Determinants
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Rules of Determinants
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More Rules of Determinants
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Cramer's Rule
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The Adjoint Matrix
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Geometrical Applications of Determinants
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[{"Name":"Determinants","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"What is a Determinant","Duration":"13m 32s","ChapterTopicVideoID":9491,"CourseChapterTopicPlaylistID":7285,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9491.jpeg","UploadDate":"2017-07-26T08:28:13.9930000","DurationForVideoObject":"PT13M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"In this clip, we start a new topic, that of determinants."},{"Start":"00:04.200 ","End":"00:06.240","Text":"More precisely, the determinant of"},{"Start":"00:06.240 ","End":"00:10.950","Text":"a matrix because it determines an answer associated with matrices."},{"Start":"00:10.950 ","End":"00:13.455","Text":"We\u0027ll start with some notation."},{"Start":"00:13.455 ","End":"00:18.060","Text":"Here, we\u0027re taking a determinant of a 2 by 2 matrix."},{"Start":"00:18.060 ","End":"00:20.715","Text":"Notice the vertical bars."},{"Start":"00:20.715 ","End":"00:28.290","Text":"Now actually, we start out with a matrix which we write with little corners here."},{"Start":"00:28.290 ","End":"00:30.135","Text":"That would be a matrix."},{"Start":"00:30.135 ","End":"00:35.355","Text":"If I drop these corners and just keep the 2 vertical lines,"},{"Start":"00:35.355 ","End":"00:42.435","Text":"then it\u0027s the determinant of that matrix and it has a value which is a number."},{"Start":"00:42.435 ","End":"00:48.505","Text":"The way we compute this number is you multiply this diagonal"},{"Start":"00:48.505 ","End":"00:51.620","Text":"and then you multiply this diagonal and you"},{"Start":"00:51.620 ","End":"00:55.370","Text":"subtract the first product minus the second product,"},{"Start":"00:55.370 ","End":"01:00.355","Text":"so that we have 1 times 4 minus 2 times 3."},{"Start":"01:00.355 ","End":"01:02.700","Text":"If you want the actual answer,"},{"Start":"01:02.700 ","End":"01:04.415","Text":"though it\u0027s less important,"},{"Start":"01:04.415 ","End":"01:06.830","Text":"then it\u0027s 4 minus 6,"},{"Start":"01:06.830 ","End":"01:08.780","Text":"which is minus 2."},{"Start":"01:08.780 ","End":"01:12.455","Text":"The determinant of this 2 by 2 matrix is minus 2."},{"Start":"01:12.455 ","End":"01:16.515","Text":"By the way, we will only be talking about square matrices,"},{"Start":"01:16.515 ","End":"01:20.210","Text":"so we\u0027re going to have n by n matrices,"},{"Start":"01:20.210 ","End":"01:22.985","Text":"plural of matrix is matrices."},{"Start":"01:22.985 ","End":"01:27.840","Text":"We\u0027ll do some 2 by 2 and then 3 by 3 and we\u0027ll"},{"Start":"01:27.840 ","End":"01:33.710","Text":"have a 4 by 4 one also in this clip, but square matrices."},{"Start":"01:33.710 ","End":"01:39.350","Text":"Now, this is not the only way of writing a determinant."},{"Start":"01:39.350 ","End":"01:40.880","Text":"It\u0027s the most common."},{"Start":"01:40.880 ","End":"01:43.879","Text":"You actually have the numbers in the bars."},{"Start":"01:43.879 ","End":"01:48.350","Text":"Sometimes we just leave the matrix form and just write the word det,"},{"Start":"01:48.350 ","End":"01:49.850","Text":"short for determinant,"},{"Start":"01:49.850 ","End":"01:52.020","Text":"in front of it."},{"Start":"01:53.510 ","End":"01:56.180","Text":"If the matrix has a name,"},{"Start":"01:56.180 ","End":"01:58.070","Text":"like matrix A,"},{"Start":"01:58.070 ","End":"02:05.210","Text":"then we would write the determinant of A with the brackets or without the brackets,"},{"Start":"02:05.210 ","End":"02:07.310","Text":"just det A,"},{"Start":"02:07.310 ","End":"02:09.570","Text":"just like here, determinant and then the matrix."},{"Start":"02:09.570 ","End":"02:13.050","Text":"Anyway, these are minor variations."},{"Start":"02:13.370 ","End":"02:16.280","Text":"That\u0027s our first example of a determinant."},{"Start":"02:16.280 ","End":"02:20.190","Text":"Now, let\u0027s go on to a 3 by 3."},{"Start":"02:20.620 ","End":"02:23.600","Text":"Note that I\u0027m just being technical here."},{"Start":"02:23.600 ","End":"02:25.935","Text":"How to compute determinants."},{"Start":"02:25.935 ","End":"02:28.990","Text":"I haven\u0027t discussed where it came from and what it\u0027s good for."},{"Start":"02:28.990 ","End":"02:32.380","Text":"At the moment, we\u0027re just learning how to compute them."},{"Start":"02:32.380 ","End":"02:35.710","Text":"Now, if you have a 3 by 3,"},{"Start":"02:35.710 ","End":"02:38.720","Text":"remember we can\u0027t have funny shaped ones,"},{"Start":"02:38.720 ","End":"02:40.400","Text":"like a 3 by 5 matrix."},{"Start":"02:40.400 ","End":"02:48.075","Text":"We had a 2 by 2 and now we\u0027re doing a 3 by 3 matrix and then finding its determinant."},{"Start":"02:48.075 ","End":"02:51.754","Text":"There\u0027s something called expanding along a row."},{"Start":"02:51.754 ","End":"02:54.110","Text":"You pick 1 of the rows,"},{"Start":"02:54.110 ","End":"02:57.170","Text":"let\u0027s say we take the first row."},{"Start":"02:57.170 ","End":"03:00.410","Text":"Now I\u0027m going to take each element at"},{"Start":"03:00.410 ","End":"03:04.085","Text":"a time and each element will give me a term in the sum."},{"Start":"03:04.085 ","End":"03:07.280","Text":"The first element, I cross out,"},{"Start":"03:07.280 ","End":"03:14.960","Text":"also the column it\u0027s in and then what I write is a product of 3 things."},{"Start":"03:14.960 ","End":"03:19.400","Text":"First of all, there\u0027s the sign and that\u0027s a plus."},{"Start":"03:19.400 ","End":"03:21.980","Text":"I\u0027ll explain more about that in just a moment."},{"Start":"03:21.980 ","End":"03:23.675","Text":"It\u0027s plus or minus."},{"Start":"03:23.675 ","End":"03:25.625","Text":"Then I put the element,"},{"Start":"03:25.625 ","End":"03:30.110","Text":"that\u0027s this 1 here and then I put the 2"},{"Start":"03:30.110 ","End":"03:35.910","Text":"by 2 determinant of what\u0027s left after I\u0027ve crossed out the row and column."},{"Start":"03:36.080 ","End":"03:40.160","Text":"This plus sign comes from a matrix,"},{"Start":"03:40.160 ","End":"03:42.140","Text":"but with signs plus,"},{"Start":"03:42.140 ","End":"03:43.740","Text":"minus, plus so on,"},{"Start":"03:43.740 ","End":"03:46.500","Text":"checkerboards tile pattern and"},{"Start":"03:46.500 ","End":"03:49.760","Text":"the top top left always begins with a plus and then you alternate plus,"},{"Start":"03:49.760 ","End":"03:52.440","Text":"minus, plus in each direction."},{"Start":"03:52.520 ","End":"03:55.775","Text":"That\u0027s the plus. Our next element,"},{"Start":"03:55.775 ","End":"03:56.870","Text":"which is going to be the 2,"},{"Start":"03:56.870 ","End":"03:58.840","Text":"will have a minus."},{"Start":"03:58.840 ","End":"04:01.545","Text":"This time we\u0027re taking the 2,"},{"Start":"04:01.545 ","End":"04:06.200","Text":"so we\u0027ve got the row and the column crossed out."},{"Start":"04:06.200 ","End":"04:12.560","Text":"Then the next term will be the sign minus from here,"},{"Start":"04:12.560 ","End":"04:17.090","Text":"then the element 2 here, and then 4,"},{"Start":"04:17.090 ","End":"04:22.570","Text":"6, 7, 9, the determinant of the remaining elements."},{"Start":"04:22.570 ","End":"04:27.730","Text":"The last one will also be a plus from this one."},{"Start":"04:28.520 ","End":"04:33.090","Text":"I forgot to cross out this column."},{"Start":"04:33.090 ","End":"04:34.960","Text":"Then the 3,"},{"Start":"04:34.960 ","End":"04:38.555","Text":"which is the element from here,"},{"Start":"04:38.555 ","End":"04:42.260","Text":"and then the determinant of the remainder,"},{"Start":"04:42.260 ","End":"04:45.015","Text":"4, 5, 7, 8."},{"Start":"04:45.015 ","End":"04:47.990","Text":"This is the example."},{"Start":"04:48.070 ","End":"04:52.070","Text":"Now we know how to compute this because we\u0027ve already talked about 2 by"},{"Start":"04:52.070 ","End":"04:59.460","Text":"2 determinants by just taking the diagonals and subtracting."},{"Start":"04:59.460 ","End":"05:02.370","Text":"If we did this, we\u0027ll get 1 times,"},{"Start":"05:02.370 ","End":"05:09.825","Text":"this is 45 minus 48 is minus 3, minus twice,"},{"Start":"05:09.825 ","End":"05:16.125","Text":"36 minus 42 is minus 6,"},{"Start":"05:16.125 ","End":"05:23.624","Text":"plus 3 times 32 minus 35 is minus 3."},{"Start":"05:23.624 ","End":"05:26.430","Text":"What I get, basically this one is plus,"},{"Start":"05:26.430 ","End":"05:27.690","Text":"it\u0027s 12 minus 3,"},{"Start":"05:27.690 ","End":"05:32.640","Text":"minus 9, it turns out to be 0."},{"Start":"05:32.640 ","End":"05:35.805","Text":"That\u0027s the numerical answer."},{"Start":"05:35.805 ","End":"05:38.420","Text":"Now I\u0027m going to introduce another concept,"},{"Start":"05:38.420 ","End":"05:41.585","Text":"another word, the minor of an element."},{"Start":"05:41.585 ","End":"05:45.545","Text":"Let\u0027s take this last computation we did, the 3."},{"Start":"05:45.545 ","End":"05:50.715","Text":"The 3 is element in Row 1, Column 3."