"04:26.150 ","End":"04:31.760","Text":"Then, I bring the 6 over to the other side, 1/6."},{"Start":"04:34.170 ","End":"04:39.680","Text":"I can put the 1/6 in the middle really once and put the A,"},{"Start":"04:39.680 ","End":"04:45.670","Text":"and then I can put 1/6 of A minus I."},{"Start":"04:45.670 ","End":"04:47.790","Text":"Here, I\u0027ve got the I."},{"Start":"04:47.790 ","End":"04:52.200","Text":"I could have put the 1/6 in front but if you multiply as"},{"Start":"04:52.200 ","End":"04:55.920","Text":"a general rule that"},{"Start":"04:55.920 ","End":"05:03.030","Text":"K times AB is the same as KA times B,"},{"Start":"05:03.030 ","End":"05:06.880","Text":"which is the same as A times KB,"},{"Start":"05:06.880 ","End":"05:10.170","Text":"meaning the constant you can put in anywhere."},{"Start":"05:13.430 ","End":"05:19.340","Text":"This bit here is our smiley,"},{"Start":"05:19.340 ","End":"05:22.715","Text":"the thing that multiplies by A to give the identity."},{"Start":"05:22.715 ","End":"05:25.505","Text":"This must be A inverse,"},{"Start":"05:25.505 ","End":"05:29.510","Text":"so we have that A inverse is equal"},{"Start":"05:29.510 ","End":"05:36.840","Text":"to 1/6 A minus I."},{"Start":"05:37.580 ","End":"05:40.954","Text":"Really should have been highlighting the answers."},{"Start":"05:40.954 ","End":"05:46.085","Text":"This is for Part B and this is for Part A,"},{"Start":"05:46.085 ","End":"05:49.680","Text":"and I guess we\u0027re done."}],"ID":14193},{"Watched":false,"Name":"Exercise 6","Duration":"7m 11s","ChapterTopicVideoID":13541,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13541.jpeg","UploadDate":"2018-09-06T10:04:07.4830000","DurationForVideoObject":"PT7M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.145","Text":"This question has 2 parts."},{"Start":"00:02.145 ","End":"00:04.455","Text":"Let\u0027s do the first part."},{"Start":"00:04.455 ","End":"00:07.860","Text":"We\u0027re given a square matrix A which satisfies"},{"Start":"00:07.860 ","End":"00:12.390","Text":"this equation and we have to prove that A is invertible,"},{"Start":"00:12.390 ","End":"00:18.725","Text":"meaning it has an inverse and to express its inverse in terms of A and I."},{"Start":"00:18.725 ","End":"00:21.955","Text":"Let me just say in general,"},{"Start":"00:21.955 ","End":"00:25.800","Text":"for any A, not just this 1,"},{"Start":"00:25.800 ","End":"00:31.020","Text":"to have an inverse means that I can find another matrix,"},{"Start":"00:31.020 ","End":"00:37.040","Text":"I\u0027ll just call it smiley and multiply by it to equal I."},{"Start":"00:37.040 ","End":"00:39.590","Text":"It could also be the other way around,"},{"Start":"00:39.590 ","End":"00:44.345","Text":"if you find something times A that\u0027s equal to I,"},{"Start":"00:44.345 ","End":"00:46.520","Text":"I, I mean the identity matrix,"},{"Start":"00:46.520 ","End":"00:49.220","Text":"if either of these is true,"},{"Start":"00:49.220 ","End":"00:54.770","Text":"then you can say that this other matrix is the inverse of A,"},{"Start":"00:54.770 ","End":"00:57.330","Text":"and that A is invertible."},{"Start":"00:57.380 ","End":"01:01.010","Text":"We are looking to try and rewrite"},{"Start":"01:01.010 ","End":"01:06.590","Text":"this equation in such a form that we get A times something equals I."},{"Start":"01:06.590 ","End":"01:08.355","Text":"Let me first copy it,"},{"Start":"01:08.355 ","End":"01:13.350","Text":"A squared minus 5A minus 2I equals 0."},{"Start":"01:13.350 ","End":"01:15.840","Text":"Now, look, we want the I on the right-hand side,"},{"Start":"01:15.840 ","End":"01:20.880","Text":"so it looks like it makes sense to take the minus 2I"},{"Start":"01:20.880 ","End":"01:24.130","Text":"and bring it over to the other side."},{"Start":"01:25.640 ","End":"01:32.390","Text":"Now we could divide both sides by 2 and multiply by 1/2,"},{"Start":"01:32.390 ","End":"01:36.725","Text":"or I would prefer for us to take A out of the brackets."},{"Start":"01:36.725 ","End":"01:45.845","Text":"Now, you would think that I could just write A minus 5 times A here,"},{"Start":"01:45.845 ","End":"01:53.115","Text":"except that I can\u0027t mix a matrix and a number and we\u0027ve seen this thing before,"},{"Start":"01:53.115 ","End":"02:01.980","Text":"we just put an I in here and then it has a matrix."},{"Start":"02:01.980 ","End":"02:03.660","Text":"If you think about it,"},{"Start":"02:03.660 ","End":"02:09.350","Text":"then A is really equal to I times A,"},{"Start":"02:09.350 ","End":"02:15.170","Text":"so that 5A is"},{"Start":"02:15.170 ","End":"02:21.515","Text":"equal to 5IA and you can put the brackets anywhere."},{"Start":"02:21.515 ","End":"02:24.590","Text":"Actually, that\u0027s not the useful rule that it will be coming up"},{"Start":"02:24.590 ","End":"02:27.800","Text":"again in a moment, that in general,"},{"Start":"02:27.800 ","End":"02:31.940","Text":"if I have a constant times the product of 2 matrices,"},{"Start":"02:31.940 ","End":"02:37.340","Text":"it\u0027s the same as if I multiply it by the first or I can"},{"Start":"02:37.340 ","End":"02:43.860","Text":"put the K together with the B,"},{"Start":"02:43.860 ","End":"02:45.930","Text":"it doesn\u0027t matter where you put the K."},{"Start":"02:45.930 ","End":"02:49.780","Text":"If I have this,"},{"Start":"02:50.030 ","End":"02:53.640","Text":"let me just write equals 2I,"},{"Start":"02:53.640 ","End":"03:00.110","Text":"then I can now multiply both sides by 1/2 or divide by 2,"},{"Start":"03:00.110 ","End":"03:06.359","Text":"and I get that 1/2A minus"},{"Start":"03:06.359 ","End":"03:13.170","Text":"5I times A is equal to I."},{"Start":"03:13.170 ","End":"03:20.885","Text":"Here I have A times something and that\u0027s like the smiley,"},{"Start":"03:20.885 ","End":"03:23.690","Text":"I guess I have it in this form,"},{"Start":"03:23.690 ","End":"03:28.185","Text":"so this is equal to the A inverse."},{"Start":"03:28.185 ","End":"03:31.395","Text":"I have that my A inverse is"},{"Start":"03:31.395 ","End":"03:39.700","Text":"1/2A minus 5I and that\u0027s part A."},{"Start":"03:40.700 ","End":"03:45.400","Text":"Now part B, which is somewhat similar to part A,"},{"Start":"03:45.400 ","End":"03:50.620","Text":"remember the idea is to find A times something equals I."},{"Start":"03:50.620 ","End":"03:58.390","Text":"Let\u0027s expand here and if I take each of these with each of these,"},{"Start":"03:58.390 ","End":"04:01.150","Text":"we get A times A,"},{"Start":"04:01.150 ","End":"04:07.940","Text":"and then plus A2I,"},{"Start":"04:07.940 ","End":"04:10.260","Text":"multiplying A by 2I,"},{"Start":"04:10.260 ","End":"04:12.525","Text":"I can put the 2 out in front,"},{"Start":"04:12.525 ","End":"04:18.660","Text":"so it\u0027s just 2A at the side."},{"Start":"04:18.660 ","End":"04:22.260","Text":"A times 2I is,"},{"Start":"04:22.260 ","End":"04:24.270","Text":"remember I said we could take constants out,"},{"Start":"04:24.270 ","End":"04:29.760","Text":"is 2AI but A times I is just A."},{"Start":"04:29.760 ","End":"04:33.140","Text":"You start not doing this without really thinking,"},{"Start":"04:33.140 ","End":"04:34.310","Text":"when you have the numbers,"},{"Start":"04:34.310 ","End":"04:40.230","Text":"you just put them anywhere and I see A times I is just A and the 2 stays."},{"Start":"04:41.030 ","End":"04:44.190","Text":"Now I have minus 3IA,"},{"Start":"04:44.190 ","End":"04:46.350","Text":"and again IA is just A,"},{"Start":"04:46.350 ","End":"04:48.780","Text":"so I have minus 3A,"},{"Start":"04:48.780 ","End":"04:55.610","Text":"and then I have minus 3I times 2I."},{"Start":"04:55.610 ","End":"04:58.070","Text":"I put all the constants in front,"},{"Start":"04:58.070 ","End":"05:02.790","Text":"it\u0027s 6II, which is just 6I."},{"Start":"05:03.260 ","End":"05:07.620","Text":"Again, if I have 3I times 2I,"},{"Start":"05:07.620 ","End":"05:14.570","Text":"I can take the constants in front and say it\u0027s 3 times 2I times I,"},{"Start":"05:14.570 ","End":"05:17.510","Text":"but I times anything is itself and in particular,"},{"Start":"05:17.510 ","End":"05:22.950","Text":"I times I is just I, equals 0."},{"Start":"05:22.950 ","End":"05:28.870","Text":"Now I\u0027ll put the 6I on the right."},{"Start":"05:29.480 ","End":"05:33.155","Text":"Here I can take A out of the brackets."},{"Start":"05:33.155 ","End":"05:34.925","Text":"You know what, I\u0027ll write 1 more step."},{"Start":"05:34.925 ","End":"05:36.625","Text":"We have A,"},{"Start":"05:36.625 ","End":"05:38.310","Text":"I could write it as A squared,"},{"Start":"05:38.310 ","End":"05:48.940","Text":"I\u0027ll leave it as AA plus 2A minus 3A is minus A is equal to 6I."},{"Start":"05:49.040 ","End":"05:53.400","Text":"Now I can take A out of the brackets,"},{"Start":"05:53.400 ","End":"05:59.355","Text":"I could say that this A is IA or AI."},{"Start":"05:59.355 ","End":"06:04.920","Text":"You know what, this time let\u0027s put the I on the right,"},{"Start":"06:04.920 ","End":"06:07.735","Text":"doesn\u0027t matter, we just imagine it there."},{"Start":"06:07.735 ","End":"06:12.515","Text":"Then I can say that A times A minus I,"},{"Start":"06:12.515 ","End":"06:14.615","Text":"that\u0027s called the distributive law,"},{"Start":"06:14.615 ","End":"06:19.520","Text":"is equal to 6I and now I want just I,"},{"Start":"06:19.520 ","End":"06:22.055","Text":"so I bring the 6 over to the other side,"},{"Start":"06:22.055 ","End":"06:24.545","Text":"multiply both sides by 1/6,"},{"Start":"06:24.545 ","End":"06:31.785","Text":"so I\u0027ve got A times A minus I."},{"Start":"06:31.785 ","End":"06:34.820","Text":"Now the 1/6 I would normally put on the left,"},{"Start":"06:34.820 ","End":"06:36.410","Text":"but as I\u0027ve said, with the constants,"},{"Start":"06:36.410 ","End":"06:37.925","Text":"you can put them anywhere,"},{"Start":"06:37.925 ","End":"06:42.300","Text":"so let me put the 1/6 in here."},{"Start":"06:42.340 ","End":"06:48.830","Text":"The reason I did that is now I have A times something equals I."