},{"Start":"05:50.715 ","End":"05:54.680","Text":"Let\u0027s suppose that the elements of this matrix,"},{"Start":"05:54.680 ","End":"05:57.600","Text":"the 3 by 3, are a_ij."},{"Start":"05:57.700 ","End":"06:02.460","Text":"This would be the element a_1,"},{"Start":"06:02.460 ","End":"06:07.890","Text":"3 and this determinant"},{"Start":"06:08.230 ","End":"06:13.010","Text":"is called the minor associated with this element,"},{"Start":"06:13.010 ","End":"06:16.605","Text":"the 3 together with its minor."},{"Start":"06:16.605 ","End":"06:23.205","Text":"The minor is labeled M and then also 1, 3."},{"Start":"06:23.205 ","End":"06:28.020","Text":"In general, M will have 2 indices for each i and j."},{"Start":"06:28.020 ","End":"06:30.810","Text":"Let\u0027s show the other examples we did."},{"Start":"06:30.810 ","End":"06:35.535","Text":"This element in Row 1, Column 2."},{"Start":"06:35.535 ","End":"06:41.945","Text":"The 2 here, its associated minor is this determinant,"},{"Start":"06:41.945 ","End":"06:47.715","Text":"which we call M_1, 2."},{"Start":"06:47.715 ","End":"06:49.260","Text":"We also know what it\u0027s equal to,"},{"Start":"06:49.260 ","End":"06:51.525","Text":"it\u0027s equal to minus 6."},{"Start":"06:51.525 ","End":"06:55.830","Text":"M_1, 3 is equal to negative 3."},{"Start":"06:55.830 ","End":"06:59.830","Text":"The determinant that we got for the first one,"},{"Start":"06:59.830 ","End":"07:06.140","Text":"it\u0027s this determinant and this is M_1, 1."},{"Start":"07:06.830 ","End":"07:12.150","Text":"That\u0027s the idea for a 3 by 3 matrix."},{"Start":"07:12.150 ","End":"07:18.455","Text":"I want to do another example along a row and then I\u0027ll do an example along a column."},{"Start":"07:18.455 ","End":"07:22.615","Text":"Here we are again with the same determinant."},{"Start":"07:22.615 ","End":"07:25.090","Text":"This time, we\u0027ll expand along"},{"Start":"07:25.090 ","End":"07:30.560","Text":"the second row and hopefully we\u0027ll get the same answer in the end."},{"Start":"07:30.560 ","End":"07:33.785","Text":"As before, we\u0027ll do them one at a time."},{"Start":"07:33.785 ","End":"07:35.330","Text":"I\u0027ll take the element 4,"},{"Start":"07:35.330 ","End":"07:40.370","Text":"so I cross out the row and column and then I look at"},{"Start":"07:40.370 ","End":"07:46.220","Text":"the sign of this and I see that it\u0027s a minus and then the 4."},{"Start":"07:46.220 ","End":"07:50.670","Text":"Then the minor, which is 2,"},{"Start":"07:50.670 ","End":"07:54.255","Text":"3, 8, 9 and here we are."},{"Start":"07:54.255 ","End":"07:56.075","Text":"Again, 3 things."},{"Start":"07:56.075 ","End":"07:59.180","Text":"The sign we take from here,"},{"Start":"07:59.180 ","End":"08:01.810","Text":"the element next,"},{"Start":"08:01.810 ","End":"08:09.300","Text":"and then the minor and the name for this minor is M,"},{"Start":"08:09.300 ","End":"08:11.085","Text":"it\u0027s Row 2,"},{"Start":"08:11.085 ","End":"08:12.915","Text":"Column 1, M_2."},{"Start":"08:12.915 ","End":"08:18.210","Text":"1, just like this element would be a_2,"},{"Start":"08:18.210 ","End":"08:22.365","Text":"1, this one here, for example."},{"Start":"08:22.365 ","End":"08:26.380","Text":"Next, we take the 5."},{"Start":"08:26.600 ","End":"08:29.805","Text":"This is highlighted now."},{"Start":"08:29.805 ","End":"08:32.265","Text":"Then the 5,"},{"Start":"08:32.265 ","End":"08:34.110","Text":"we take together with the sign,"},{"Start":"08:34.110 ","End":"08:35.580","Text":"which will be plus."},{"Start":"08:35.580 ","End":"08:37.170","Text":"Then 1, 3, 7,"},{"Start":"08:37.170 ","End":"08:41.565","Text":"9 and here we are, the plus,"},{"Start":"08:41.565 ","End":"08:44.960","Text":"the 5, and the minor,"},{"Start":"08:44.960 ","End":"08:48.625","Text":"which in this case is M_2, 2."},{"Start":"08:48.625 ","End":"08:51.200","Text":"When you\u0027re actually computing them,"},{"Start":"08:51.200 ","End":"08:52.910","Text":"you don\u0027t have to write these Ms."},{"Start":"08:52.910 ","End":"08:56.699","Text":"This is just for the education purposes."},{"Start":"08:56.699 ","End":"09:00.420","Text":"Then the last one on this row."},{"Start":"09:00.450 ","End":"09:05.170","Text":"There we are. We take the 6."},{"Start":"09:05.170 ","End":"09:07.040","Text":"Before that, we take the sign,"},{"Start":"09:07.040 ","End":"09:10.040","Text":"which the corresponding place here is a minus."},{"Start":"09:10.040 ","End":"09:19.900","Text":"Then 1, 2, 7, 8 and this minor has index 2, 3."},{"Start":"09:19.900 ","End":"09:23.630","Text":"This time I\u0027ll leave it up to you to check the computation."},{"Start":"09:23.630 ","End":"09:25.160","Text":"It should come out 0."},{"Start":"09:25.160 ","End":"09:27.720","Text":"If not, then something is very wrong."},{"Start":"09:27.720 ","End":"09:35.690","Text":"Now I said I\u0027d give you another example this time with a column,"},{"Start":"09:35.690 ","End":"09:38.960","Text":"which is going to work pretty much the same way."},{"Start":"09:38.960 ","End":"09:43.160","Text":"I just want to keep this table, this checkerboard pattern."},{"Start":"09:43.160 ","End":"09:49.805","Text":"This time I\u0027ll choose the third column and expand along a column."},{"Start":"09:49.805 ","End":"09:51.350","Text":"If you think about it,"},{"Start":"09:51.350 ","End":"09:53.420","Text":"there\u0027s actually 6 possibilities."},{"Start":"09:53.420 ","End":"09:57.305","Text":"There are 3 columns and 3 rows."},{"Start":"09:57.305 ","End":"09:58.895","Text":"You have 6 choices."},{"Start":"09:58.895 ","End":"10:03.215","Text":"In practice, you would tend to choose a row or a column"},{"Start":"10:03.215 ","End":"10:07.970","Text":"which has 1 or more zeros and then the work is easier"},{"Start":"10:07.970 ","End":"10:10.630","Text":"but all 6 are possible."},{"Start":"10:10.630 ","End":"10:13.440","Text":"We\u0027ll first take the 3, then the 6, then the 9."},{"Start":"10:13.440 ","End":"10:17.314","Text":"First the 3, cross out the row and the column."},{"Start":"10:17.314 ","End":"10:22.055","Text":"We take the sign corresponding to this is a plus,"},{"Start":"10:22.055 ","End":"10:23.330","Text":"then the 3,"},{"Start":"10:23.330 ","End":"10:27.500","Text":"and then the minor. Here we are."},{"Start":"10:27.500 ","End":"10:29.315","Text":"The plus with the 3,"},{"Start":"10:29.315 ","End":"10:30.590","Text":"with the 4,"},{"Start":"10:30.590 ","End":"10:32.815","Text":"5, 7, 8 determinant."},{"Start":"10:32.815 ","End":"10:35.670","Text":"Next comes the 6,"},{"Start":"10:35.670 ","End":"10:38.139","Text":"the row and the column."},{"Start":"10:38.590 ","End":"10:41.165","Text":"The minus from here,"},{"Start":"10:41.165 ","End":"10:42.940","Text":"because the 6 is in that position."},{"Start":"10:42.940 ","End":"10:48.455","Text":"Then the 6 itself and then the minor belonging to 6."},{"Start":"10:48.455 ","End":"10:52.290","Text":"Then finally the 9,"},{"Start":"10:52.290 ","End":"11:00.420","Text":"which is this row and column and the sign is a plus with the 9 and with the 1,"},{"Start":"11:00.420 ","End":"11:07.140","Text":"2, 4, 5 minor in a determinant, like so."},{"Start":"11:07.140 ","End":"11:10.430","Text":"This time you see I didn\u0027t bother writing these Ms."},{"Start":"11:10.430 ","End":"11:12.545","Text":"Anyway, if we check the result,"},{"Start":"11:12.545 ","End":"11:15.685","Text":"you should get 0."},{"Start":"11:15.685 ","End":"11:21.490","Text":"Now we;\u0027re going to do a 4 by 4 example."},{"Start":"11:22.660 ","End":"11:25.760","Text":"This case works similarly."},{"Start":"11:25.760 ","End":"11:32.640","Text":"We expand along a row or a column and I\u0027ll go for the first row."},{"Start":"11:32.640 ","End":"11:36.625","Text":"Now, we haven\u0027t got the checkerboard pattern printed out"},{"Start":"11:36.625 ","End":"11:37.870","Text":"but as I said,"},{"Start":"11:37.870 ","End":"11:40.915","Text":"all you have to remember is the top left is a plus."},{"Start":"11:40.915 ","End":"11:43.120","Text":"We only need the first row,"},{"Start":"11:43.120 ","End":"11:44.815","Text":"so it\u0027s plus, minus,"},{"Start":"11:44.815 ","End":"11:49.670","Text":"plus, minus and that\u0027s all we need to know, we\u0027re expanding this."},{"Start":"11:49.670 ","End":"11:53.700","Text":"Let\u0027s do the first element."},{"Start":"11:53.700 ","End":"12:01.845","Text":"Cross out the column and the row and then we have the plus with the 1,"},{"Start":"12:01.845 ","End":"12:07.015","Text":"with the minor, the 3 by 3 determinant this time."},{"Start":"12:07.015 ","End":"12:10.375","Text":"Next we take the Element 2,"},{"Start":"12:10.375 ","End":"12:13.495","Text":"row and column crossed out."},{"Start":"12:13.495 ","End":"12:17.695","Text":"It\u0027s a minus, is the minus,"},{"Start":"12:17.695 ","End":"12:19.540","Text":"the 2 from here."},{"Start":"12:19.540 ","End":"12:24.950","Text":"The minor, which is 5,"},{"Start":"12:24.950 ","End":"12:26.105","Text":"7, 8, whatever,"},{"Start":"12:26.105 ","End":"12:29.015","Text":"wasn\u0027t crossed out, 3 by 3."},{"Start":"12:29.015 ","End":"12:32.935","Text":"Then we move on to the 3,"},{"Start":"12:32.935 ","End":"12:36.550","Text":"which is a plus term."},{"Start":"12:37.040 ","End":"12:40.905","Text":"Here\u0027s the plus, here\u0027s the 3,"},{"Start":"12:40.905 ","End":"12:47.120","Text":"and here\u0027s this bit is what we get when we cross out the row and the column,"},{"Start":"12:47.120 ","End":"12:50.085","Text":"the 5, 6, 8 and so on."},{"Start":"12:50.085 ","End":"12:53.385","Text":"Now the last one,"},{"Start":"12:53.385 ","End":"12:55.560","Text":"this 4th column,"},{"Start":"12:55.560 ","End":"13:00.390","Text":"where the element is 4 and the sign is a minus."},{"Start":"13:00.390 ","End":"13:03.885","Text":"Here is the minus,"},{"Start":"13:03.885 ","End":"13:05.310","Text":"here\u0027s the 4,"},{"Start":"13:05.310 ","End":"13:07.875","Text":"and here\u0027s the minor,"},{"Start":"13:07.875 ","End":"13:10.265","Text":"the 5, 6, 7 and so on."},{"Start":"13:10.265 ","End":"13:13.680","Text":"Now of course, there\u0027s still more work to do"},{"Start":"13:13.680 ","End":"13:16.890","Text":"but we know how to do 3 by 3 determinants,"},{"Start":"13:16.890 ","End":"13:21.095","Text":"as before and there\u0027s nothing difficult, it\u0027s just tedious."},{"Start":"13:21.095 ","End":"13:23.165","Text":"Later on, much later,"},{"Start":"13:23.165 ","End":"13:29.315","Text":"we\u0027ll learn other methods other than expanding along a row or a column."},{"Start":"13:29.315 ","End":"13:32.700","Text":"Meanwhile, for this clip we\u0027re done."}],"ID":9880},{"Watched":false,"Name":"Exercises 1-3","Duration":"1m 16s","ChapterTopicVideoID":9494,"CourseChapterTopicPlaylistID":7285,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9494.jpeg","UploadDate":"2017-07-26T08:28:44.9400000","DurationForVideoObject":"PT1M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.905","Text":"In this exercise, we have to evaluate the following determinants,"},{"Start":"00:04.905 ","End":"00:06.525","Text":"the 3 of them."},{"Start":"00:06.525 ","End":"00:11.220","Text":"The first one, it can actually evaluated as a number"},{"Start":"00:11.220 ","End":"00:13.515","Text":"but you can get an expression."},{"Start":"00:13.515 ","End":"00:20.355","Text":"Remember it\u0027s the product of this diagonal minus the product of this diagonal,"},{"Start":"00:20.355 ","End":"00:23.820","Text":"meaning a, d minus b,"},{"Start":"00:23.820 ","End":"00:26.955","Text":"c and that\u0027s it."},{"Start":"00:26.955 ","End":"00:30.930","Text":"In this one, we actually have numbers."},{"Start":"00:30.930 ","End":"00:37.845","Text":"We need to multiply 5 times 3 and then subtract minus 7 times 2,"},{"Start":"00:37.845 ","End":"00:47.805","Text":"like so or 2 times minus 7 and the answer comes out to be 29 because it\u0027s 15 plus 14."},{"Start":"00:47.805 ","End":"00:50.450","Text":"Similarly, the last one this times this,"},{"Start":"00:50.450 ","End":"00:55.955","Text":"minus this times this and after we evaluate it,"},{"Start":"00:55.955 ","End":"00:59.440","Text":"we get minus 1."},{"Start":"00:59.440 ","End":"01:04.030","Text":"It\u0027s very important that you should remember this formula."},{"Start":"01:04.030 ","End":"01:06.680","Text":"You don\u0027t have to remember it with a, b, c, d,"},{"Start":"01:06.680 ","End":"01:09.905","Text":"just remember that it\u0027s the main diagonal,"},{"Start":"01:09.905 ","End":"01:17.490","Text":"the product and take away the product of the other diagonal. That\u0027s it."}],"ID":9881},{"Watched":false,"Name":"Exercises 4-6","Duration":"4m 54s","ChapterTopicVideoID":9495,"CourseChapterTopicPlaylistID":7285,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9495.jpeg","UploadDate":"2017-07-26T08:28:56.7470000","DurationForVideoObject":"PT4M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.395","Text":"In this exercise, we have some determinants to compute."},{"Start":"00:04.395 ","End":"00:07.290","Text":"The only technique we\u0027ve learned is,"},{"Start":"00:07.290 ","End":"00:10.845","Text":"expanding along the row or a column."},{"Start":"00:10.845 ","End":"00:14.640","Text":"You might remember, I suggested choosing"},{"Start":"00:14.640 ","End":"00:18.180","Text":"a row or column with a lot of zeros as much as possible."},{"Start":"00:18.180 ","End":"00:23.220","Text":"I can see that this column has 2 zeros in it,"},{"Start":"00:23.220 ","End":"00:25.230","Text":"so I expand along it."},{"Start":"00:25.230 ","End":"00:27.150","Text":"I want to remind you what we do."},{"Start":"00:27.150 ","End":"00:30.750","Text":"We go along this column,"},{"Start":"00:30.750 ","End":"00:32.550","Text":"and for each element,"},{"Start":"00:32.550 ","End":"00:36.645","Text":"we have a term which consists of"},{"Start":"00:36.645 ","End":"00:46.110","Text":"a sign times the entry times a minor."},{"Start":"00:46.110 ","End":"00:51.380","Text":"The sign, this is a plus or minus."},{"Start":"00:51.380 ","End":"00:54.215","Text":"I brought in a picture I found on the Internet."},{"Start":"00:54.215 ","End":"00:56.720","Text":"In this case we have a 3 by 3,"},{"Start":"00:56.720 ","End":"01:01.335","Text":"so we just take it up to here."},{"Start":"01:01.335 ","End":"01:04.950","Text":"It\u0027s an alternating checkerboard sign."},{"Start":"01:04.950 ","End":"01:09.170","Text":"In this case, the 0 here would be a minus,"},{"Start":"01:09.170 ","End":"01:10.895","Text":"the 1 would go with the plus,"},{"Start":"01:10.895 ","End":"01:13.570","Text":"and the 0 go with a minus."},{"Start":"01:13.570 ","End":"01:16.950","Text":"The entry means the entry itself."},{"Start":"01:16.950 ","End":"01:24.530","Text":"The minor is what we get when we cross out the row and column."},{"Start":"01:24.530 ","End":"01:30.785","Text":"For the entry 0, the minor is the determinant of what\u0027s left"},{"Start":"01:30.785 ","End":"01:33.200","Text":"but since we are going to be multiplying by 0,"},{"Start":"01:33.200 ","End":"01:35.259","Text":"it doesn\u0027t really matter."},{"Start":"01:35.259 ","End":"01:42.665","Text":"It\u0027s going to be minus 0 times some determinant the minor won\u0027t contribute anything."},{"Start":"01:42.665 ","End":"01:45.185","Text":"Only the 1 here,"},{"Start":"01:45.185 ","End":"01:48.990","Text":"is significant because we get a 0 again."},{"Start":"01:49.220 ","End":"01:54.095","Text":"This is the row that I need to cross out."},{"Start":"01:54.095 ","End":"02:00.195","Text":"The sign is a plus the entry is 1,"},{"Start":"02:00.195 ","End":"02:04.100","Text":"and the minor is the determinant of 1, 2,"},{"Start":"02:04.100 ","End":"02:08.580","Text":"2, 3, and here it is, once again."},{"Start":"02:08.580 ","End":"02:12.990","Text":"Plus from the sign, from this 1."},{"Start":"02:12.990 ","End":"02:16.380","Text":"The 1 is that the entry,"},{"Start":"02:16.380 ","End":"02:17.940","Text":"and the 1, 2, 2,"},{"Start":"02:17.940 ","End":"02:23.850","Text":"3 determinant is from there, the minor."},{"Start":"02:23.850 ","End":"02:26.725","Text":"The 2 by 2 determinant,"},{"Start":"02:26.725 ","End":"02:32.510","Text":"I\u0027m sure you remember this diagonal product minus this diagonal,"},{"Start":"02:32.510 ","End":"02:35.990","Text":"1 times 3 minus 2 times 2."},{"Start":"02:35.990 ","End":"02:39.120","Text":"The answer is minus 1."},{"Start":"02:39.920 ","End":"02:46.325","Text":"I want to emphasize again that we choose a column with as many zeros as possible,"},{"Start":"02:46.325 ","End":"02:48.829","Text":"because each time we have a 0 entry,"},{"Start":"02:48.829 ","End":"02:50.465","Text":"we won\u0027t get anything."},{"Start":"02:50.465 ","End":"02:53.090","Text":"This really should have been the sum of 3 things."},{"Start":"02:53.090 ","End":"02:55.370","Text":"Should have been a 0 plus this,"},{"Start":"02:55.370 ","End":"02:58.910","Text":"plus a 0, or a minus a 0."},{"Start":"02:58.910 ","End":"03:03.275","Text":"In the next example, if we look,"},{"Start":"03:03.275 ","End":"03:06.935","Text":"you see that the third row,"},{"Start":"03:06.935 ","End":"03:09.825","Text":"this 1 here has 2 zeros in it."},{"Start":"03:09.825 ","End":"03:12.675","Text":"We\u0027ll expand along the third row."},{"Start":"03:12.675 ","End":"03:14.840","Text":"As we saw before, the zeros,"},{"Start":"03:14.840 ","End":"03:18.110","Text":"we don\u0027t have to take into account because they don\u0027t contribute anything."},{"Start":"03:18.110 ","End":"03:23.205","Text":"It\u0027s just this entry here, the 1."},{"Start":"03:23.205 ","End":"03:25.080","Text":"We need the sign,"},{"Start":"03:25.080 ","End":"03:26.780","Text":"we consult the table,"},{"Start":"03:26.780 ","End":"03:29.569","Text":"the pattern, it\u0027s a plus,"},{"Start":"03:29.569 ","End":"03:33.110","Text":"there\u0027s a 1, and we need the determinant of this bit,"},{"Start":"03:33.110 ","End":"03:35.310","Text":"which is the minor,"},{"Start":"03:35.570 ","End":"03:37.995","Text":"This is what we get."},{"Start":"03:37.995 ","End":"03:41.780","Text":"Computing the 2-by-2 determinant,"},{"Start":"03:41.780 ","End":"03:44.830","Text":"this product minus this product."},{"Start":"03:44.830 ","End":"03:48.450","Text":"We get minus 1 minus 2,"},{"Start":"03:48.450 ","End":"03:50.700","Text":"which is minus 3 times the plus 1,"},{"Start":"03:50.700 ","End":"03:54.030","Text":"which is still minus 3, is the answer."},{"Start":"03:54.030 ","End":"03:55.875","Text":"In the last one,"},{"Start":"03:55.875 ","End":"03:57.465","Text":"let\u0027s take a look."},{"Start":"03:57.465 ","End":"04:05.625","Text":"Once again, it\u0027s the third row that is got most zeros."},{"Start":"04:05.625 ","End":"04:09.040","Text":"We only need this entry."},{"Start":"04:11.240 ","End":"04:14.560","Text":"We can look at the checkerboard and see it\u0027s a minus"},{"Start":"04:14.560 ","End":"04:17.290","Text":"but usually we just start off and it\u0027s always"},{"Start":"04:17.290 ","End":"04:20.380","Text":"plus in the top-left corner so we just alternate,"},{"Start":"04:20.380 ","End":"04:23.045","Text":"plus, minus, plus, minus."