},{"Start":"06:48.830 ","End":"06:52.750","Text":"The A in the first part I called this the smiley,"},{"Start":"06:52.750 ","End":"06:55.520","Text":"but whatever it is A times it is I,"},{"Start":"06:55.520 ","End":"06:58.895","Text":"so it must be the inverse of A."},{"Start":"06:58.895 ","End":"07:04.710","Text":"The inverse of A is 1/6A minus I"},{"Start":"07:04.710 ","End":"07:08.580","Text":"and that\u0027s the answer to part B"},{"Start":"07:08.580 ","End":"07:10.750","Text":"and we\u0027re done."}],"ID":14194},{"Watched":false,"Name":"Exercise 7 - Part a","Duration":"9m 13s","ChapterTopicVideoID":13542,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13542.jpeg","UploadDate":"2018-09-06T10:05:35.2800000","DurationForVideoObject":"PT9M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this exercise, we\u0027re given a matrix and we\u0027re given"},{"Start":"00:04.170 ","End":"00:09.880","Text":"a polynomial and we\u0027re asked to compute p of A."},{"Start":"00:09.880 ","End":"00:12.165","Text":"Now what does this even mean?"},{"Start":"00:12.165 ","End":"00:14.715","Text":"I mean this is normally"},{"Start":"00:14.715 ","End":"00:23.540","Text":"an expression where we can substitute x for a number."},{"Start":"00:23.540 ","End":"00:27.080","Text":"But in principle, symbolically at least we can"},{"Start":"00:27.080 ","End":"00:32.420","Text":"substitute x equals a matrix, and that\u0027s what we do."},{"Start":"00:32.420 ","End":"00:36.935","Text":"There\u0027s a small variation and you\u0027ll see in a moment."},{"Start":"00:36.935 ","End":"00:39.730","Text":"If I said p of A,"},{"Start":"00:39.730 ","End":"00:42.750","Text":"I would say, we know what A cubed is;"},{"Start":"00:42.750 ","End":"00:44.415","Text":"this A times A times A."},{"Start":"00:44.415 ","End":"00:46.685","Text":"We know to multiply a matrix by a number,"},{"Start":"00:46.685 ","End":"00:49.160","Text":"A squared is A times A,"},{"Start":"00:49.160 ","End":"00:51.560","Text":"20 times a matrix is okay."},{"Start":"00:51.560 ","End":"00:53.405","Text":"Just the last bit."},{"Start":"00:53.405 ","End":"00:55.055","Text":"When it\u0027s just a number,"},{"Start":"00:55.055 ","End":"00:56.675","Text":"it\u0027s understood to be"},{"Start":"00:56.675 ","End":"00:59.179","Text":"a number times the identity matrix"},{"Start":"00:59.179 ","End":"01:01.430","Text":"and that\u0027s how we define a polynomial"},{"Start":"01:01.430 ","End":"01:04.370","Text":"on a matrix instead of a number."},{"Start":"01:04.370 ","End":"01:07.660","Text":"Now in our case if we want to compute it,"},{"Start":"01:07.660 ","End":"01:09.450","Text":"we know what I is."},{"Start":"01:09.450 ","End":"01:14.440","Text":"It\u0027s the identity 3 by 3 and 1s along the main diagonal."},{"Start":"01:14.440 ","End":"01:16.375","Text":"We know what A is, it\u0027s given here."},{"Start":"01:16.375 ","End":"01:18.250","Text":"We need A squared and A cubed,"},{"Start":"01:18.250 ","End":"01:19.915","Text":"so we just have to compute."},{"Start":"01:19.915 ","End":"01:24.635","Text":"A squared, which is A times A,"},{"Start":"01:24.635 ","End":"01:29.315","Text":"is equal to, I\u0027ll just copy it."},{"Start":"01:29.315 ","End":"01:32.685","Text":"It\u0027s minus 1, 3, 0,"},{"Start":"01:32.685 ","End":"01:35.295","Text":"then 3 minus 1, 0,"},{"Start":"01:35.295 ","End":"01:40.860","Text":"then minus 2 minus 2, 6."},{"Start":"01:40.860 ","End":"01:43.364","Text":"I need the same thing again,"},{"Start":"01:43.364 ","End":"01:44.670","Text":"minus 1, 3, 0,"},{"Start":"01:44.670 ","End":"01:46.590","Text":"3 minus 1, 0,"},{"Start":"01:46.590 ","End":"01:50.235","Text":"minus 2 minus 2, 6."},{"Start":"01:50.235 ","End":"01:57.810","Text":"Now it\u0027s a 3 by 3 times a 3 by 3 and what we\u0027re getting is also a 3 by 3."},{"Start":"01:57.810 ","End":"02:04.500","Text":"Let\u0027s start with first row here and first column here"},{"Start":"02:04.500 ","End":"02:06.170","Text":"then we\u0027ll get this entry here,"},{"Start":"02:06.170 ","End":"02:12.780","Text":"which will be plus 1 plus 9 plus 0, that\u0027s 10."},{"Start":"02:12.800 ","End":"02:16.650","Text":"Next 1, this column."},{"Start":"02:16.650 ","End":"02:22.380","Text":"Minus 3, minus 3, and 0, that\u0027s minus 6."},{"Start":"02:22.380 ","End":"02:26.535","Text":"The last column, this 1."},{"Start":"02:26.535 ","End":"02:30.180","Text":"With this, the minus 1 hits 0,"},{"Start":"02:30.180 ","End":"02:32.550","Text":"the 3 hits 0, the 0 hits 6."},{"Start":"02:32.550 ","End":"02:34.455","Text":"They\u0027re all products of 0,"},{"Start":"02:34.455 ","End":"02:39.045","Text":"so the sum is going to be 0."},{"Start":"02:39.045 ","End":"02:41.820","Text":"Now on to the middle row."},{"Start":"02:41.820 ","End":"02:44.490","Text":"It\u0027s this with each 1 of these."},{"Start":"02:44.490 ","End":"02:47.985","Text":"This with this is what minus 3,"},{"Start":"02:47.985 ","End":"02:51.120","Text":"minus 3, and then a 0."},{"Start":"02:51.120 ","End":"02:54.735","Text":"It\u0027s going to be minus 6."},{"Start":"02:54.735 ","End":"02:56.460","Text":"Then with this column,"},{"Start":"02:56.460 ","End":"03:00.435","Text":"it\u0027ll be 3 times 3 is 9 plus 1 is 10,"},{"Start":"03:00.435 ","End":"03:06.990","Text":"and then a 0, that\u0027s 10 in the last column."},{"Start":"03:06.990 ","End":"03:09.420","Text":"Well, it\u0027s going to be 0."},{"Start":"03:09.420 ","End":"03:10.650","Text":"The 3 is going to hit a 0,"},{"Start":"03:10.650 ","End":"03:13.020","Text":"the minus 1 is and then there\u0027s a 0,"},{"Start":"03:13.020 ","End":"03:17.655","Text":"so this 1 is 0. The last row."},{"Start":"03:17.655 ","End":"03:19.785","Text":"First, this with this,"},{"Start":"03:19.785 ","End":"03:24.150","Text":"we get 2 minus 6,"},{"Start":"03:24.150 ","End":"03:30.990","Text":"minus 12, 2 minus 18 minus 16."},{"Start":"03:30.990 ","End":"03:41.470","Text":"Next, minus 6 plus 2 minus 12,"},{"Start":"03:42.410 ","End":"03:46.770","Text":"that\u0027s minus 6, minus 10."},{"Start":"03:46.770 ","End":"03:49.950","Text":"It\u0027s minus 16 again."},{"Start":"03:49.950 ","End":"03:54.300","Text":"Last 1, this and this are going to hit the 0,"},{"Start":"03:54.300 ","End":"03:58.020","Text":"6 times 6 is 36."},{"Start":"03:58.020 ","End":"04:00.940","Text":"That\u0027s the A squared."},{"Start":"04:00.950 ","End":"04:04.000","Text":"Now I want A cubed,"},{"Start":"04:04.000 ","End":"04:12.245","Text":"which I can just do as either A squared times A or A times A squared."},{"Start":"04:12.245 ","End":"04:15.435","Text":"I\u0027ll go with it this way."},{"Start":"04:15.435 ","End":"04:24.700","Text":"I copy-pasted A squared here and I copied A from here."},{"Start":"04:24.950 ","End":"04:28.890","Text":"Let\u0027s see what A cubed is."},{"Start":"04:28.890 ","End":"04:31.780","Text":"I know this is tedious."},{"Start":"04:31.780 ","End":"04:33.560","Text":"Let\u0027s go a bit quicker."},{"Start":"04:33.560 ","End":"04:37.595","Text":"I\u0027ll just take this first row and we\u0027ll just mentally imagine the columns."},{"Start":"04:37.595 ","End":"04:41.405","Text":"First column, I\u0027ve got minus 10,"},{"Start":"04:41.405 ","End":"04:45.000","Text":"minus 18 and 0,"},{"Start":"04:45.000 ","End":"04:53.040","Text":"so that\u0027s minus 28."},{"Start":"04:53.040 ","End":"04:55.455","Text":"Then with this column,"},{"Start":"04:55.455 ","End":"05:03.315","Text":"it\u0027s 30 plus 6 and then 0, so that\u0027s 36."},{"Start":"05:03.315 ","End":"05:05.500","Text":"With the last column,"},{"Start":"05:05.500 ","End":"05:14.970","Text":"it just comes out to be 0 because 1 of the factors in each pair is a 0."},{"Start":"05:18.890 ","End":"05:22.845","Text":"Middle row, the first column."},{"Start":"05:22.845 ","End":"05:30.920","Text":"Plus 6, plus 30, 0, that\u0027s 36."},{"Start":"05:30.920 ","End":"05:32.975","Text":"Then with this 1,"},{"Start":"05:32.975 ","End":"05:39.595","Text":"we have minus 18, minus 10, and 0."},{"Start":"05:39.595 ","End":"05:43.500","Text":"That\u0027s minus 28."},{"Start":"05:43.500 ","End":"05:45.650","Text":"Then with the last 1,"},{"Start":"05:45.650 ","End":"05:49.650","Text":"well clearly we get a 0 again."},{"Start":"05:49.820 ","End":"05:52.170","Text":"Last row."},{"Start":"05:52.170 ","End":"05:54.290","Text":"With this column,"},{"Start":"05:54.290 ","End":"06:00.395","Text":"we have plus 16 minus 48,"},{"Start":"06:00.395 ","End":"06:09.300","Text":"that\u0027s minus 32 minus 72 is minus 104."},{"Start":"06:09.300 ","End":"06:13.325","Text":"Then with this column still with this row,"},{"Start":"06:13.325 ","End":"06:17.090","Text":"minus 48, minus 16,"},{"Start":"06:17.090 ","End":"06:20.280","Text":"that\u0027s down to minus 64."},{"Start":"06:27.290 ","End":"06:31.865","Text":"The minus with a minus is plus is minus 48 plus 16,"},{"Start":"06:31.865 ","End":"06:40.200","Text":"which is minus 32 then minus 72 is again minus 104."},{"Start":"06:40.200 ","End":"06:44.540","Text":"The last 1, which is this row with this column,"},{"Start":"06:44.540 ","End":"06:50.915","Text":"we\u0027re only going to get a contribution from the 36 times 6, which is 216."},{"Start":"06:50.915 ","End":"06:52.835","Text":"We have A,"},{"Start":"06:52.835 ","End":"06:54.530","Text":"we have A squared,"},{"Start":"06:54.530 ","End":"06:56.345","Text":"and we have A cubed."},{"Start":"06:56.345 ","End":"07:00.310","Text":"Now we can compute this expression."},{"Start":"07:00.610 ","End":"07:09.120","Text":"What we get is the polynomial of matrix A is A cubed."},{"Start":"07:09.120 ","End":"07:16.325","Text":"I copy from here or I could have done a copy-paste, never mind."},{"Start":"07:16.325 ","End":"07:21.155","Text":"It\u0027s pretty quick to just copy minus 28, 0,"},{"Start":"07:21.155 ","End":"07:27.555","Text":"minus 104, minus 104, 216,"},{"Start":"07:27.555 ","End":"07:33.280","Text":"minus 4 times A squared I\u0027m looking here."