},{"Start":"04:23.045 ","End":"04:24.595","Text":"Doesn\u0027t matter which way you go,"},{"Start":"04:24.595 ","End":"04:27.790","Text":"plus, minus, plus, minus."},{"Start":"04:27.790 ","End":"04:34.005","Text":"Once again, the sign with the entry and then the minus is the sign,"},{"Start":"04:34.005 ","End":"04:35.819","Text":"2 is the entry."},{"Start":"04:35.819 ","End":"04:40.210","Text":"The minor is what\u0027s not been crossed out,"},{"Start":"04:40.210 ","End":"04:44.140","Text":"2, 1, 3, 5."},{"Start":"04:47.600 ","End":"04:50.280","Text":"We do the arithmetic,"},{"Start":"04:50.280 ","End":"04:54.370","Text":"we get minus 14, and we\u0027re done."}],"ID":9882},{"Watched":false,"Name":"Exercises 7-9","Duration":"5m 43s","ChapterTopicVideoID":9496,"CourseChapterTopicPlaylistID":7285,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9496.jpeg","UploadDate":"2017-07-26T08:29:18.7000000","DurationForVideoObject":"PT5M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"In this exercise, we have determinants to"},{"Start":"00:02.550 ","End":"00:07.635","Text":"compute and it\u0027ll be a bit of work because they\u0027re are 4 by 4,"},{"Start":"00:07.635 ","End":"00:12.760","Text":"but not as much as you might think because there\u0027s a lot of 0s."},{"Start":"00:12.920 ","End":"00:18.315","Text":"If we look carefully for row or column with the most 0s here,"},{"Start":"00:18.315 ","End":"00:24.090","Text":"we actually have all 0s except for this 1."},{"Start":"00:24.090 ","End":"00:26.490","Text":"I\u0027m jumping they are going a bit that I\u0027m looking"},{"Start":"00:26.490 ","End":"00:28.875","Text":"ahead and I see here that it\u0027s going to be"},{"Start":"00:28.875 ","End":"00:34.395","Text":"the second column which only has a single non-zero entry."},{"Start":"00:34.395 ","End":"00:37.800","Text":"Anyway, back here."},{"Start":"00:37.800 ","End":"00:41.720","Text":"They brought this little picture in as a reminder,"},{"Start":"00:41.720 ","End":"00:45.005","Text":"I think is last time I\u0027ll do that for the plus minus,"},{"Start":"00:45.005 ","End":"00:47.255","Text":"because you can always just count."},{"Start":"00:47.255 ","End":"00:50.900","Text":"Anyway, in this 1, there\u0027s only going to be,"},{"Start":"00:50.900 ","End":"00:57.110","Text":"1 non-zero entry, this 1 strikeout the column as well."},{"Start":"00:57.110 ","End":"01:01.410","Text":"Now we have the 1 which is the entry,"},{"Start":"01:01.410 ","End":"01:03.935","Text":"we need to multiply it by the sine."},{"Start":"01:03.935 ","End":"01:05.240","Text":"The sine is a plus,"},{"Start":"01:05.240 ","End":"01:08.090","Text":"it\u0027s always a plus in the top left corner."},{"Start":"01:08.090 ","End":"01:10.175","Text":"Then we also have a minor,"},{"Start":"01:10.175 ","End":"01:12.980","Text":"which is the determinant of this."},{"Start":"01:12.980 ","End":"01:14.960","Text":"This is what we have,"},{"Start":"01:14.960 ","End":"01:19.820","Text":"the only 1 term because the other 3 terms are all 0."},{"Start":"01:19.820 ","End":"01:27.005","Text":"Now we have a 3 by 3 determinant to compute and we take a look."},{"Start":"01:27.005 ","End":"01:33.514","Text":"I guess I should have mentioned that here I also could have chosen the fourth column."},{"Start":"01:33.514 ","End":"01:37.850","Text":"It also has only a single non-zero."},{"Start":"01:37.850 ","End":"01:41.840","Text":"But the numbers here are smaller or 1 is less than a 4."},{"Start":"01:41.840 ","End":"01:43.670","Text":"I\u0027m saying this here again,"},{"Start":"01:43.670 ","End":"01:46.565","Text":"we have a choice we could take the first row."},{"Start":"01:46.565 ","End":"01:50.460","Text":"It has only 1 non-zero entry."},{"Start":"01:50.460 ","End":"01:53.365","Text":"The last column also,"},{"Start":"01:53.365 ","End":"01:58.505","Text":"I prefer rows and I prefer smaller numbers we\u0027ll go with this 1."},{"Start":"01:58.505 ","End":"02:03.915","Text":"It only has the 1 non-zero entry which is here."},{"Start":"02:03.915 ","End":"02:12.120","Text":"Continuing we have the plus 1 and then again we have a plus 2 with this minor."},{"Start":"02:12.470 ","End":"02:16.710","Text":"From here we get the plus 2 times this determinant."},{"Start":"02:16.710 ","End":"02:22.885","Text":"Then I have to multiply this diagonal minus this diagonal,"},{"Start":"02:22.885 ","End":"02:28.740","Text":"3 times 4 minus 0 times whatever."},{"Start":"02:29.530 ","End":"02:32.270","Text":"The answer came out, 24."},{"Start":"02:32.270 ","End":"02:35.300","Text":"Now the next 1,"},{"Start":"02:35.300 ","End":"02:39.695","Text":"we already chose the column with it,"},{"Start":"02:39.695 ","End":"02:43.830","Text":"the only choice with a single non-zero."},{"Start":"02:43.830 ","End":"02:46.370","Text":"Of course you could choose any row or column,"},{"Start":"02:46.370 ","End":"02:52.020","Text":"just saying that we choose this with a lot of zeros just to make life easier."},{"Start":"02:53.110 ","End":"02:55.690","Text":"Only the 3 counts,"},{"Start":"02:55.690 ","End":"02:58.800","Text":"cross this out also and plus,"},{"Start":"02:58.800 ","End":"03:02.315","Text":"minus, plus, minus so we can refer here."},{"Start":"03:02.315 ","End":"03:04.855","Text":"This is the minus,"},{"Start":"03:04.855 ","End":"03:10.110","Text":"the minus with the entry 3 and the corresponding minor."},{"Start":"03:10.110 ","End":"03:15.525","Text":"Now we have to compute a 3 by 3 determinant."},{"Start":"03:15.525 ","End":"03:19.840","Text":"Let\u0027s take a look and see where we have a lot of 0s."},{"Start":"03:19.840 ","End":"03:23.690","Text":"There\u0027s only a single 0 here so the best we could"},{"Start":"03:23.690 ","End":"03:28.385","Text":"do would be to take either the middle row or the last column."},{"Start":"03:28.385 ","End":"03:33.875","Text":"I\u0027ll go with the last column and we\u0027ll have 2 entries this time."},{"Start":"03:33.875 ","End":"03:38.010","Text":"First we\u0027ll go with the 5."},{"Start":"03:39.170 ","End":"03:43.520","Text":"This will give us plus 5 and times the minor,"},{"Start":"03:43.520 ","End":"03:46.055","Text":"which is the determinant of this bit."},{"Start":"03:46.055 ","End":"03:52.305","Text":"Then the last bit comes from the 44."},{"Start":"03:52.305 ","End":"03:54.930","Text":"It\u0027s also a plus, plus, minus,"},{"Start":"03:54.930 ","End":"03:57.750","Text":"plus, minus, plus so look at the table."},{"Start":"03:57.750 ","End":"04:00.530","Text":"Then we have this determinant,"},{"Start":"04:00.530 ","End":"04:04.460","Text":"which is the minor of the element."},{"Start":"04:04.460 ","End":"04:06.995","Text":"Then 2 by 2,"},{"Start":"04:06.995 ","End":"04:13.210","Text":"we know how to compute this diagonal multiplied minus this diagonal and so on."},{"Start":"04:13.210 ","End":"04:18.920","Text":"Here I\u0027ve written minus 2 times 5 is minus 10,"},{"Start":"04:18.920 ","End":"04:21.110","Text":"the minus 6, and the 2 is minus 12,"},{"Start":"04:21.110 ","End":"04:23.280","Text":"and so on and so on."},{"Start":"04:24.100 ","End":"04:30.180","Text":"Finally we get 234."},{"Start":"04:30.640 ","End":"04:33.575","Text":"Let\u0027s do another 1."},{"Start":"04:33.575 ","End":"04:36.050","Text":"Again, 4 by 4."},{"Start":"04:36.050 ","End":"04:44.625","Text":"We take a look to see if we can get many 0s as we can, the third row."},{"Start":"04:44.625 ","End":"04:47.330","Text":"Highlight this."},{"Start":"04:47.330 ","End":"04:52.460","Text":"There\u0027s only 1 entry that\u0027s not 0 so we cross out"},{"Start":"04:52.460 ","End":"04:59.935","Text":"the column and we don\u0027t need the checkerboard because we can count plus, minus, plus."},{"Start":"04:59.935 ","End":"05:04.580","Text":"Here we are the plus is the sign, then the entry,"},{"Start":"05:04.580 ","End":"05:06.305","Text":"and then the minor,"},{"Start":"05:06.305 ","End":"05:10.400","Text":"which is the 3 by 3 determinant that\u0027s left."},{"Start":"05:10.400 ","End":"05:17.205","Text":"Here we see that if I expand by the first row,"},{"Start":"05:17.205 ","End":"05:22.395","Text":"then I only have 1 single non-zero, this 1."},{"Start":"05:22.395 ","End":"05:25.140","Text":"This is what we get, plus 4,"},{"Start":"05:25.140 ","End":"05:28.725","Text":"and this 5 is also a plus, plus, minus, plus."},{"Start":"05:28.725 ","End":"05:36.080","Text":"Then this determinant, this product minus this product and these 2 are here,"},{"Start":"05:36.080 ","End":"05:37.655","Text":"and this is the calculation,"},{"Start":"05:37.655 ","End":"05:43.650","Text":"and it comes out minus 300. We\u0027re now done."}],"ID":9883},{"Watched":false,"Name":"Exercise 10","Duration":"2m 32s","ChapterTopicVideoID":9492,"CourseChapterTopicPlaylistID":7285,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9492.jpeg","UploadDate":"2017-07-26T08:28:26.2500000","DurationForVideoObject":"PT2M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.675","Text":"In this exercise, we have to compute a 5 by 5 determinant."},{"Start":"00:06.675 ","End":"00:11.445","Text":"We look for a row or column with a lot of 0s."},{"Start":"00:11.445 ","End":"00:18.465","Text":"It\u0027s down to 2. It\u0027s either the last row or the fourth column."},{"Start":"00:18.465 ","End":"00:22.275","Text":"I\u0027ll go with the last row,"},{"Start":"00:22.275 ","End":"00:23.490","Text":"doesn\u0027t make much difference."},{"Start":"00:23.490 ","End":"00:25.590","Text":"I prefer a 1, 2 or 3."},{"Start":"00:25.590 ","End":"00:27.960","Text":"They\u0027re easier to compute."},{"Start":"00:27.960 ","End":"00:31.350","Text":"There\u0027s going to be 5 terms,"},{"Start":"00:31.350 ","End":"00:32.520","Text":"but 4 of them are 0."},{"Start":"00:32.520 ","End":"00:36.000","Text":"Only this 1 matters."},{"Start":"00:36.000 ","End":"00:41.375","Text":"I\u0027ll cross out the column and I need to know it\u0027s sign."},{"Start":"00:41.375 ","End":"00:44.195","Text":"We can just count starting from here."},{"Start":"00:44.195 ","End":"00:48.755","Text":"Plus, minus, plus, minus, plus, minus, plus."},{"Start":"00:48.755 ","End":"00:51.300","Text":"This will be a plus."},{"Start":"00:51.650 ","End":"00:54.420","Text":"Here the plus is the sign,"},{"Start":"00:54.420 ","End":"00:56.520","Text":"the 1 is the entry."},{"Start":"00:56.520 ","End":"01:00.110","Text":"This determinant is the minor of this entry"},{"Start":"01:00.110 ","End":"01:04.250","Text":"which we get by striking out the row and column."},{"Start":"01:04.250 ","End":"01:07.280","Text":"Now, we have a 4 by 4 determinant."},{"Start":"01:07.280 ","End":"01:10.835","Text":"The 1 with the most 0s is this."},{"Start":"01:10.835 ","End":"01:14.555","Text":"It\u0027s the only 1 that has 3 0s in it."},{"Start":"01:14.555 ","End":"01:18.370","Text":"We\u0027re going to expand according to this and we\u0027ll only get"},{"Start":"01:18.370 ","End":"01:24.615","Text":"1 entry that\u0027s not 0 so I\u0027ll strike this out."},{"Start":"01:24.615 ","End":"01:27.730","Text":"Plus, minus, plus."},{"Start":"01:28.040 ","End":"01:34.935","Text":"Plus 3, then we have left a 3 by 3 determinant."},{"Start":"01:34.935 ","End":"01:37.870","Text":"Then let\u0027s look at this 1."},{"Start":"01:38.310 ","End":"01:41.020","Text":"The single 0 here."},{"Start":"01:41.020 ","End":"01:43.990","Text":"I could take the first row or the middle column."},{"Start":"01:43.990 ","End":"01:48.005","Text":"I\u0027ll go with the top row."},{"Start":"01:48.005 ","End":"01:50.880","Text":"Then I\u0027ll have 2 terms."},{"Start":"01:50.880 ","End":"01:53.355","Text":"1 with the 3 and 1 with the 2."},{"Start":"01:53.355 ","End":"01:57.505","Text":"They\u0027re both plus by the way, plus, minus, plus."},{"Start":"01:57.505 ","End":"02:02.899","Text":"The first time I\u0027ll cross out this column."},{"Start":"02:02.899 ","End":"02:06.215","Text":"That gives us up to here,"},{"Start":"02:06.215 ","End":"02:15.615","Text":"the plus 3 and then this minor and this bit comes from the 2 element."},{"Start":"02:15.615 ","End":"02:21.640","Text":"Obviously, there\u0027s also a plus and then this is the minor this determinant."},{"Start":"02:22.460 ","End":"02:26.225","Text":"Then it\u0027s just a bit of computation."},{"Start":"02:26.225 ","End":"02:32.070","Text":"The answer comes out to be 9. We\u0027re done."}],"ID":9884},{"Watched":false,"Name":"Exercise 11","Duration":"2m 35s","ChapterTopicVideoID":9493,"CourseChapterTopicPlaylistID":7285,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9493.jpeg","UploadDate":"2017-07-26T08:28:41.0800000","DurationForVideoObject":"PT2M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"Here we have to evaluate this determinant and it\u0027s a 5 by 5."},{"Start":"00:05.280 ","End":"00:10.440","Text":"What we usually do is look for a row or column with a lot of 0s."},{"Start":"00:10.440 ","End":"00:17.370","Text":"I see that I could go with the second row or the second column."},{"Start":"00:17.370 ","End":"00:19.560","Text":"I\u0027ll go with the second column."},{"Start":"00:19.560 ","End":"00:23.640","Text":"No particular reason may be because 2 is smaller than 3."},{"Start":"00:23.640 ","End":"00:27.180","Text":"If we expand by this column,"},{"Start":"00:27.180 ","End":"00:31.515","Text":"the only non-zero is this 1,"},{"Start":"00:31.515 ","End":"00:35.525","Text":"so we\u0027ll only get 1 entry."},{"Start":"00:35.525 ","End":"00:37.700","Text":"We need to check its sign."},{"Start":"00:37.700 ","End":"00:41.030","Text":"Plus, minus, plus, minus. It\u0027s a minus."},{"Start":"00:41.030 ","End":"00:44.710","Text":"We\u0027ll have minus 2 times the minor,"},{"Start":"00:44.710 ","End":"00:47.325","Text":"which is this the minus,"},{"Start":"00:47.325 ","End":"00:50.555","Text":"when we remove the row and the column,"},{"Start":"00:50.555 ","End":"00:53.000","Text":"now we\u0027ve got a 4 by 4 determinant,"},{"Start":"00:53.000 ","End":"00:56.615","Text":"and we look again for a lot of 0s."},{"Start":"00:56.615 ","End":"01:03.280","Text":"To me, it looks like the second row is the best we can do."},{"Start":"01:03.560 ","End":"01:05.895","Text":"In this second row,"},{"Start":"01:05.895 ","End":"01:10.140","Text":"only the 3 will be non-zero,"},{"Start":"01:10.140 ","End":"01:16.410","Text":"so I\u0027ll just get a single term again at plus, minus, plus."},{"Start":"01:16.410 ","End":"01:18.840","Text":"The minor is the 4, 5,"},{"Start":"01:18.840 ","End":"01:21.765","Text":"0, 3, 2 minus 1 and so on."},{"Start":"01:21.765 ","End":"01:25.825","Text":"Now 3 by 3 determinant. Let\u0027s see."},{"Start":"01:25.825 ","End":"01:27.880","Text":"There is only a single 0,"},{"Start":"01:27.880 ","End":"01:33.055","Text":"so I got to choose either the top row or the right column."},{"Start":"01:33.055 ","End":"01:38.350","Text":"In our particular reason I\u0027ll choose the top row."},{"Start":"01:38.350 ","End":"01:39.490","Text":"Then we\u0027ll have 2 entries."},{"Start":"01:39.490 ","End":"01:42.000","Text":"We\u0027ll have the 4 and the 5."},{"Start":"01:42.000 ","End":"01:43.905","Text":"Let\u0027s start with the 4."},{"Start":"01:43.905 ","End":"01:46.605","Text":"That will be as a plus."},{"Start":"01:46.605 ","End":"01:51.795","Text":"Then the minor will be this determinant."},{"Start":"01:51.795 ","End":"01:56.580","Text":"That gives us the first part of this."},{"Start":"01:56.580 ","End":"02:00.535","Text":"Here is the 2 minus 1 minus 3 2 here."},{"Start":"02:00.535 ","End":"02:05.550","Text":"The other bit comes from the 5,"},{"Start":"02:05.550 ","End":"02:07.730","Text":"and this 1 is a minus,"},{"Start":"02:07.730 ","End":"02:09.500","Text":"so it\u0027s minus 5."},{"Start":"02:09.500 ","End":"02:15.975","Text":"The minor is 3 minus 4 minus 5 2, like this."},{"Start":"02:15.975 ","End":"02:19.595","Text":"If we just compute it,"},{"Start":"02:19.595 ","End":"02:22.970","Text":"we just get the oldest arithmetic to do."},{"Start":"02:22.970 ","End":"02:25.265","Text":"Remember the 2 by 2 determinant,"},{"Start":"02:25.265 ","End":"02:29.790","Text":"multiply this diagonal and subtract a part of this diagonal."},{"Start":"02:29.790 ","End":"02:32.045","Text":"This minus this, and so on."},{"Start":"02:32.045 ","End":"02:35.370","Text":"The answer is 6, and we\u0027re done."}],"ID":9885}],"Thumbnail":null,"ID":7285},{"Name":"Rules of Determinants","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 Parts 1-3","Duration":"5m 31s","ChapterTopicVideoID":9598,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9598.jpeg","UploadDate":"2017-07-26T08:37:19.0900000","DurationForVideoObject":"PT5M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.295","Text":"Size is going to be 3,"},{"Start":"00:02.295 ","End":"00:10.390","Text":"4 by 4 determinants and we have to compute them using row operations."},{"Start":"00:10.640 ","End":"00:13.710","Text":"Here\u0027s the first 1."},{"Start":"00:13.710 ","End":"00:19.315","Text":"The idea is to try and get them into row echelon form."},{"Start":"00:19.315 ","End":"00:24.980","Text":"What we can do is add multiples of this first row to"},{"Start":"00:24.980 ","End":"00:30.615","Text":"the other rows and get 0s in the rest of this column."},{"Start":"00:30.615 ","End":"00:38.270","Text":"We\u0027re going to add twice the first row to the second row. That\u0027s this."},{"Start":"00:38.270 ","End":"00:43.460","Text":"Then we\u0027ll subtract 3 times the first row from the third row"},{"Start":"00:43.460 ","End":"00:48.245","Text":"and put that into the row and that will give us a 0 here."},{"Start":"00:48.245 ","End":"00:54.695","Text":"Then with the last row will subtract twice the first row from it."},{"Start":"00:54.695 ","End":"00:58.420","Text":"If we do all these 3 operations,"},{"Start":"00:58.420 ","End":"01:00.720","Text":"we will get this."},{"Start":"01:00.720 ","End":"01:04.760","Text":"Each of these operations doesn\u0027t change the determinant."},{"Start":"01:04.760 ","End":"01:08.105","Text":"Adding a multiple of 1 row to another doesn\u0027t change it."},{"Start":"01:08.105 ","End":"01:10.490","Text":"I won\u0027t go through all the details, but let\u0027s just,"},{"Start":"01:10.490 ","End":"01:13.960","Text":"for example, show you how I got this third row."},{"Start":"01:13.960 ","End":"01:19.219","Text":"I do 3 minus 3 times 1 is 0."},{"Start":"01:19.219 ","End":"01:23.255","Text":"I\u0027m talking about this part here."},{"Start":"01:23.255 ","End":"01:28.700","Text":"Then if I do 5 minus 3 times 3 is negative 4,"},{"Start":"01:28.700 ","End":"01:31.430","Text":"2 minus 3 times 0 is 2."},{"Start":"01:31.430 ","End":"01:34.550","Text":"1 minus 3 times 2 is minus 5."