},{"Start":"07:33.920 ","End":"07:36.900","Text":"10 minus 6, 0,"},{"Start":"07:36.900 ","End":"07:39.000","Text":"minus 6, 10, 0,"},{"Start":"07:39.000 ","End":"07:43.425","Text":"minus 16, minus 16, 36,"},{"Start":"07:43.425 ","End":"07:46.260","Text":"then minus 20,"},{"Start":"07:46.260 ","End":"07:49.305","Text":"and we need A itself."},{"Start":"07:49.305 ","End":"07:53.295","Text":"A was this 1 here;"},{"Start":"07:53.295 ","End":"07:55.995","Text":"minus 1, 3, 0,"},{"Start":"07:55.995 ","End":"07:58.110","Text":"3 minus, 1, 0,"},{"Start":"07:58.110 ","End":"08:00.915","Text":"minus 2, minus 2, 6."},{"Start":"08:00.915 ","End":"08:03.940","Text":"Finally, 48I."},{"Start":"08:04.580 ","End":"08:14.110","Text":"I is just 1s here and everywhere else, 0."},{"Start":"08:16.520 ","End":"08:19.125","Text":"I\u0027ll make room for it."},{"Start":"08:19.125 ","End":"08:23.790","Text":"It\u0027ll be a 3 by 3"},{"Start":"08:23.790 ","End":"08:30.680","Text":"and we\u0027ll do each of the 9 positions as a separate exercise."},{"Start":"08:30.680 ","End":"08:35.290","Text":"Let\u0027s take the first row, first column."},{"Start":"08:35.290 ","End":"08:44.540","Text":"If we do the arithmetic to the first 2 terms are negative; minus 28 minus 40 is minus 68."},{"Start":"08:44.540 ","End":"08:54.140","Text":"These 2 are positive plus 20 plus 48 is plus 68 cancels with the minus 68 so we get a 0."},{"Start":"08:54.140 ","End":"08:57.395","Text":"There\u0027s 9 entries, I\u0027m not going to do each 1."},{"Start":"08:57.395 ","End":"09:02.735","Text":"Just check that we really do get 0 in all 9 entries,"},{"Start":"09:02.735 ","End":"09:06.310","Text":"which means that the answer is the 0 matrix."},{"Start":"09:06.310 ","End":"09:09.745","Text":"I\u0027ll just make it a bold 0 or maybe underline it."},{"Start":"09:09.745 ","End":"09:12.860","Text":"That\u0027s the answer to Part 1."}],"ID":14195},{"Watched":false,"Name":"Exercise 7 - Part b","Duration":"3m 27s","ChapterTopicVideoID":13543,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13543.jpeg","UploadDate":"2018-09-06T10:06:06.7670000","DurationForVideoObject":"PT3M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.870","Text":"This is a continuation of part 1 where we had to compute p of A,"},{"Start":"00:06.870 ","End":"00:14.265","Text":"and what we concluded was that p of A was equal to,"},{"Start":"00:14.265 ","End":"00:17.040","Text":"there was a matrix 3 by 3, all 0s,"},{"Start":"00:17.040 ","End":"00:21.225","Text":"sometimes we call that the 0 matrix as opposed to 0 number,"},{"Start":"00:21.225 ","End":"00:24.370","Text":"but it\u0027s given the same symbol."},{"Start":"00:24.410 ","End":"00:29.475","Text":"We have to use this now to show that A is invertible."},{"Start":"00:29.475 ","End":"00:31.050","Text":"What it means, well,"},{"Start":"00:31.050 ","End":"00:36.125","Text":"we already computed p of A."},{"Start":"00:36.125 ","End":"00:45.960","Text":"We expressed it as A cubed minus 4A squared minus 20A plus 48,"},{"Start":"00:45.960 ","End":"00:47.475","Text":"but that\u0027s not it,"},{"Start":"00:47.475 ","End":"00:49.485","Text":"we have to add the I."},{"Start":"00:49.485 ","End":"00:52.985","Text":"Remember we take p of a matrix,"},{"Start":"00:52.985 ","End":"00:57.460","Text":"the number becomes that number times the matrix I."},{"Start":"00:57.460 ","End":"01:02.345","Text":"Now because this is equal to the 0 matrix,"},{"Start":"01:02.345 ","End":"01:03.710","Text":"I\u0027ll make it a bit bolder,"},{"Start":"01:03.710 ","End":"01:09.005","Text":"we can now use the technique of some of the previous exercises."},{"Start":"01:09.005 ","End":"01:15.640","Text":"We need to find A times something,"},{"Start":"01:15.640 ","End":"01:18.210","Text":"that something, you can call it smiley,"},{"Start":"01:18.210 ","End":"01:19.390","Text":"is equal to I,"},{"Start":"01:19.390 ","End":"01:20.690","Text":"or the other way around,"},{"Start":"01:20.690 ","End":"01:24.935","Text":"it could be, it doesn\u0027t matter either 1,"},{"Start":"01:24.935 ","End":"01:28.834","Text":"if you get A times something to be the identity matrix,"},{"Start":"01:28.834 ","End":"01:34.800","Text":"then that something is your inverse of A."},{"Start":"01:34.800 ","End":"01:42.510","Text":"Here what I\u0027m going to do is bring the 48I to the other side,"},{"Start":"01:42.510 ","End":"01:49.420","Text":"but it\u0027ll be with a minus."},{"Start":"01:51.620 ","End":"01:56.805","Text":"Here, what we can do is on the first 3 terms,"},{"Start":"01:56.805 ","End":"01:59.945","Text":"we can take A outside the brackets,"},{"Start":"01:59.945 ","End":"02:01.865","Text":"let\u0027s say on the right,"},{"Start":"02:01.865 ","End":"02:07.410","Text":"so I\u0027ve got that"},{"Start":"02:07.410 ","End":"02:14.500","Text":"A squared minus 4A minus 20,"},{"Start":"02:18.200 ","End":"02:20.570","Text":"A is equal to this,"},{"Start":"02:20.570 ","End":"02:24.530","Text":"but I left a gap here because we cannot just write minus 20,"},{"Start":"02:24.530 ","End":"02:26.880","Text":"we write minus 20I."},{"Start":"02:29.140 ","End":"02:32.300","Text":"If I want to have just I on the right,"},{"Start":"02:32.300 ","End":"02:35.390","Text":"I just have to divide by minus 48,"},{"Start":"02:35.390 ","End":"02:40.895","Text":"so I\u0027ve got minus 1/48 times this thing here,"},{"Start":"02:40.895 ","End":"02:48.590","Text":"A squared minus 4A minus 20I times"},{"Start":"02:48.590 ","End":"02:51.500","Text":"A equals just I."},{"Start":"02:51.500 ","End":"02:56.825","Text":"I said it\u0027s good the other way around to have something,"},{"Start":"02:56.825 ","End":"03:00.590","Text":"this times A is equal to I,"},{"Start":"03:00.590 ","End":"03:03.470","Text":"so this thing is the inverse of A."},{"Start":"03:03.470 ","End":"03:08.720","Text":"I can now answer that the inverse of A is minus 1/48."},{"Start":"03:08.720 ","End":"03:10.070","Text":"There\u0027s no need to multiply,"},{"Start":"03:10.070 ","End":"03:11.465","Text":"you could leave it like this."},{"Start":"03:11.465 ","End":"03:16.330","Text":"A squared minus 4A minus 20I."},{"Start":"03:16.330 ","End":"03:23.965","Text":"It\u0027s expressed in terms of A and I"},{"Start":"03:23.965 ","End":"03:27.010","Text":"and we\u0027re done."}],"ID":14196},{"Watched":false,"Name":"Exercise 8","Duration":"6m 44s","ChapterTopicVideoID":13544,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13544.jpeg","UploadDate":"2018-09-06T10:07:09.0130000","DurationForVideoObject":"PT6M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"In this exercise, we\u0027re given"},{"Start":"00:02.520 ","End":"00:08.190","Text":"a square matrix A and it has the property that A to the 4th is 0."},{"Start":"00:08.190 ","End":"00:10.380","Text":"When I say 0 here,"},{"Start":"00:10.380 ","End":"00:14.235","Text":"this of course means the 0 matrix,"},{"Start":"00:14.235 ","End":"00:19.335","Text":"not the number 0, it\u0027s matrix with all 0s in it."},{"Start":"00:19.335 ","End":"00:21.360","Text":"There\u0027s 2 parts."},{"Start":"00:21.360 ","End":"00:23.895","Text":"Let\u0027s first of all, just look at A."},{"Start":"00:23.895 ","End":"00:28.990","Text":"In A, we have to prove that A is not invertible."},{"Start":"00:29.960 ","End":"00:34.490","Text":"There\u0027s a property I want to mention,"},{"Start":"00:34.490 ","End":"00:38.960","Text":"I\u0027m not 100 percent sure I mentioned it in the tutorial,"},{"Start":"00:38.960 ","End":"00:41.720","Text":"but if not then I\u0027m mentioning it now,"},{"Start":"00:41.720 ","End":"00:49.130","Text":"and that is that the product of invertible matrices is also invertible."},{"Start":"00:49.130 ","End":"00:51.680","Text":"It may be, we didn\u0027t say it in that form,"},{"Start":"00:51.680 ","End":"00:58.439","Text":"but we did have a formula that AB inverse is B inverse,"},{"Start":"00:58.439 ","End":"01:02.190","Text":"A inverse, which means that if A has an inverse,"},{"Start":"01:02.190 ","End":"01:05.319","Text":"and B has an inverse and AB does have an inverse."},{"Start":"01:05.319 ","End":"01:07.550","Text":"If it\u0027s true for AB,"},{"Start":"01:07.550 ","End":"01:10.295","Text":"I mean it\u0027s also true in general."},{"Start":"01:10.295 ","End":"01:14.670","Text":"We can generalize it to more than 2."},{"Start":"01:14.670 ","End":"01:16.550","Text":"In this case, I\u0027d like to have 4"},{"Start":"01:16.550 ","End":"01:20.880","Text":"this inverse would also be D minus 1,"},{"Start":"01:20.880 ","End":"01:23.965","Text":"C minus 1, B minus 1, A minus 1."},{"Start":"01:23.965 ","End":"01:25.580","Text":"The formula is not important."},{"Start":"01:25.580 ","End":"01:28.730","Text":"The point is that the product is still invertible,"},{"Start":"01:28.730 ","End":"01:32.060","Text":"not just of 2, but of 4, or any number."},{"Start":"01:32.060 ","End":"01:42.360","Text":"In particular, what I want to say is that if A is invertible,"},{"Start":"01:42.360 ","End":"01:48.000","Text":"then so is A times A times A times A,"},{"Start":"01:48.000 ","End":"01:50.760","Text":"which is A to the 4th."},{"Start":"01:50.760 ","End":"01:55.430","Text":"What we\u0027re doing here is basically a proof by contradiction,"},{"Start":"01:55.430 ","End":"01:57.050","Text":"and what we call in mathematics."},{"Start":"01:57.050 ","End":"01:59.290","Text":"If I had to prove that it\u0027s non-invertible,"},{"Start":"01:59.290 ","End":"02:05.390","Text":"I assume on the contrary that it is invertible and show that I get a contradiction,"},{"Start":"02:05.390 ","End":"02:08.680","Text":"and that will prove that it\u0027s non-invertible."},{"Start":"02:08.680 ","End":"02:11.560","Text":"If A is invertible,"},{"Start":"02:11.560 ","End":"02:15.650","Text":"then A to the 4th is invertible."},{"Start":"02:15.810 ","End":"02:18.295","Text":"If A is invertible,"},{"Start":"02:18.295 ","End":"02:25.420","Text":"i.e A to the 4th is invertible,"},{"Start":"02:25.970 ","End":"02:29.020","Text":"rather use an arrow."},{"Start":"02:29.300 ","End":"02:33.285","Text":"If A is invertible, A to the 4th is invertible,"},{"Start":"02:33.285 ","End":"02:40.920","Text":"which means that the matrix 0 is invertible."},{"Start":"02:40.920 ","End":"02:45.080","Text":"You might say, the matrix 0 might be invertible."},{"Start":"02:45.080 ","End":"02:46.820","Text":"Who says it isn\u0027t?"},{"Start":"02:46.820 ","End":"02:50.410","Text":"Well, it certainly is not,"},{"Start":"02:50.410 ","End":"02:53.090","Text":"and I\u0027ll show you why."},{"Start":"02:53.090 ","End":"02:55.580","Text":"There\u0027s several ways to do this."},{"Start":"02:55.580 ","End":"02:58.985","Text":"1 is just to say suppose it has an inverse."},{"Start":"02:58.985 ","End":"03:03.130","Text":"Suppose that the inverse is,"},{"Start":"03:03.130 ","End":"03:08.810","Text":"Let\u0027s call it X."},{"Start":"03:08.810 ","End":"03:10.460","Text":"For it to be an inverse,"},{"Start":"03:10.460 ","End":"03:18.920","Text":"then we must have that the matrix times its inverse or the other way around,"},{"Start":"03:18.920 ","End":"03:21.500","Text":"have got to equal the identity,"},{"Start":"03:21.500 ","End":"03:28.550","Text":"but matrix 0 times X is just 0 matrix."},{"Start":"03:28.550 ","End":"03:31.625","Text":"I\u0027m talking about all square matrices of the same size here."},{"Start":"03:31.625 ","End":"03:34.765","Text":"That means that 0 equals I,"},{"Start":"03:34.765 ","End":"03:42.425","Text":"and that\u0027s not true because this has 1s along the diagonal and 0s everywhere."},{"Start":"03:42.425 ","End":"03:45.260","Text":"If A were invertible,"},{"Start":"03:45.260 ","End":"03:48.245","Text":"then we\u0027d get that 0 is invertible,"},{"Start":"03:48.245 ","End":"03:52.415","Text":"which is not true because 0 does not equal I,"},{"Start":"03:52.415 ","End":"03:56.160","Text":"and so that proves part a."},{"Start":"03:56.900 ","End":"04:00.430","Text":"In part b,"},{"Start":"04:00.840 ","End":"04:06.070","Text":"we have to show that I minus A is invertible."},{"Start":"04:06.070 ","End":"04:09.470","Text":"Not only that to find its inverse."},{"Start":"04:13.400 ","End":"04:15.990","Text":"Just remembering that,"},{"Start":"04:15.990 ","End":"04:19.359","Text":"just write down what we had before"},{"Start":"04:19.359 ","End":"04:25.330","Text":"from the given that A to the 4th equals 0 matrix."},{"Start":"04:25.330 ","End":"04:29.335","Text":"That\u0027s just scrolled off."},{"Start":"04:29.335 ","End":"04:34.000","Text":"I\u0027m going to use a formula from Algebra"},{"Start":"04:34.000 ","End":"04:43.390","Text":"that 1 minus x to the 4th."},{"Start":"04:43.390 ","End":"04:45.750","Text":"We don\u0027t need the brackets,"},{"Start":"04:45.750 ","End":"04:48.870","Text":"is equal to 1 minus x,"},{"Start":"04:48.870 ","End":"04:55.120","Text":"1 plus x plus x squared plus x cubed."},{"Start":"04:55.120 ","End":"04:57.130","Text":"Actually, it works for any n,"},{"Start":"04:57.130 ","End":"04:58.335","Text":"not just 4,"},{"Start":"04:58.335 ","End":"05:00.540","Text":"1 minus x to the n,"},{"Start":"05:00.540 ","End":"05:02.975","Text":"is 1 minus x times 1 plus x,"},{"Start":"05:02.975 ","End":"05:06.040","Text":"and so on, up to x to the n minus 1."},{"Start":"05:06.040 ","End":"05:09.340","Text":"It works in general, we know it certainly for x equals 2,"},{"Start":"05:09.340 ","End":"05:12.480","Text":"that 1 minus x squared is 1 minus x, 1 plus x,"},{"Start":"05:12.480 ","End":"05:16.145","Text":"but this is a general formula."},{"Start":"05:16.145 ","End":"05:23.460","Text":"This would mean in our case that I minus A to"},{"Start":"05:23.460 ","End":"05:29.190","Text":"the 4th is equal to I minus A."},{"Start":"05:29.190 ","End":"05:35.660","Text":"I, numbers are replaced by the number times I,"},{"Start":"05:35.660 ","End":"05:38.390","Text":"when we go over from numbers to matrices,"},{"Start":"05:38.390 ","End":"05:44.340","Text":"plus A, plus A squared plus A cubed."},{"Start":"05:44.420 ","End":"05:47.955","Text":"A to the 4th is 0,"},{"Start":"05:47.955 ","End":"05:51.870","Text":"we\u0027ve written it here, so we\u0027ve got 0."},{"Start":"05:51.870 ","End":"05:54.800","Text":"Let me just write it the other way around,"},{"Start":"05:54.800 ","End":"05:56.645","Text":"so I\u0027ve got that,"},{"Start":"05:56.645 ","End":"06:06.810","Text":"I minus A times I plus A plus A squared plus A cubed"},{"Start":"06:06.810 ","End":"06:14.550","Text":"is equal to I, because like I said, this is 0."},{"Start":"06:15.380 ","End":"06:19.815","Text":"We want to show that I minus A is invertible,"},{"Start":"06:19.815 ","End":"06:23.745","Text":"I minus A times this something is I,"},{"Start":"06:23.745 ","End":"06:27.115","Text":"so that has to be the inverse of I minus A."},{"Start":"06:27.115 ","End":"06:31.160","Text":"We know that I minus A inverse exists"},{"Start":"06:31.160 ","End":"06:39.430","Text":"and that it\u0027s equal to I plus A plus A squared plus A cubed."},{"Start":"06:39.430 ","End":"06:44.340","Text":"That answers the question and we\u0027re done."}],"ID":14197},{"Watched":false,"Name":"Exercise 9","Duration":"2m 54s","ChapterTopicVideoID":13545,"CourseChapterTopicPlaylistID":7284,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13545.jpeg","UploadDate":"2018-09-06T10:07:34.9200000","DurationForVideoObject":"PT2M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.730","Text":"In this exercise, we\u0027re given 2 equations or identities."},{"Start":"00:11.150 ","End":"00:14.400","Text":"P^minus 1 AP is B,"},{"Start":"00:14.400 ","End":"00:16.560","Text":"Q^minus 1 BQ is C,"},{"Start":"00:16.560 ","End":"00:20.520","Text":"presumably all the matrices A, B, P,"},{"Start":"00:20.520 ","End":"00:25.830","Text":"Q, C are all square matrices, the same size."},{"Start":"00:25.830 ","End":"00:34.710","Text":"We want to show that there is a matrix D such that D^minus 1 AD is C."},{"Start":"00:34.710 ","End":"00:44.445","Text":"Now the thing to notice here is that I have a B here and a B here."},{"Start":"00:44.445 ","End":"00:49.710","Text":"I can substitute the B here as to what it equals."},{"Start":"00:49.710 ","End":"00:50.790","Text":"In other words,"},{"Start":"00:50.790 ","End":"00:57.575","Text":"the bottom 1 can be rewritten as Q inverse,"},{"Start":"00:57.575 ","End":"00:59.900","Text":"and then instead of B,"},{"Start":"00:59.900 ","End":"01:06.065","Text":"I can write P^minus 1 AP."},{"Start":"01:06.065 ","End":"01:10.985","Text":"Then, I have another Q it\u0027s equal to"},{"Start":"01:10.985 ","End":"01:17.650","Text":"C. Let me just break this up differently because I\u0027m trying to get to this form."},{"Start":"01:17.650 ","End":"01:23.270","Text":"Let me write it as first the Q^minus 1 and then the p^minus 1,"},{"Start":"01:23.270 ","End":"01:26.980","Text":"and then the A and then the PQ,"},{"Start":"01:27.530 ","End":"01:30.800","Text":"and that is equal to C. Now,"},{"Start":"01:30.800 ","End":"01:32.765","Text":"this starting to look like this,"},{"Start":"01:32.765 ","End":"01:40.130","Text":"I would like to let this be D. But if I want this to be D, what about this?"},{"Start":"01:40.130 ","End":"01:42.050","Text":"Is it equal to D^minus 1?"},{"Start":"01:42.050 ","End":"01:49.350","Text":"Well, yes, because D^minus 1 is PQ inverse."},{"Start":"01:49.350 ","End":"01:51.035","Text":"Remember when we take an inverse,"},{"Start":"01:51.035 ","End":"01:52.640","Text":"we take the inverse of each 1,"},{"Start":"01:52.640 ","End":"01:54.860","Text":"but we reverse the order."},{"Start":"01:54.860 ","End":"01:59.830","Text":"This is equal to Q^minus 1, P^minus 1."},{"Start":"01:59.830 ","End":"02:01.650","Text":"That\u0027s what we have here."},{"Start":"02:01.650 ","End":"02:04.755","Text":"If this is D, this is really D inverse,"},{"Start":"02:04.755 ","End":"02:11.910","Text":"and so we\u0027ve written that D^minus 1 AD is equal to C."},{"Start":"02:11.910 ","End":"02:22.165","Text":"If we just let D be equal to P times Q,"},{"Start":"02:22.165 ","End":"02:26.570","Text":"and that\u0027s our invertible matrix D."},{"Start":"02:26.570 ","End":"02:29.595","Text":"Why is D invertible?"},{"Start":"02:29.595 ","End":"02:33.420","Text":"Because P is invertible,"},{"Start":"02:33.420 ","End":"02:35.495","Text":"and so is Q."},{"Start":"02:35.495 ","End":"02:38.240","Text":"If we have P inverse and Q inverse,"},{"Start":"02:38.240 ","End":"02:40.490","Text":"then D is invertible."},{"Start":"02:40.490 ","End":"02:45.625","Text":"We see anyway that D^minus 1 exists."},{"Start":"02:45.625 ","End":"02:47.460","Text":"This times this is I."},{"Start":"02:47.460 ","End":"02:54.600","Text":"D is invertible and satisfies the condition and then that\u0027s it."}],"ID":14198}],"Thumbnail":null,"ID":7284},{"Name":"Elementary Matrices and LU Decomposition","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Elementary Matrices, Introduction","Duration":"5m 40s","ChapterTopicVideoID":21391,"CourseChapterTopicPlaylistID":99467,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21391.jpeg","UploadDate":"2020-04-16T10:22:37.3300000","DurationForVideoObject":"PT5M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"This clip is an introduction to the concept of an elementary matrix."},{"Start":"00:05.280 ","End":"00:07.050","Text":"But first, we need to go back"},{"Start":"00:07.050 ","End":"00:11.130","Text":"and remember what an elementary operation is."},{"Start":"00:11.130 ","End":"00:14.610","Text":"Well, elementary row operation,"},{"Start":"00:14.610 ","End":"00:17.780","Text":"because there are column operations,"},{"Start":"00:17.780 ","End":"00:21.065","Text":"but we\u0027ll just be concerned with row operations in this course."},{"Start":"00:21.065 ","End":"00:24.890","Text":"An elementary operation on a matrix is 1 of the following."},{"Start":"00:24.890 ","End":"00:31.760","Text":"It\u0027s multiplying a row by a non-zero constant or interchanging 2 rows,"},{"Start":"00:31.760 ","End":"00:32.960","Text":"swapping them around,"},{"Start":"00:32.960 ","End":"00:37.305","Text":"or adding a multiple of 1 row to another row."