},{"Start":"01:34.550 ","End":"01:36.560","Text":"Try it with the last row,"},{"Start":"01:36.560 ","End":"01:38.210","Text":"subtracting twice the first row,"},{"Start":"01:38.210 ","End":"01:39.845","Text":"you\u0027ll see that you get this."},{"Start":"01:39.845 ","End":"01:44.840","Text":"Now, when 2 rows are the same in a determinant,"},{"Start":"01:44.840 ","End":"01:47.915","Text":"then the determinant is just 0."},{"Start":"01:47.915 ","End":"01:51.330","Text":"That\u0027s Part 1 done."},{"Start":"01:51.800 ","End":"01:57.310","Text":"Let\u0027s continue to Part 2."},{"Start":"01:57.380 ","End":"02:04.320","Text":"In this 1, I just need 2-row operations to get the first column to be 1"},{"Start":"02:04.320 ","End":"02:11.465","Text":"and 0s to subtract twice this from this,"},{"Start":"02:11.465 ","End":"02:12.905","Text":"and then add this,"},{"Start":"02:12.905 ","End":"02:15.325","Text":"the first row to the fourth row."},{"Start":"02:15.325 ","End":"02:20.000","Text":"This is what I said in words just written in coded form."},{"Start":"02:20.000 ","End":"02:24.870","Text":"If we do that, you can check and see that this is"},{"Start":"02:24.870 ","End":"02:31.280","Text":"the determinant we get on each of these raw operations doesn\u0027t change the value."},{"Start":"02:31.280 ","End":"02:40.810","Text":"Finally, we just note that this row is the same as this row."},{"Start":"02:40.910 ","End":"02:48.680","Text":"You can also say that this row is a multiple of this row and also works,"},{"Start":"02:48.680 ","End":"02:54.220","Text":"but it\u0027s just easier to see that the second and fourth rows are equal."},{"Start":"02:54.220 ","End":"02:56.940","Text":"This is also 0."},{"Start":"02:56.940 ","End":"02:59.850","Text":"Now let\u0027s get onto the next 1."},{"Start":"02:59.850 ","End":"03:03.590","Text":"Here\u0027s the third and last determinant in"},{"Start":"03:03.590 ","End":"03:09.020","Text":"this clip and we proceed as usual with row operations."},{"Start":"03:09.020 ","End":"03:13.225","Text":"We want to try and get old 0s below this 1."},{"Start":"03:13.225 ","End":"03:15.125","Text":"So we\u0027ll do 3 things."},{"Start":"03:15.125 ","End":"03:17.870","Text":"Subtract this row from this row,"},{"Start":"03:17.870 ","End":"03:20.225","Text":"add this row to this row,"},{"Start":"03:20.225 ","End":"03:24.250","Text":"and subtract 3 times this row from this row."},{"Start":"03:24.250 ","End":"03:28.860","Text":"Here\u0027s what I just said in a more technical language."},{"Start":"03:29.330 ","End":"03:35.790","Text":"What I want to do next is get 0s below this 1."},{"Start":"03:35.790 ","End":"03:40.565","Text":"If I subtract this row from this row and leave it here,"},{"Start":"03:40.565 ","End":"03:44.680","Text":"and also twice this 1 from this 1."},{"Start":"03:44.680 ","End":"03:48.365","Text":"Here it is in more precise terms."},{"Start":"03:48.365 ","End":"03:53.925","Text":"Then what we will get is this,"},{"Start":"03:53.925 ","End":"03:58.610","Text":"and it\u0027s close to being in echelon form."},{"Start":"03:58.610 ","End":"04:06.004","Text":"In fact, I notice that if we just swap the 2 last rows around,"},{"Start":"04:06.004 ","End":"04:10.265","Text":"then it will be in echelon form."},{"Start":"04:10.265 ","End":"04:14.465","Text":"This is what I\u0027m suggesting in technical terms."},{"Start":"04:14.465 ","End":"04:21.500","Text":"But the thing is that previously the row operations didn\u0027t change the determinant."},{"Start":"04:21.500 ","End":"04:24.650","Text":"Adding a multiple of 1 row to another doesn\u0027t change,"},{"Start":"04:24.650 ","End":"04:29.200","Text":"but swapping 2 rows around makes it minus."},{"Start":"04:29.200 ","End":"04:33.335","Text":"What we get if we swap these 2, is what it looks like."},{"Start":"04:33.335 ","End":"04:37.045","Text":"But notice that I put a minus here."},{"Start":"04:37.045 ","End":"04:46.615","Text":"Now, what we have is not just echelon form, it\u0027s upper triangular."},{"Start":"04:46.615 ","End":"04:52.280","Text":"Here\u0027s the main diagonal and everything above it,"},{"Start":"04:52.280 ","End":"04:56.045","Text":"and below this, it\u0027s all 0s."},{"Start":"04:56.045 ","End":"05:01.665","Text":"This part is all 0s and that\u0027s what it means to be upper triangular."},{"Start":"05:01.665 ","End":"05:04.010","Text":"When we have that case,"},{"Start":"05:04.010 ","End":"05:11.645","Text":"all we have to do is multiply the entries on the main diagonal."},{"Start":"05:11.645 ","End":"05:14.560","Text":"But of course, there\u0027s still a minus here."},{"Start":"05:14.560 ","End":"05:19.430","Text":"What we end up with is the minus from here and then 1, 1,"},{"Start":"05:19.430 ","End":"05:22.430","Text":"3, sorry, it\u0027s a minus 3 there,"},{"Start":"05:22.430 ","End":"05:25.040","Text":"and then a 1."},{"Start":"05:25.040 ","End":"05:31.050","Text":"The answer is just 3. Now we\u0027re done."}],"ID":9886},{"Watched":false,"Name":"Exercise 1 Parts 4-6","Duration":"10m 36s","ChapterTopicVideoID":9599,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9599.jpeg","UploadDate":"2017-07-26T08:37:48.0930000","DurationForVideoObject":"PT10M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"In this exercise, we\u0027re going to evaluate some 5 by 5 determinants."},{"Start":"00:04.830 ","End":"00:06.855","Text":"There will be 3 of them."},{"Start":"00:06.855 ","End":"00:10.425","Text":"We\u0027re going to be using row operations."},{"Start":"00:10.425 ","End":"00:13.350","Text":"Here\u0027s the first one."},{"Start":"00:13.350 ","End":"00:19.725","Text":"We\u0027re going to do row operations to make everything 0 below this 1."},{"Start":"00:19.725 ","End":"00:24.060","Text":"We\u0027re just going to be adding multiples or subtracting multiples"},{"Start":"00:24.060 ","End":"00:28.860","Text":"of this first row from the next 4 rows."},{"Start":"00:28.860 ","End":"00:32.145","Text":"Like we\u0027re going to subtract it once from here,"},{"Start":"00:32.145 ","End":"00:34.330","Text":"and then we\u0027re going to add it twice here,"},{"Start":"00:34.330 ","End":"00:37.400","Text":"subtract it 3 times from here and from here."},{"Start":"00:37.400 ","End":"00:41.280","Text":"I\u0027ll summarize what I said here. I\u0027ll just take 1 of them."},{"Start":"00:41.280 ","End":"00:51.120","Text":"For example, this 1 says that we take row 4 and subtract from it 3 times row 1,"},{"Start":"00:51.120 ","End":"00:54.105","Text":"and it stays in row 4."},{"Start":"00:54.105 ","End":"00:59.165","Text":"Each of these will not change the determinant,"},{"Start":"00:59.165 ","End":"01:02.725","Text":"when you add or subtract a multiple of 1 row from another,"},{"Start":"01:02.725 ","End":"01:05.630","Text":"it doesn\u0027t change the determinant."},{"Start":"01:07.240 ","End":"01:15.205","Text":"This is what we get and what we want now is to make 0s here and here."},{"Start":"01:15.205 ","End":"01:23.080","Text":"The way to do that would be to add the second row to the fourth row."},{"Start":"01:23.080 ","End":"01:28.630","Text":"That will make this 0 and add twice the second row to the fifth row,"},{"Start":"01:28.630 ","End":"01:30.395","Text":"to the last one."},{"Start":"01:30.395 ","End":"01:32.310","Text":"Here I wrote it down."},{"Start":"01:32.310 ","End":"01:34.695","Text":"This one says that in to row 4,"},{"Start":"01:34.695 ","End":"01:37.660","Text":"we put what was row 4 plus row 2,"},{"Start":"01:37.660 ","End":"01:43.245","Text":"and row 5 gets what it was plus twice row 2."},{"Start":"01:43.245 ","End":"01:46.510","Text":"If we do those operations,"},{"Start":"01:46.510 ","End":"01:49.870","Text":"you can check that we get this determinant."},{"Start":"01:49.870 ","End":"01:52.480","Text":"I\u0027ll just show you, for example,"},{"Start":"01:52.480 ","End":"01:55.580","Text":"row 4 plus row 2."},{"Start":"01:55.580 ","End":"01:59.984","Text":"This plus 0 plus 0 is 0,"},{"Start":"01:59.984 ","End":"02:01.800","Text":"2 and minus 2,"},{"Start":"02:01.800 ","End":"02:03.435","Text":"that gives me this 0."},{"Start":"02:03.435 ","End":"02:07.180","Text":"Minus 4 and 0 is still minus 4,"},{"Start":"02:07.180 ","End":"02:10.040","Text":"minus 1 and 8 is 7,"},{"Start":"02:10.040 ","End":"02:12.975","Text":"minus 6 and minus 2 is minus 8."},{"Start":"02:12.975 ","End":"02:21.135","Text":"Similarly, for the last one and so on now which almost in row echelon form."},{"Start":"02:21.135 ","End":"02:25.100","Text":"If I could just switch these 2 rows around,"},{"Start":"02:25.100 ","End":"02:28.720","Text":"put this minus 4 up there,"},{"Start":"02:28.720 ","End":"02:32.345","Text":"this 0 would go down here, that would be good."},{"Start":"02:32.345 ","End":"02:35.345","Text":"Well, we can change 2 rows around,"},{"Start":"02:35.345 ","End":"02:37.460","Text":"but if we do that,"},{"Start":"02:37.460 ","End":"02:39.260","Text":"we also have to change the sign."},{"Start":"02:39.260 ","End":"02:42.830","Text":"It\u0027s 1 of those operations that does make a difference."},{"Start":"02:42.830 ","End":"02:45.350","Text":"We write this as follows."},{"Start":"02:45.350 ","End":"02:49.100","Text":"That row 3 changes places with row 4."},{"Start":"02:49.100 ","End":"02:51.580","Text":"What it gives us,"},{"Start":"02:51.580 ","End":"02:55.205","Text":"this is what we get after we swap rows 3 and 4."},{"Start":"02:55.205 ","End":"03:00.845","Text":"Notice that now it\u0027s not just row echelon form,"},{"Start":"03:00.845 ","End":"03:09.550","Text":"it\u0027s actually an upper triangular determinant or matrix."