},{"Start":"00:37.305 ","End":"00:39.255","Text":"Of course, in column operations,"},{"Start":"00:39.255 ","End":"00:41.210","Text":"you just change the word row with column,"},{"Start":"00:41.210 ","End":"00:43.550","Text":"but we won\u0027t be concerned with those."},{"Start":"00:43.550 ","End":"00:47.165","Text":"Now that we remember what an elementary operation is,"},{"Start":"00:47.165 ","End":"00:49.865","Text":"the elementary matrix is 1"},{"Start":"00:49.865 ","End":"00:54.410","Text":"which is obtained from the identity matrix"},{"Start":"00:54.410 ","End":"00:58.370","Text":"by performing an elementary operation on it."},{"Start":"00:58.370 ","End":"01:01.385","Text":"We\u0027ll give some examples."},{"Start":"01:01.385 ","End":"01:05.960","Text":"First example, take the identity matrix."},{"Start":"01:05.960 ","End":"01:11.885","Text":"The elementary row operation can be multiplying the first row by 2."},{"Start":"01:11.885 ","End":"01:13.790","Text":"Write that as follows,"},{"Start":"01:13.790 ","End":"01:17.510","Text":"twice the first row into the first row."},{"Start":"01:17.510 ","End":"01:21.060","Text":"Then just double the first row."},{"Start":"01:21.060 ","End":"01:21.915","Text":"We get this."},{"Start":"01:21.915 ","End":"01:24.320","Text":"This is an elementary matrix corresponding"},{"Start":"01:24.320 ","End":"01:26.840","Text":"to this elementary operation."},{"Start":"01:26.840 ","End":"01:29.900","Text":"Next example, the identity matrix."},{"Start":"01:29.900 ","End":"01:35.160","Text":"This time, this is swapping Row 2 and Row 3."},{"Start":"01:35.160 ","End":"01:39.860","Text":"We end up with this just the third row here, second row here."},{"Start":"01:39.860 ","End":"01:41.975","Text":"Another example."},{"Start":"01:41.975 ","End":"01:45.800","Text":"Start with the identity matrix and their operation will be,"},{"Start":"01:45.800 ","End":"01:52.805","Text":"subtract 3 times Row 2 from Row 1 and put it into Row 1."},{"Start":"01:52.805 ","End":"01:58.040","Text":"We get this minus 3 times this into here."},{"Start":"01:58.040 ","End":"02:00.875","Text":"It just gives us a minus 3 here."},{"Start":"02:00.875 ","End":"02:09.005","Text":"Next example, their operation is adding twice the second row to the first row."},{"Start":"02:09.005 ","End":"02:14.795","Text":"We start with the identity matrix and then add twice this to this,"},{"Start":"02:14.795 ","End":"02:16.010","Text":"and we get this."},{"Start":"02:16.010 ","End":"02:17.630","Text":"1 more example."},{"Start":"02:17.630 ","End":"02:20.880","Text":"Let\u0027s take 4 by 4."},{"Start":"02:20.930 ","End":"02:24.170","Text":"Here we have the identity matrix."},{"Start":"02:24.170 ","End":"02:29.100","Text":"Their operation would be swapping Row 2 with Row 4."},{"Start":"02:29.600 ","End":"02:33.650","Text":"This row goes down here and this row up here,"},{"Start":"02:33.650 ","End":"02:35.390","Text":"and this is what we get."},{"Start":"02:35.390 ","End":"02:40.515","Text":"Enough for the examples, now the theorem."},{"Start":"02:40.515 ","End":"02:43.465","Text":"Let\u0027s say we have a matrix A."},{"Start":"02:43.465 ","End":"02:49.070","Text":"Then the result of applying an elementary row operation to A"},{"Start":"02:49.070 ","End":"02:52.970","Text":"is the same as multiplying A on the left"},{"Start":"02:52.970 ","End":"02:56.870","Text":"by the appropriate elementary matrix."},{"Start":"02:56.870 ","End":"02:59.990","Text":"I\u0027ll give you some examples because this may"},{"Start":"02:59.990 ","End":"03:02.360","Text":"seem a bit abstract before the examples"},{"Start":"03:02.360 ","End":"03:06.620","Text":"I\u0027ll just mention that there is another theorem"},{"Start":"03:06.620 ","End":"03:08.480","Text":"or part of the same theorem that"},{"Start":"03:08.480 ","End":"03:11.750","Text":"if you replace the word row with column,"},{"Start":"03:11.750 ","End":"03:13.835","Text":"so we\u0027re talking about column operations,"},{"Start":"03:13.835 ","End":"03:20.405","Text":"then you multiply A on the right by the appropriate elementary column matrix."},{"Start":"03:20.405 ","End":"03:23.375","Text":"Anyway, we\u0027d just be concerned with row matrices,"},{"Start":"03:23.375 ","End":"03:25.495","Text":"but I thought I\u0027d mention it."},{"Start":"03:25.495 ","End":"03:27.735","Text":"Now some examples."},{"Start":"03:27.735 ","End":"03:31.250","Text":"Let\u0027s say we get from this matrix to"},{"Start":"03:31.250 ","End":"03:36.105","Text":"this matrix using the following elementary row operation."},{"Start":"03:36.105 ","End":"03:39.370","Text":"We multiply Row 1 by 4."},{"Start":"03:39.370 ","End":"03:40.980","Text":"That\u0027s basically what it says here."},{"Start":"03:40.980 ","End":"03:44.320","Text":"Instead of 1, 2, 3, we have 4, 8, 12."},{"Start":"03:44.320 ","End":"03:46.480","Text":"We take the row operation"},{"Start":"03:46.480 ","End":"03:51.730","Text":"and find the corresponding elementary matrix,"},{"Start":"03:51.730 ","End":"03:55.705","Text":"which means that we apply this to the identity matrix."},{"Start":"03:55.705 ","End":"03:57.805","Text":"This is our elementary matrix."},{"Start":"03:57.805 ","End":"04:02.020","Text":"We multiplied on the left by this matrix here,"},{"Start":"04:02.020 ","End":"04:03.820","Text":"we get the 1 here."},{"Start":"04:03.820 ","End":"04:08.709","Text":"I\u0027ll leave you to check the multiplication that it actually works."},{"Start":"04:08.709 ","End":"04:11.740","Text":"Now another example."},{"Start":"04:11.740 ","End":"04:20.845","Text":"Suppose that we swap Row 1 with Row 3 on this matrix to get this matrix."},{"Start":"04:20.845 ","End":"04:25.015","Text":"We just get the 7, 8, 9 here and the 1, 2, 3 down here."},{"Start":"04:25.015 ","End":"04:28.010","Text":"Now, swapping Row 1 with Row 3."},{"Start":"04:28.010 ","End":"04:30.470","Text":"If we do that to the identity matrix,"},{"Start":"04:30.470 ","End":"04:33.725","Text":"we get this and this is our elementary matrix,"},{"Start":"04:33.725 ","End":"04:37.150","Text":"multiply it by this to get this."},{"Start":"04:37.150 ","End":"04:45.665","Text":"This elementary matrix times this matrix here gives us this here."},{"Start":"04:45.665 ","End":"04:51.230","Text":"Once again, a row operation is equivalent to multiplication"},{"Start":"04:51.230 ","End":"04:54.640","Text":"by an elementary matrix on the left."},{"Start":"04:54.640 ","End":"04:59.120","Text":"Another example, we get from here to here"},{"Start":"04:59.120 ","End":"05:04.685","Text":"by subtracting 3 times Row 1 from Row 2."},{"Start":"05:04.685 ","End":"05:07.805","Text":"Take this multiply by 3, that\u0027s 36."},{"Start":"05:07.805 ","End":"05:09.725","Text":"Subtract 36 from here,"},{"Start":"05:09.725 ","End":"05:15.305","Text":"we get 0 minus 2 is going to be equivalent to a matrix multiplication."},{"Start":"05:15.305 ","End":"05:19.145","Text":"Take this, get the elementary matrix,"},{"Start":"05:19.145 ","End":"05:23.090","Text":"which is what you get if you apply this to the identity,"},{"Start":"05:23.090 ","End":"05:27.095","Text":"subtract 3 times 1, 0 from 0, 1, we get this."},{"Start":"05:27.095 ","End":"05:29.840","Text":"Now, multiply this by this to get this."},{"Start":"05:29.840 ","End":"05:35.050","Text":"I\u0027ll leave you to check that this actually works."},{"Start":"05:35.180 ","End":"05:40.320","Text":"With this example, we\u0027ll conclude this clip."}],"ID":22254},{"Watched":false,"Name":"Elementary Matrices, Theorem","Duration":"6m 22s","ChapterTopicVideoID":21392,"CourseChapterTopicPlaylistID":99467,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21392.jpeg","UploadDate":"2020-04-16T10:23:42.6830000","DurationForVideoObject":"PT6M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.785","Text":"In the previous clip,"},{"Start":"00:01.785 ","End":"00:04.650","Text":"we talked about elementary matrices,"},{"Start":"00:04.650 ","End":"00:07.785","Text":"or to be precise, elementary row matrices."},{"Start":"00:07.785 ","End":"00:11.415","Text":"Now here\u0027s the main theorem concerning these,"},{"Start":"00:11.415 ","End":"00:15.705","Text":"and it says that every invertible that\u0027s important"},{"Start":"00:15.705 ","End":"00:22.275","Text":"square matrix can be written as the product of elementary matrices."},{"Start":"00:22.275 ","End":"00:24.900","Text":"I\u0027m not going to prove this in general,"},{"Start":"00:24.900 ","End":"00:26.430","Text":"we\u0027ll prove it through an example."},{"Start":"00:26.430 ","End":"00:30.540","Text":"It\u0027s such a typical example that in general it will work the same way."},{"Start":"00:30.540 ","End":"00:35.640","Text":"It\u0027s actually a constructive proof for these constructive example,"},{"Start":"00:35.640 ","End":"00:40.910","Text":"how exactly we get matrix A as a product of elementary matrices."},{"Start":"00:40.910 ","End":"00:42.485","Text":"I will start with this example,"},{"Start":"00:42.485 ","End":"00:43.910","Text":"a 3 by 3,"},{"Start":"00:43.910 ","End":"00:45.485","Text":"where A is this,"},{"Start":"00:45.485 ","End":"00:50.310","Text":"and we want to write it as the product of elementary matrices. We do it in steps."},{"Start":"00:50.310 ","End":"00:52.550","Text":"The first step is to apply"},{"Start":"00:52.550 ","End":"00:58.130","Text":"elementary row operations to A until you get to the identity matrix,"},{"Start":"00:58.130 ","End":"01:00.335","Text":"and I\u0027ll show you the steps."},{"Start":"01:00.335 ","End":"01:02.515","Text":"Here\u0027s the starting point,"},{"Start":"01:02.515 ","End":"01:05.240","Text":"now what we want to do is subtract multiples of"},{"Start":"01:05.240 ","End":"01:09.635","Text":"the first row from the second and the third."},{"Start":"01:09.635 ","End":"01:13.565","Text":"First of all, we\u0027ll take away 4 times this from this,"},{"Start":"01:13.565 ","End":"01:15.745","Text":"and we end up with this."},{"Start":"01:15.745 ","End":"01:20.120","Text":"Then we want to take twice this from this,"},{"Start":"01:20.120 ","End":"01:22.700","Text":"and we end up with this."},{"Start":"01:22.700 ","End":"01:26.395","Text":"Next thing you want to do is get a 0 here,"},{"Start":"01:26.395 ","End":"01:33.350","Text":"so if we just add the second and the third rows and put the result in the third row,"},{"Start":"01:33.350 ","End":"01:36.330","Text":"then we\u0027ve got a 0 here."},{"Start":"01:37.400 ","End":"01:45.235","Text":"Now we want to get rid of this 2 here by subtracting the third row from the first row."},{"Start":"01:45.235 ","End":"01:47.250","Text":"It\u0027s almost the identity matrix,"},{"Start":"01:47.250 ","End":"01:48.830","Text":"we just have to adjust the diagonal,"},{"Start":"01:48.830 ","End":"01:53.465","Text":"multiply this by minus 1 and this by 1/2."},{"Start":"01:53.465 ","End":"01:56.635","Text":"Multiplying by 1/2,"},{"Start":"01:56.635 ","End":"02:03.590","Text":"the third row we get this and negating the second row we get this."},{"Start":"02:03.590 ","End":"02:04.880","Text":"Sorry, there\u0027s a typo,"},{"Start":"02:04.880 ","End":"02:09.335","Text":"it should be minus row 2 into row 2."},{"Start":"02:09.335 ","End":"02:12.920","Text":"That\u0027s obvious. That\u0027s the end of the first step."},{"Start":"02:12.920 ","End":"02:14.495","Text":"In the second step,"},{"Start":"02:14.495 ","End":"02:18.140","Text":"we make a list of all these row operations that we performed,"},{"Start":"02:18.140 ","End":"02:19.520","Text":"so we just copy them."},{"Start":"02:19.520 ","End":"02:21.980","Text":"This is 1, 2, 3, 4,"},{"Start":"02:21.980 ","End":"02:26.395","Text":"5, and 6, like so."},{"Start":"02:26.395 ","End":"02:32.790","Text":"Then step 3 is to alter these 6 according to certain rules."},{"Start":"02:32.790 ","End":"02:36.725","Text":"What we\u0027re actually going to do is take the inverse of each of them."},{"Start":"02:36.725 ","End":"02:40.370","Text":"Well, you\u0027ll see instead of like here,"},{"Start":"02:40.370 ","End":"02:42.980","Text":"subtracting 4 times row 1 from row 2,"},{"Start":"02:42.980 ","End":"02:46.210","Text":"we\u0027d be adding 4 times row 1 to row 2."},{"Start":"02:46.210 ","End":"02:52.974","Text":"In general, if we have k times row j plus or minus,"},{"Start":"02:52.974 ","End":"02:58.230","Text":"and then add it to our i and put it into our i,"},{"Start":"02:58.230 ","End":"03:00.650","Text":"we just reverse the plus or minus."},{"Start":"03:00.650 ","End":"03:01.970","Text":"If it was plus it\u0027s minus,"},{"Start":"03:01.970 ","End":"03:05.060","Text":"and if it was minus it\u0027s plus, that\u0027s one rule."},{"Start":"03:05.060 ","End":"03:13.010","Text":"The second rule is that if we multiply a row by k and k is non-zero,"},{"Start":"03:13.010 ","End":"03:19.230","Text":"that\u0027s part of the elementary row operations multiplication by non-zero, then we divide."},{"Start":"03:19.230 ","End":"03:21.080","Text":"Here we multiply by 3,"},{"Start":"03:21.080 ","End":"03:23.495","Text":"here we divide by 3."},{"Start":"03:23.495 ","End":"03:28.040","Text":"The last one basically is unchanged."},{"Start":"03:28.040 ","End":"03:29.944","Text":"If we swap 2 rows,"},{"Start":"03:29.944 ","End":"03:31.445","Text":"we swap them back,"},{"Start":"03:31.445 ","End":"03:33.695","Text":"which is just like swapping,"},{"Start":"03:33.695 ","End":"03:36.590","Text":"so you don\u0027t change swaps."},{"Start":"03:36.590 ","End":"03:42.690","Text":"What happens for these 6 rules is that we get 6 new roles as follows,"},{"Start":"03:42.690 ","End":"03:44.160","Text":"like instead of a minus 4,"},{"Start":"03:44.160 ","End":"03:45.780","Text":"we have a plus 4,"},{"Start":"03:45.780 ","End":"03:49.410","Text":"instead of a minus 2 here we have a plus 2."},{"Start":"03:49.410 ","End":"03:54.090","Text":"Like here, we took 1/2 of row 3,"},{"Start":"03:54.090 ","End":"03:58.440","Text":"so the reciprocal of a 1/5 is 2, that goes here."},{"Start":"03:58.440 ","End":"04:02.720","Text":"Here the reciprocal of minus is minus,"},{"Start":"04:02.720 ","End":"04:05.380","Text":"1 over minus 1 is minus 1."},{"Start":"04:05.380 ","End":"04:10.475","Text":"I don\u0027t think we had a swapping of 2 rows here to illustrate,"},{"Start":"04:10.475 ","End":"04:13.115","Text":"but you would leave those unchanged."},{"Start":"04:13.115 ","End":"04:15.145","Text":"That was step 3."},{"Start":"04:15.145 ","End":"04:17.050","Text":"Now in step 4,"},{"Start":"04:17.050 ","End":"04:23.885","Text":"what we do is we take the elementary matrix for each row operation."},{"Start":"04:23.885 ","End":"04:26.480","Text":"We learned in the previous clip that corresponding"},{"Start":"04:26.480 ","End":"04:29.060","Text":"to every row operation there is a matrix,"},{"Start":"04:29.060 ","End":"04:32.350","Text":"which if you multiply on the left has the same effect."},{"Start":"04:32.350 ","End":"04:34.720","Text":"Let\u0027s first remember what these were,"},{"Start":"04:34.720 ","End":"04:37.925","Text":"I just copy pasted these 6,"},{"Start":"04:37.925 ","End":"04:44.030","Text":"and now this row operation corresponds to this elementary matrix."},{"Start":"04:44.030 ","End":"04:47.795","Text":"We add 4 times the first row to the second row."},{"Start":"04:47.795 ","End":"04:49.595","Text":"So that corresponds to this."},{"Start":"04:49.595 ","End":"04:54.020","Text":"Here, we add twice the first row to the third row,"},{"Start":"04:54.020 ","End":"04:57.175","Text":"so we get this 2 here,"},{"Start":"04:57.175 ","End":"04:59.535","Text":"and so on 4,"},{"Start":"04:59.535 ","End":"05:01.110","Text":"5, and 6."},{"Start":"05:01.110 ","End":"05:04.380","Text":"The last one is just inverting row 2,"},{"Start":"05:04.380 ","End":"05:06.390","Text":"putting minus row 2 into row 2,"},{"Start":"05:06.390 ","End":"05:09.285","Text":"so we have the 6 matrices."},{"Start":"05:09.285 ","End":"05:13.790","Text":"Lastly, you could call this step 5 or part of step 4."},{"Start":"05:13.790 ","End":"05:16.205","Text":"Just multiply them in this order,"},{"Start":"05:16.205 ","End":"05:18.560","Text":"E_1, E_2, E_3 however many there are,"},{"Start":"05:18.560 ","End":"05:23.895","Text":"multiply them together, that will give us our result. Let\u0027s check."},{"Start":"05:23.895 ","End":"05:26.120","Text":"You could believe me that this times this times this times"},{"Start":"05:26.120 ","End":"05:28.400","Text":"this times this is our original A."},{"Start":"05:28.400 ","End":"05:32.975","Text":"Let\u0027s check it. It\u0027s a bit of calculation here."},{"Start":"05:32.975 ","End":"05:36.325","Text":"Here I\u0027ve written E_6,"},{"Start":"05:36.325 ","End":"05:37.920","Text":"here\u0027s E_5, E_4,"},{"Start":"05:37.920 ","End":"05:39.510","Text":"E_3, E_2, E_1."},{"Start":"05:39.510 ","End":"05:43.320","Text":"Each time I multiply this by this on the left,"},{"Start":"05:43.320 ","End":"05:44.670","Text":"the E_5 is on the left,"},{"Start":"05:44.670 ","End":"05:46.230","Text":"we get E_5, E_6."},{"Start":"05:46.230 ","End":"05:48.975","Text":"E_4 times this is this."},{"Start":"05:48.975 ","End":"05:52.365","Text":"E_3 times this is this."},{"Start":"05:52.365 ","End":"05:55.260","Text":"This times this is this."},{"Start":"05:55.260 ","End":"05:58.860","Text":"Finally, this times this is this,"},{"Start":"05:58.860 ","End":"06:00.780","Text":"and I\u0027ll leave you to do the checking,"},{"Start":"06:00.780 ","End":"06:02.430","Text":"do it in that detail."},{"Start":"06:02.430 ","End":"06:05.630","Text":"Finally, we get this, which is correct."},{"Start":"06:05.630 ","End":"06:07.835","Text":"It is our matrix A."},{"Start":"06:07.835 ","End":"06:09.920","Text":"If you go back, you rewind and look,"},{"Start":"06:09.920 ","End":"06:13.550","Text":"this was our original 3 by 3 matrix."},{"Start":"06:13.550 ","End":"06:16.174","Text":"This technique works in general,"},{"Start":"06:16.174 ","End":"06:22.590","Text":"any invertible matrix can be broken up this way. We\u0027re done."}],"ID":22255},{"Watched":false,"Name":"LU Decomposition","Duration":"8m 30s","ChapterTopicVideoID":21393,"CourseChapterTopicPlaylistID":99467,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21393.jpeg","UploadDate":"2020-04-16T10:25:07.6300000","DurationForVideoObject":"PT8M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.995","Text":"In this clip, we\u0027ll learn about LU decomposition of a square matrix."},{"Start":"00:06.995 ","End":"00:10.045","Text":"If we have a square matrix A,"},{"Start":"00:10.045 ","End":"00:16.365","Text":"an LU decomposition of A means writing A as the product of 2 square matrices."},{"Start":"00:16.365 ","End":"00:18.720","Text":"L, which is lower triangular,"},{"Start":"00:18.720 ","End":"00:21.285","Text":"and U, which is upper triangular."},{"Start":"00:21.285 ","End":"00:25.530","Text":"L stands for lower and U for upper."},{"Start":"00:25.530 ","End":"00:29.385","Text":"In a 3-by-3 case, just to illustrate,"},{"Start":"00:29.385 ","End":"00:36.165","Text":"A would be any 3-by-3 matrix and L is lower triangular,"},{"Start":"00:36.165 ","End":"00:42.120","Text":"meaning that everything above the main diagonal is 0."},{"Start":"00:42.120 ","End":"00:44.810","Text":"U is an upper triangular matrix,"},{"Start":"00:44.810 ","End":"00:50.185","Text":"which means everything below the main diagonal is 0."},{"Start":"00:50.185 ","End":"00:52.445","Text":"Here\u0027s an actual example."},{"Start":"00:52.445 ","End":"00:55.280","Text":"A is this matrix."},{"Start":"00:55.280 ","End":"01:00.080","Text":"You can check that it\u0027s equal to L times U, like here."},{"Start":"01:00.080 ","End":"01:07.485","Text":"L really is lower diagonal and U is upper diagonal."},{"Start":"01:07.485 ","End":"01:13.250","Text":"By the way, not every square matrix has an LU decomposition."},{"Start":"01:13.250 ","End":"01:17.675","Text":"The way of saying that is not every square matrix is LU decomposable."},{"Start":"01:17.675 ","End":"01:20.030","Text":"Just another way of phrasing it."