},{"Start":"03:09.550 ","End":"03:16.524","Text":"Notice that we added the minus because we switched rows around."},{"Start":"03:16.524 ","End":"03:19.510","Text":"Now when you have an upper triangular,"},{"Start":"03:19.510 ","End":"03:25.180","Text":"then all you have to do is multiply the entries on the diagonal."},{"Start":"03:25.180 ","End":"03:28.955","Text":"Here\u0027s the minus and then the diagonal here,"},{"Start":"03:28.955 ","End":"03:34.075","Text":"and it\u0027s just 1 times 2 times minus 4 times 3 times 1."},{"Start":"03:34.075 ","End":"03:38.240","Text":"The answer comes out to be 24."},{"Start":"03:38.240 ","End":"03:41.465","Text":"Let\u0027s move on to the next one."},{"Start":"03:41.465 ","End":"03:45.800","Text":"Here\u0027s the second 5 by 5 determinant."},{"Start":"03:45.800 ","End":"03:48.820","Text":"We start off with usual, row operations."},{"Start":"03:48.820 ","End":"03:51.535","Text":"I see that there\u0027s a 1 here which is good."},{"Start":"03:51.535 ","End":"03:54.655","Text":"But I want all 0s and here I have a 3."},{"Start":"03:54.655 ","End":"03:59.210","Text":"What I\u0027m going to do is subtract 3 times this row from this row."},{"Start":"03:59.210 ","End":"04:01.400","Text":"This is how we write it."},{"Start":"04:01.400 ","End":"04:07.445","Text":"Second row minus 3 times the first row is what goes in the second row."},{"Start":"04:07.445 ","End":"04:14.255","Text":"What we get is the same determinant."},{"Start":"04:14.255 ","End":"04:18.110","Text":"They\u0027re equal because this row operation where we add a multiple"},{"Start":"04:18.110 ","End":"04:21.740","Text":"of 1 row to another doesn\u0027t change the determinant."},{"Start":"04:21.740 ","End":"04:27.210","Text":"But in a moment, we\u0027ll do something a bit different because what I have"},{"Start":"04:27.210 ","End":"04:33.060","Text":"now is I\u0027ve got the 0s here that\u0027s fine,"},{"Start":"04:33.060 ","End":"04:36.290","Text":"but here is where I have a problem."},{"Start":"04:36.290 ","End":"04:41.850","Text":"I want to get rid of the minus 3 and the 5 to make these both 0."},{"Start":"04:42.190 ","End":"04:48.260","Text":"One way to do that would be to multiply this by"},{"Start":"04:48.260 ","End":"04:54.575","Text":"2 and add 3 times this because it get minus 6 plus 6 is 0."},{"Start":"04:54.575 ","End":"04:59.109","Text":"Similarly here, if I double this row, that will give me 10."},{"Start":"04:59.109 ","End":"05:02.090","Text":"Then I can subtract 5 times this row."},{"Start":"05:02.090 ","End":"05:04.400","Text":"That will also give me 0."},{"Start":"05:04.400 ","End":"05:11.570","Text":"But doubling a row does change the determinant."},{"Start":"05:11.570 ","End":"05:15.945","Text":"Well, first let\u0027s just write down what I said in more technical language."},{"Start":"05:15.945 ","End":"05:19.875","Text":"I take row 4 and double it,"},{"Start":"05:19.875 ","End":"05:23.885","Text":"and then I add 3 times row 3,"},{"Start":"05:23.885 ","End":"05:26.290","Text":"and this all stays in row 4."},{"Start":"05:26.290 ","End":"05:29.510","Text":"I double this and then add 3 times this."},{"Start":"05:29.510 ","End":"05:33.180","Text":"Now the doubling is the problem because this"},{"Start":"05:33.180 ","End":"05:36.995","Text":"is not one of those operations that leaves the determinant unchanged."},{"Start":"05:36.995 ","End":"05:41.540","Text":"This actually causes the determinant to be doubled."},{"Start":"05:41.540 ","End":"05:44.960","Text":"Similarly, when I do it on the last row,"},{"Start":"05:44.960 ","End":"05:50.945","Text":"I double it and then add or subtract a multiple of this row."},{"Start":"05:50.945 ","End":"05:53.000","Text":"That\u0027s also going to double the determinants."},{"Start":"05:53.000 ","End":"05:59.220","Text":"We have to compensate by dividing by 2 and dividing by 2 again."},{"Start":"05:59.510 ","End":"06:02.410","Text":"Note that after doing these operations,"},{"Start":"06:02.410 ","End":"06:06.955","Text":"I do indeed get a 0 and a 0 here and these numbers have changed."},{"Start":"06:06.955 ","End":"06:10.390","Text":"But I also, because of this 2 put a 1/2,"},{"Start":"06:10.390 ","End":"06:11.680","Text":"and because of this 2,"},{"Start":"06:11.680 ","End":"06:15.520","Text":"I put a 1/2 to compensate because multiplying"},{"Start":"06:15.520 ","End":"06:21.020","Text":"a row by a number has the effect of multiplying the determinant."},{"Start":"06:21.960 ","End":"06:24.865","Text":"Now we\u0027re going to continue."},{"Start":"06:24.865 ","End":"06:28.570","Text":"What we have to do next is"},{"Start":"06:28.570 ","End":"06:35.880","Text":"to get a 0 here which,"},{"Start":"06:35.880 ","End":"06:40.780","Text":"of course, I\u0027ll get by just adding the fourth row to the fifth row."},{"Start":"06:40.780 ","End":"06:42.240","Text":"This is how we write it."},{"Start":"06:42.240 ","End":"06:48.449","Text":"Row 5 plus row 4 go into the row 5."},{"Start":"06:48.449 ","End":"06:52.260","Text":"This doesn\u0027t change the determinant."},{"Start":"06:52.260 ","End":"06:58.620","Text":"Here\u0027s what we have. I just combined the 1/2 with the 1/2 to give a 1/4."},{"Start":"07:17.090 ","End":"07:25.489","Text":"Notice that what we have here is an upper triangular matrix with determinant."},{"Start":"07:25.489 ","End":"07:34.180","Text":"We have a diagonal and everything below the diagonal is 0."},{"Start":"07:34.180 ","End":"07:39.890","Text":"We have an upper triangular matrix and the determinant of"},{"Start":"07:39.890 ","End":"07:46.480","Text":"such a thing is just the product of all the elements along the diagonal."},{"Start":"07:46.480 ","End":"07:50.465","Text":"What we\u0027re left with is the quarter from here."},{"Start":"07:50.465 ","End":"07:55.070","Text":"Then all these 5 elements multiplied together."},{"Start":"07:55.070 ","End":"07:59.700","Text":"That gives us the answer of 44."},{"Start":"07:59.840 ","End":"08:05.295","Text":"Now we come to the third of these 5 by 5 determinants."},{"Start":"08:05.295 ","End":"08:10.830","Text":"I\u0027m going to show you a new trick in this one."},{"Start":"08:10.830 ","End":"08:14.960","Text":"This is breaking the matrix up into blocks."},{"Start":"08:14.960 ","End":"08:18.230","Text":"It\u0027s best explained with illustration."},{"Start":"08:18.230 ","End":"08:22.960","Text":"I\u0027ll draw some lines and I\u0027ll show you where the blocks are."},{"Start":"08:22.960 ","End":"08:25.050","Text":"These 2 columns,"},{"Start":"08:25.050 ","End":"08:26.855","Text":"it\u0027s like combined into 1."},{"Start":"08:26.855 ","End":"08:28.819","Text":"Similarly, the top 2 rows,"},{"Start":"08:28.819 ","End":"08:32.605","Text":"it\u0027s like combined into 1 and the bottom 2 rows."},{"Start":"08:32.605 ","End":"08:36.000","Text":"It\u0027s like I have 9 elements,"},{"Start":"08:36.000 ","End":"08:37.260","Text":"a 3 by 3,"},{"Start":"08:37.260 ","End":"08:38.930","Text":"if I just look at it as blocks,"},{"Start":"08:38.930 ","End":"08:40.820","Text":"this is a block and this is a block."},{"Start":"08:40.820 ","End":"08:43.205","Text":"If you take a block view,"},{"Start":"08:43.205 ","End":"08:46.340","Text":"then this is really"},{"Start":"08:46.340 ","End":"08:55.290","Text":"an upper diagonal matrix because I have 0s below the diagonal."},{"Start":"08:55.910 ","End":"09:03.710","Text":"The diagonal is here and these 3 blocks,"},{"Start":"09:03.710 ","End":"09:06.260","Text":"this, this, and this are on the diagonal."},{"Start":"09:06.260 ","End":"09:08.645","Text":"I\u0027ll add a bit of color,"},{"Start":"09:08.645 ","End":"09:13.590","Text":"and they\u0027ll add the lines back in."},{"Start":"09:15.950 ","End":"09:23.120","Text":"Then you can see that each color is like the elements on the diagonal."},{"Start":"09:23.120 ","End":"09:32.210","Text":"The upper triangle is all this."},{"Start":"09:32.210 ","End":"09:34.630","Text":"Didn\u0027t do that so great."},{"Start":"09:34.630 ","End":"09:39.690","Text":"The diagonal, these 3,"},{"Start":"09:39.690 ","End":"09:47.050","Text":"this 1, this 1, and this 1."},{"Start":"09:48.770 ","End":"09:52.750","Text":"Normally, we just multiply the elements on the diagonal,"},{"Start":"09:52.750 ","End":"09:54.655","Text":"but because it\u0027s in block form,"},{"Start":"09:54.655 ","End":"09:56.575","Text":"it\u0027s not exactly elements."},{"Start":"09:56.575 ","End":"10:02.865","Text":"We multiply determinants of 2 by 2 determinant times a 1 by 1 times a 2 by 2."},{"Start":"10:02.865 ","End":"10:05.330","Text":"From here we get this."},{"Start":"10:05.330 ","End":"10:07.145","Text":"Maybe I\u0027ll get rid of the color,"},{"Start":"10:07.145 ","End":"10:09.590","Text":"then we can see it a bit better."},{"Start":"10:09.590 ","End":"10:13.790","Text":"Then we just actually do the multiplication."},{"Start":"10:13.790 ","End":"10:18.585","Text":"From here I get 1 times 5 minus 3 times 1,"},{"Start":"10:18.585 ","End":"10:19.890","Text":"and then the 2,"},{"Start":"10:19.890 ","End":"10:23.655","Text":"and then 4 times 7 minus 1 times 2."},{"Start":"10:23.655 ","End":"10:27.540","Text":"We compute this, it comes out to 104."},{"Start":"10:27.540 ","End":"10:35.880","Text":"That\u0027s a trick with blocks. We are done."