},{"Start":"01:20.030 ","End":"01:25.610","Text":"Next, a proposition relating to a sufficient condition."},{"Start":"01:25.610 ","End":"01:28.580","Text":"When a matrix is LU decomposable."},{"Start":"01:28.580 ","End":"01:31.850","Text":"If square matrix A can be brought to"},{"Start":"01:31.850 ","End":"01:37.340","Text":"an upper triangular matrix using only row operations of the type."},{"Start":"01:37.340 ","End":"01:44.660","Text":"What\u0027s written here, which is that you add a multiple of row j to row i."},{"Start":"01:44.660 ","End":"01:48.155","Text":"Requirement is that j is less than I."},{"Start":"01:48.155 ","End":"01:51.395","Text":"You add a multiple of a row that\u0027s above it."},{"Start":"01:51.395 ","End":"01:54.065","Text":"If you can do this,"},{"Start":"01:54.065 ","End":"01:56.360","Text":"then it\u0027s guaranteed that A is"},{"Start":"01:56.360 ","End":"02:00.890","Text":"LU decomposable and it won\u0027t prove the proposition but it\u0027ll give you"},{"Start":"02:00.890 ","End":"02:04.250","Text":"an example that\u0027s so typical that it\u0027s just like a proof as"},{"Start":"02:04.250 ","End":"02:08.555","Text":"a method to actually construct the LU decomposition."},{"Start":"02:08.555 ","End":"02:13.220","Text":"Once you\u0027ve shown that you can use row operations of the above to get it"},{"Start":"02:13.220 ","End":"02:18.185","Text":"into upper triangular form and it will take this matrix as an example."},{"Start":"02:18.185 ","End":"02:25.415","Text":"Step 1 is to apply row operations to A until we get the upper triangular matrix U."},{"Start":"02:25.415 ","End":"02:28.445","Text":"This is the matrix from here."},{"Start":"02:28.445 ","End":"02:35.690","Text":"The first step is we\u0027ll subtract 4 times the first row from the second row."},{"Start":"02:35.690 ","End":"02:37.700","Text":"That\u0027s written this way,"},{"Start":"02:37.700 ","End":"02:39.919","Text":"and that will give us this."},{"Start":"02:39.919 ","End":"02:46.165","Text":"Next thing we\u0027ll do is to take away twice the first row from the third row."},{"Start":"02:46.165 ","End":"02:52.235","Text":"That\u0027s this. In each case I\u0027m subtracting a row that\u0027s above it, multiple of it."},{"Start":"02:52.235 ","End":"02:59.990","Text":"Next, we\u0027ll take the third row and add it to this row."},{"Start":"02:59.990 ","End":"03:02.455","Text":"We get this."},{"Start":"03:02.455 ","End":"03:07.715","Text":"This turns out to be the U from the LU decomposition."},{"Start":"03:07.715 ","End":"03:10.115","Text":"Now, how do we get the L part?"},{"Start":"03:10.115 ","End":"03:12.350","Text":"Well, that\u0027s step 2."},{"Start":"03:12.350 ","End":"03:14.930","Text":"Make a list of all the row operations you\u0027ve performed."},{"Start":"03:14.930 ","End":"03:17.400","Text":"That\u0027s 1, 2 and 3."},{"Start":"03:17.510 ","End":"03:20.445","Text":"Just copy these 3 here."},{"Start":"03:20.445 ","End":"03:22.920","Text":"I\u0027m noticing the k,"},{"Start":"03:22.920 ","End":"03:25.320","Text":"I highlighted them, those are important."},{"Start":"03:25.320 ","End":"03:30.665","Text":"Step 3 is to make a new list by altering the row operations as follows."},{"Start":"03:30.665 ","End":"03:35.720","Text":"Instead of row i plus or minus k row j,"},{"Start":"03:35.720 ","End":"03:38.495","Text":"we write the opposite minus or plus,"},{"Start":"03:38.495 ","End":"03:42.020","Text":"as long as you negate k. In our case instead of minus 4,"},{"Start":"03:42.020 ","End":"03:43.040","Text":"we have plus 4."},{"Start":"03:43.040 ","End":"03:46.680","Text":"Minus 2 is plus 2 and plus 1 is minus 1."},{"Start":"03:46.680 ","End":"03:48.165","Text":"We don\u0027t really need the 1."},{"Start":"03:48.165 ","End":"03:51.360","Text":"Here are our 3-row operations."},{"Start":"03:51.360 ","End":"03:56.725","Text":"Now, step 4 is to build L from this as follows."},{"Start":"03:56.725 ","End":"04:01.085","Text":"We start by taking an identity matrix,"},{"Start":"04:01.085 ","End":"04:06.125","Text":"but replacing below the diagonal with these symbols,"},{"Start":"04:06.125 ","End":"04:07.230","Text":"not 1, the number,"},{"Start":"04:07.230 ","End":"04:09.180","Text":"just the step numbers."},{"Start":"04:09.180 ","End":"04:12.240","Text":"Step number 1 goes in row 2,"},{"Start":"04:12.240 ","End":"04:14.675","Text":"column 1 according to these indices,"},{"Start":"04:14.675 ","End":"04:16.340","Text":"row 2, column 1."},{"Start":"04:16.340 ","End":"04:21.780","Text":"Then step number 2 goes into row 3,"},{"Start":"04:21.780 ","End":"04:24.315","Text":"column 1, so that\u0027s the 2 here."},{"Start":"04:24.315 ","End":"04:28.850","Text":"Then step 3 is indicated in row 3,"},{"Start":"04:28.850 ","End":"04:31.910","Text":"column 2, so that\u0027s here, that\u0027s the 3."},{"Start":"04:31.910 ","End":"04:33.725","Text":"Now, we interpret this."},{"Start":"04:33.725 ","End":"04:39.690","Text":"Well, I\u0027ll show you. This 1 here has a k of plus 4."},{"Start":"04:39.690 ","End":"04:41.805","Text":"We put a plus 4 here."},{"Start":"04:41.805 ","End":"04:45.450","Text":"This 2 corresponds to a 2 here,"},{"Start":"04:45.450 ","End":"04:47.325","Text":"so we put a 2 here."},{"Start":"04:47.325 ","End":"04:50.760","Text":"This 3 corresponds to a minus 1,"},{"Start":"04:50.760 ","End":"04:52.980","Text":"so we put a minus 1 here."},{"Start":"04:52.980 ","End":"04:55.290","Text":"This is exactly the L we need."},{"Start":"04:55.290 ","End":"04:58.830","Text":"We already found U, which was here."},{"Start":"04:58.830 ","End":"05:00.315","Text":"We just write the answer."},{"Start":"05:00.315 ","End":"05:05.600","Text":"The A that we were given is equal to the L from here and the U from above."},{"Start":"05:05.600 ","End":"05:10.985","Text":"That\u0027s it. What is this LU decomposition good for?"},{"Start":"05:10.985 ","End":"05:15.920","Text":"Well, 1 use is in solving systems of linear equations."},{"Start":"05:15.920 ","End":"05:22.415","Text":"Here\u0027s an example, a system of 3 equations in 3 unknowns over the reals."},{"Start":"05:22.415 ","End":"05:26.420","Text":"We can write this in matrix form as follows."},{"Start":"05:26.420 ","End":"05:31.010","Text":"Now the idea is to decompose A as L times U,"},{"Start":"05:31.010 ","End":"05:33.590","Text":"but this is exactly the example we just saw."},{"Start":"05:33.590 ","End":"05:36.140","Text":"We have this decomposition already."},{"Start":"05:36.140 ","End":"05:39.895","Text":"Here\u0027s L and here\u0027s U copied from above."},{"Start":"05:39.895 ","End":"05:48.205","Text":"What we\u0027re going to do first is replace this that\u0027s colored by y_1, y_2, y_3."},{"Start":"05:48.205 ","End":"05:50.235","Text":"We\u0027ll solve for y,"},{"Start":"05:50.235 ","End":"05:55.575","Text":"and then we\u0027ll solve that Ux equals y at the end."},{"Start":"05:55.575 ","End":"06:00.345","Text":"Let\u0027s write this as y_1, y_2, y_3."},{"Start":"06:00.345 ","End":"06:03.045","Text":"We have this system of linear equations."},{"Start":"06:03.045 ","End":"06:08.210","Text":"Because this is a lower triangular matrix,"},{"Start":"06:08.210 ","End":"06:09.964","Text":"it\u0027ll be easy to solve."},{"Start":"06:09.964 ","End":"06:12.545","Text":"Write it in equation form."},{"Start":"06:12.545 ","End":"06:13.970","Text":"We have this."},{"Start":"06:13.970 ","End":"06:17.065","Text":"From here we see that y_1 is 1."},{"Start":"06:17.065 ","End":"06:19.380","Text":"Substitute y_1 into here,"},{"Start":"06:19.380 ","End":"06:23.480","Text":"and we get that y_2 is minus 4."},{"Start":"06:23.480 ","End":"06:27.390","Text":"Then substitute both y_1 and y_2 into here,"},{"Start":"06:27.390 ","End":"06:31.665","Text":"and we get that y_3 is 0."},{"Start":"06:31.665 ","End":"06:33.450","Text":"We found y,"},{"Start":"06:33.450 ","End":"06:36.090","Text":"and we have this,"},{"Start":"06:36.090 ","End":"06:39.830","Text":"that U times x equals y as here."},{"Start":"06:39.830 ","End":"06:42.125","Text":"Now we have another system to solve,"},{"Start":"06:42.125 ","End":"06:46.565","Text":"this time with an upper triangular matrix to get the x\u0027s."},{"Start":"06:46.565 ","End":"06:51.145","Text":"We write this in equation form."},{"Start":"06:51.145 ","End":"06:55.010","Text":"Here we solve it from the bottom row upward."},{"Start":"06:55.010 ","End":"06:57.594","Text":"We have x_3 first is 0,"},{"Start":"06:57.594 ","End":"07:00.295","Text":"then x_2 is 4,"},{"Start":"07:00.295 ","End":"07:02.000","Text":"then substitute in here,"},{"Start":"07:02.000 ","End":"07:06.655","Text":"and we get that x_1 is equal to 1."},{"Start":"07:06.655 ","End":"07:08.985","Text":"That solves the system."},{"Start":"07:08.985 ","End":"07:13.430","Text":"Now another example with the same matrix,"},{"Start":"07:13.430 ","End":"07:17.880","Text":"just different vector column b. I want you to"},{"Start":"07:17.880 ","End":"07:22.990","Text":"note how quickly we solve this example now that we\u0027ve done 1."},{"Start":"07:22.990 ","End":"07:28.435","Text":"In fact, the LU decomposition is particularly helpful when we have several systems"},{"Start":"07:28.435 ","End":"07:33.670","Text":"with the same matrix because it may take a little bit of time to do the LU decomposition,"},{"Start":"07:33.670 ","End":"07:36.590","Text":"but once we have it, it goes very quickly observe."},{"Start":"07:36.590 ","End":"07:41.740","Text":"Just like before, we replace the matrix A by LU,"},{"Start":"07:41.740 ","End":"07:46.360","Text":"we let U times x represent a column vector y."},{"Start":"07:46.360 ","End":"07:50.680","Text":"Then we solve L times y equals b."},{"Start":"07:50.680 ","End":"07:53.190","Text":"This is a lower triangular matrix,"},{"Start":"07:53.190 ","End":"07:56.200","Text":"so we solve the system from the top-down."},{"Start":"07:56.200 ","End":"07:57.610","Text":"y_1 is 7."},{"Start":"07:57.610 ","End":"08:00.400","Text":"Substitute here we get y_2 is minus 2."},{"Start":"08:00.400 ","End":"08:02.680","Text":"Substitute y_1 and y_2 here,"},{"Start":"08:02.680 ","End":"08:05.015","Text":"and we\u0027ve got y_3 is 6."},{"Start":"08:05.015 ","End":"08:09.200","Text":"Now that we have y, we can solve for x"},{"Start":"08:09.200 ","End":"08:13.760","Text":"using this and this 1 we should write the equations,"},{"Start":"08:13.760 ","End":"08:16.010","Text":"solve them from the bottom up."},{"Start":"08:16.010 ","End":"08:21.115","Text":"x_3 is 3, 6 over 2. x_2 is plus 2."},{"Start":"08:21.115 ","End":"08:23.490","Text":"Plug in x_3 here,"},{"Start":"08:23.490 ","End":"08:28.785","Text":"x_1 plus 6 is 7. x_1 is equal to 1, and that\u0027s it."},{"Start":"08:28.785 ","End":"08:31.090","Text":"We\u0027re done with this clip."}],"ID":22256},{"Watched":false,"Name":"Exercise LU1","Duration":"2m 39s","ChapterTopicVideoID":21394,"CourseChapterTopicPlaylistID":99467,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21394.jpeg","UploadDate":"2020-04-16T10:25:36.7100000","DurationForVideoObject":"PT2M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"In this exercise, we have a 3 by 3 matrix A"},{"Start":"00:03.690 ","End":"00:06.795","Text":"and we want to compute its LU decomposition."},{"Start":"00:06.795 ","End":"00:09.450","Text":"We follow the steps as in the tutorial."},{"Start":"00:09.450 ","End":"00:13.080","Text":"The first step is to apply row operations"},{"Start":"00:13.080 ","End":"00:18.220","Text":"to the matrix A until we get an upper triangular matrix U."},{"Start":"00:18.410 ","End":"00:23.160","Text":"First of all, let\u0027s subtract 3 times the first row from"},{"Start":"00:23.160 ","End":"00:28.095","Text":"the second row and that gives us this and we have a 0 here."},{"Start":"00:28.095 ","End":"00:36.720","Text":"Next, we want a 0 here so subtract twice the first row from the third row, like so"},{"Start":"00:36.720 ","End":"00:39.410","Text":"and that gives us a 0 here."},{"Start":"00:39.410 ","End":"00:44.660","Text":"Now we want a 0 here so subtract the second row"},{"Start":"00:44.660 ","End":"00:47.960","Text":"from the third row and that gives us this,"},{"Start":"00:47.960 ","End":"00:51.320","Text":"and now we have an upper triangular matrix U."},{"Start":"00:51.320 ","End":"00:57.095","Text":"The next step is just to make a list of these row operations."},{"Start":"00:57.095 ","End":"00:59.240","Text":"Write them like this."},{"Start":"00:59.240 ","End":"01:01.640","Text":"Instead of R_3 minus R_2,"},{"Start":"01:01.640 ","End":"01:04.550","Text":"you can think of it as R_3 minus 1R_2,"},{"Start":"01:04.550 ","End":"01:06.505","Text":"so as not to leave a blank there."},{"Start":"01:06.505 ","End":"01:09.100","Text":"Then in Step 3,"},{"Start":"01:09.100 ","End":"01:11.580","Text":"we want to invert these operations."},{"Start":"01:11.580 ","End":"01:13.520","Text":"Instead of plus or minus,"},{"Start":"01:13.520 ","End":"01:16.660","Text":"we write minus or plus respectively."},{"Start":"01:16.660 ","End":"01:21.240","Text":"What we have now is instead of a minus 3 here, a plus 3."},{"Start":"01:21.240 ","End":"01:22.530","Text":"Instead of minus 2, plus 2."},{"Start":"01:22.530 ","End":"01:24.645","Text":"Instead of minus 1, plus 1."},{"Start":"01:24.645 ","End":"01:26.870","Text":"The last step, Step 4,"},{"Start":"01:26.870 ","End":"01:31.400","Text":"is to build L from this list as follows."},{"Start":"01:31.400 ","End":"01:34.940","Text":"We start with an identity matrix,"},{"Start":"01:34.940 ","End":"01:40.340","Text":"and each 1 of these replaces something in the lower diagonal."},{"Start":"01:40.340 ","End":"01:46.515","Text":"We place a 1 in Row 2, Column 1 using these indices here."},{"Start":"01:46.515 ","End":"01:48.915","Text":"Row 2, Column 1, you put a 1."},{"Start":"01:48.915 ","End":"01:51.285","Text":"Then Row 3, Column 1,"},{"Start":"01:51.285 ","End":"01:53.235","Text":"we put a 2, that\u0027s this."},{"Start":"01:53.235 ","End":"01:57.315","Text":"Row 3 column 2, there\u0027s a 3."},{"Start":"01:57.315 ","End":"02:01.115","Text":"Then we use this to get the final answer."},{"Start":"02:01.115 ","End":"02:02.675","Text":"These are just symbolic."},{"Start":"02:02.675 ","End":"02:04.910","Text":"In place of this symbol 1,"},{"Start":"02:04.910 ","End":"02:08.660","Text":"we look at the number here, plus 3, and we put a 3 here."},{"Start":"02:08.660 ","End":"02:13.455","Text":"Instead of the 2, we put a plus 2, or just 2 here."},{"Start":"02:13.455 ","End":"02:18.315","Text":"Instead of this 3, we put a 1 here."},{"Start":"02:18.315 ","End":"02:20.550","Text":"That gives us our L."},{"Start":"02:20.550 ","End":"02:25.430","Text":"Then finally, we just copy L and U,"},{"Start":"02:25.430 ","End":"02:29.275","Text":"L from here and U from here."},{"Start":"02:29.275 ","End":"02:32.680","Text":"That gives us our LU decomposition of A,"},{"Start":"02:32.680 ","End":"02:37.100","Text":"and you should really multiply it out to verify that you really do get this."},{"Start":"02:37.100 ","End":"02:39.690","Text":"That\u0027s it."}],"ID":22257},{"Watched":false,"Name":"Exercise LU2","Duration":"2m 47s","ChapterTopicVideoID":21395,"CourseChapterTopicPlaylistID":99467,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21395.jpeg","UploadDate":"2020-04-16T10:26:05.1300000","DurationForVideoObject":"PT2M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.530","Text":"In this exercise, we want to compute the LU decomposition of matrix A,"},{"Start":"00:05.530 ","End":"00:07.820","Text":"which is this 4 by 4."},{"Start":"00:07.820 ","End":"00:10.810","Text":"We\u0027ll do it according to the steps in that tutorial."},{"Start":"00:10.810 ","End":"00:16.525","Text":"The first step is to apply row operations until we get an upper triangular matrix."},{"Start":"00:16.525 ","End":"00:18.340","Text":"Let\u0027s do them a few at a time."},{"Start":"00:18.340 ","End":"00:22.670","Text":"I\u0027ll subtract multiples of the 1st row from the 2nd, 3rd, 4th."},{"Start":"00:22.670 ","End":"00:26.445","Text":"Subtract twice the first row from the 2nd,"},{"Start":"00:26.445 ","End":"00:28.185","Text":"1 time from the 3rd,"},{"Start":"00:28.185 ","End":"00:30.590","Text":"3 times from the 3rd."},{"Start":"00:30.590 ","End":"00:33.585","Text":"That\u0027s written out here."},{"Start":"00:33.585 ","End":"00:35.085","Text":"This is what we get."},{"Start":"00:35.085 ","End":"00:36.745","Text":"We\u0027ve got 0s here."},{"Start":"00:36.745 ","End":"00:40.925","Text":"Next you want to 0 out below the minus 1 here."},{"Start":"00:40.925 ","End":"00:48.730","Text":"We\u0027ll add 1 time the 2nd row to the 3rd row and twice the 2nd row to the 4th row."},{"Start":"00:48.730 ","End":"00:52.400","Text":"That will give us this."},{"Start":"00:52.400 ","End":"00:54.470","Text":"Next, we need a 0 here,"},{"Start":"00:54.470 ","End":"00:59.405","Text":"so subtract twice the 3rd row from the 4th row."},{"Start":"00:59.405 ","End":"01:02.780","Text":"This gives us the matrix U."},{"Start":"01:02.780 ","End":"01:07.460","Text":"Next, we just make a list of all these row operations,"},{"Start":"01:07.460 ","End":"01:08.845","Text":"there\u0027s 6 of them."},{"Start":"01:08.845 ","End":"01:12.330","Text":"We just write them out like so."},{"Start":"01:12.330 ","End":"01:16.230","Text":"Step 3 is to invert the numbers I\u0027ve highlighted here,"},{"Start":"01:16.230 ","End":"01:20.700","Text":"instead of minus 2 plus 2, and so on."},{"Start":"01:20.700 ","End":"01:22.380","Text":"We get a plus 2 here,"},{"Start":"01:22.380 ","End":"01:24.780","Text":"plus 1 instead of minus 1 and so on,"},{"Start":"01:24.780 ","End":"01:27.420","Text":"up to instead of minus 2, we have a plus 2."},{"Start":"01:27.420 ","End":"01:28.970","Text":"Then from this list,"},{"Start":"01:28.970 ","End":"01:31.940","Text":"we build L as follows."},{"Start":"01:31.940 ","End":"01:35.000","Text":"We start off with the identity matrix."},{"Start":"01:35.000 ","End":"01:42.005","Text":"Then we place these numbers in the brackets into this matrix according to the indices."},{"Start":"01:42.005 ","End":"01:45.410","Text":"Row 2, column 1, we put a 1."},{"Start":"01:45.410 ","End":"01:48.305","Text":"Row 3, column 1, we put a 2."},{"Start":"01:48.305 ","End":"01:50.830","Text":"Row 4, column 1, a 3."},{"Start":"01:50.830 ","End":"01:53.910","Text":"Row 3, column 2, a 4."},{"Start":"01:53.910 ","End":"01:55.770","Text":"Row 4, column 2,"},{"Start":"01:55.770 ","End":"01:57.330","Text":"we have a 5."},{"Start":"01:57.330 ","End":"01:58.860","Text":"Finally, row 4,"},{"Start":"01:58.860 ","End":"02:02.580","Text":"column 3, we have a 3."},{"Start":"02:02.580 ","End":"02:08.015","Text":"Then we replace these symbols by the numbers that I have colored here."},{"Start":"02:08.015 ","End":"02:09.290","Text":"Instead of this 1,"},{"Start":"02:09.290 ","End":"02:12.380","Text":"we put this plus 2 here."},{"Start":"02:12.380 ","End":"02:13.865","Text":"Instead of the 2,"},{"Start":"02:13.865 ","End":"02:15.490","Text":"we have a 1."},{"Start":"02:15.490 ","End":"02:17.070","Text":"Instead of the 3,"},{"Start":"02:17.070 ","End":"02:19.230","Text":"we have a 3."},{"Start":"02:19.230 ","End":"02:22.090","Text":"Here we have a minus 1,"},{"Start":"02:22.090 ","End":"02:24.785","Text":"then a minus 2,"},{"Start":"02:24.785 ","End":"02:27.260","Text":"and then a plus 2."},{"Start":"02:27.260 ","End":"02:32.040","Text":"That gives us our L. We already have U above,"},{"Start":"02:32.040 ","End":"02:35.750","Text":"so what we have to do is write it out L here,"},{"Start":"02:35.750 ","End":"02:37.370","Text":"U copied from before,"},{"Start":"02:37.370 ","End":"02:39.335","Text":"and that gives us our decomposition."},{"Start":"02:39.335 ","End":"02:41.690","Text":"As a final step, you should multiply these"},{"Start":"02:41.690 ","End":"02:44.645","Text":"out just to verify that you really do get this."},{"Start":"02:44.645 ","End":"02:47.650","Text":"Other than that, we are done."}],"ID":22258}],"Thumbnail":null,"ID":99467}]

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