}],"ID":9887},{"Watched":false,"Name":"Exercise 2","Duration":"3m 11s","ChapterTopicVideoID":9600,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9600.jpeg","UploadDate":"2017-07-26T08:37:57.5670000","DurationForVideoObject":"PT3M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.650","Text":"Here we have to evaluate the following 4 by 4 determinant."},{"Start":"00:03.650 ","End":"00:06.560","Text":"Up to now we\u0027ve had 2 main techniques,"},{"Start":"00:06.560 ","End":"00:13.125","Text":"and that was expanding along a row or column."},{"Start":"00:13.125 ","End":"00:18.000","Text":"We also had the row operations or"},{"Start":"00:18.000 ","End":"00:26.505","Text":"column operations as column row ups to try and bring it or get closer to echelon form."},{"Start":"00:26.505 ","End":"00:30.840","Text":"What we\u0027re going to do here, and in general,"},{"Start":"00:30.840 ","End":"00:32.160","Text":"it\u0027s a good idea,"},{"Start":"00:32.160 ","End":"00:36.390","Text":"is to use a combination of both techniques."},{"Start":"00:36.390 ","End":"00:38.565","Text":"That usually yields the best results."},{"Start":"00:38.565 ","End":"00:40.590","Text":"Let\u0027s do it here."},{"Start":"00:40.590 ","End":"00:49.240","Text":"Now, notice that this column could be brought to be mostly 0s or 5 or 3 0s."},{"Start":"00:49.240 ","End":"00:51.695","Text":"If I just got this to be 0,"},{"Start":"00:51.695 ","End":"00:53.449","Text":"which I can do with their operation."},{"Start":"00:53.449 ","End":"01:00.880","Text":"If I subtract 3 times the top row from the bottom row, or more precisely,"},{"Start":"01:00.880 ","End":"01:07.860","Text":"this row 4 subtract 3 times row 1 and put that into row 4,"},{"Start":"01:07.860 ","End":"01:13.130","Text":"and that brings us here like 15 minus 3 times 5 is 0,"},{"Start":"01:13.130 ","End":"01:17.045","Text":"minus 7 takeaway minus 9 is 2,"},{"Start":"01:17.045 ","End":"01:21.410","Text":"minus 2 takeaway 3 times minus 1 is 1."},{"Start":"01:21.410 ","End":"01:23.120","Text":"This is what we get."},{"Start":"01:23.120 ","End":"01:32.380","Text":"Now, what I would suggest is expanding along the second column."},{"Start":"01:32.380 ","End":"01:34.250","Text":"Since it\u0027s mostly zeros,"},{"Start":"01:34.250 ","End":"01:37.435","Text":"we only have to consider the 5 element."},{"Start":"01:37.435 ","End":"01:40.105","Text":"I\u0027ll cross out the row,"},{"Start":"01:40.105 ","End":"01:46.655","Text":"and notice that this is a minus on the checkerboard pattern plus minus."},{"Start":"01:46.655 ","End":"01:50.959","Text":"We have a minus times the 5 times the minor,"},{"Start":"01:50.959 ","End":"01:54.520","Text":"which is this determinant, 3 by 3."},{"Start":"01:54.520 ","End":"01:56.210","Text":"Now looking at this,"},{"Start":"01:56.210 ","End":"02:00.230","Text":"I see there if I add twice the top row to the middle row,"},{"Start":"02:00.230 ","End":"02:02.290","Text":"I can get a 0 here,"},{"Start":"02:02.290 ","End":"02:03.919","Text":"and to be precise,"},{"Start":"02:03.919 ","End":"02:10.310","Text":"I place in row 2 the sum of row 2 plus twice row"},{"Start":"02:10.310 ","End":"02:17.490","Text":"1 minus 6 plus twice 3 is 0 and here I add twice 1 minus 2 here,"},{"Start":"02:17.490 ","End":"02:19.710","Text":"I add twice minus 3, I get 3."},{"Start":"02:19.710 ","End":"02:23.360","Text":"At this point I have a couple of options."},{"Start":"02:23.360 ","End":"02:27.765","Text":"I could expand along the first column,"},{"Start":"02:27.765 ","End":"02:30.920","Text":"but there\u0027s something even easier to do because if I"},{"Start":"02:30.920 ","End":"02:34.850","Text":"just add the second row to the last row,"},{"Start":"02:34.850 ","End":"02:37.355","Text":"I\u0027ll get a diagonal matrix,"},{"Start":"02:37.355 ","End":"02:39.810","Text":"which I will write more precisely like this,"},{"Start":"02:39.810 ","End":"02:43.130","Text":"and I prefer to go for a diagonal matrix."},{"Start":"02:43.130 ","End":"02:47.180","Text":"See, this is what I get after I add this to this."},{"Start":"02:47.180 ","End":"02:49.460","Text":"Then it is diagonal,"},{"Start":"02:49.460 ","End":"02:51.575","Text":"upper diagonal to be precise."},{"Start":"02:51.575 ","End":"02:53.780","Text":"With a diagonal matrix,"},{"Start":"02:53.780 ","End":"02:58.375","Text":"all we do is multiply the elements along the diagonal."},{"Start":"02:58.375 ","End":"03:05.705","Text":"That gives me, well, I had the minus 5 and then 3 times minus 2 times 4,"},{"Start":"03:05.705 ","End":"03:11.640","Text":"and that gives me 120 and we\u0027re done."}],"ID":9888},{"Watched":false,"Name":"Exercise 3","Duration":"2m 44s","ChapterTopicVideoID":16864,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/16864.jpeg","UploadDate":"2019-02-18T06:01:17.8130000","DurationForVideoObject":"PT2M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"In this exercise, we have to evaluate the following 4 by 4 determinant."},{"Start":"00:03.945 ","End":"00:05.910","Text":"Like in the previous exercise,"},{"Start":"00:05.910 ","End":"00:08.850","Text":"we will use a combination of methods."},{"Start":"00:08.850 ","End":"00:14.220","Text":"We\u0027ll use expansion along a row or column,"},{"Start":"00:14.220 ","End":"00:22.320","Text":"and we\u0027ll also use row operations or column operations to try to bring to echelon form."},{"Start":"00:22.320 ","End":"00:28.605","Text":"The first thing I do is I look at it and I see I have a 2 here and I have 4s here."},{"Start":"00:28.605 ","End":"00:35.130","Text":"I can make all these 4s disappear by adding twice this row to each of these."},{"Start":"00:35.130 ","End":"00:39.970","Text":"Here is what I just said in more precise terms like the first 1 says,"},{"Start":"00:39.970 ","End":"00:42.140","Text":"row 2, instead of it,"},{"Start":"00:42.140 ","End":"00:48.545","Text":"you put row 2 minus twice row 1 that will give us 4 minus twice 2 is 0 here,"},{"Start":"00:48.545 ","End":"00:51.155","Text":"and so on for the other 2."},{"Start":"00:51.155 ","End":"00:54.380","Text":"What we get in the end is this 4 by 4,"},{"Start":"00:54.380 ","End":"01:00.900","Text":"but the second column contains just single nonzero element."},{"Start":"01:00.900 ","End":"01:03.530","Text":"What we\u0027re going to do now is expand along"},{"Start":"01:03.530 ","End":"01:09.880","Text":"the second column and then only the 2 will count."},{"Start":"01:09.880 ","End":"01:16.610","Text":"I\u0027ll just remind you of the formula that we take the sign from the checkerboard."},{"Start":"01:16.610 ","End":"01:20.540","Text":"Then we take the element and then we take"},{"Start":"01:20.540 ","End":"01:24.880","Text":"the minor of that element and then we add the sign."},{"Start":"01:24.880 ","End":"01:28.000","Text":"Checkerboard means plus, minus, plus, minus."},{"Start":"01:28.000 ","End":"01:30.005","Text":"This 1 is a minus."},{"Start":"01:30.005 ","End":"01:33.905","Text":"This 2 really is a minus 2 and then"},{"Start":"01:33.905 ","End":"01:38.315","Text":"the minor is what we get when we delete what I just highlighted."},{"Start":"01:38.315 ","End":"01:41.255","Text":"It\u0027s 5 minus 3 0 and so on."},{"Start":"01:41.255 ","End":"01:44.320","Text":"We have a 3 by 3 determinant."},{"Start":"01:44.320 ","End":"01:46.730","Text":"I see there is a 0 here,"},{"Start":"01:46.730 ","End":"01:53.530","Text":"I\u0027m happy enough just to expand along the first row or you can do second column."},{"Start":"01:53.530 ","End":"01:56.090","Text":"Then we\u0027ll have 2 terms, 1,"},{"Start":"01:56.090 ","End":"02:01.675","Text":"for the first column and then for the second column, 1 at a time."},{"Start":"02:01.675 ","End":"02:06.010","Text":"The 5 has a plus in the checkerboard."},{"Start":"02:06.020 ","End":"02:08.205","Text":"Minus 2 was here."},{"Start":"02:08.205 ","End":"02:13.525","Text":"Now we have plus 5 and the minor is this 1 which is here."},{"Start":"02:13.525 ","End":"02:17.439","Text":"The minus 3 is a minus on the checkerboard,"},{"Start":"02:17.439 ","End":"02:20.365","Text":"so it\u0027s minus minus 3."},{"Start":"02:20.365 ","End":"02:23.290","Text":"If I strike out this,"},{"Start":"02:23.290 ","End":"02:24.820","Text":"the minor is 7,"},{"Start":"02:24.820 ","End":"02:27.185","Text":"6, 5, 3 as here."},{"Start":"02:27.185 ","End":"02:29.690","Text":"If we do the computation,"},{"Start":"02:29.690 ","End":"02:31.700","Text":"2 by 2 determinants are easy,"},{"Start":"02:31.700 ","End":"02:35.240","Text":"0 times 3 minus 1 times 6 is minus 6."},{"Start":"02:35.240 ","End":"02:45.370","Text":"7 times 3 is 21 minus 30 minus 14 and that gives us 114, and we\u0027re done."}],"ID":17611},{"Watched":false,"Name":"Exercise 4","Duration":"2m 13s","ChapterTopicVideoID":9597,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9597.jpeg","UploadDate":"2017-07-26T08:37:05.0070000","DurationForVideoObject":"PT2M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.695","Text":"We need to compute the following 4-by-4 determinant."},{"Start":"00:04.695 ","End":"00:07.110","Text":"I\u0027m going to use a combination of techniques,"},{"Start":"00:07.110 ","End":"00:09.750","Text":"either row and column operations,"},{"Start":"00:09.750 ","End":"00:15.825","Text":"or expansion along a row or, a column."},{"Start":"00:15.825 ","End":"00:18.375","Text":"The easiest thing to do, I think,"},{"Start":"00:18.375 ","End":"00:25.110","Text":"is to get rid of this 3 by adding 3 times the top row to the second row,"},{"Start":"00:25.110 ","End":"00:28.515","Text":"and then we\u0027ll get a column with multiple 0s."},{"Start":"00:28.515 ","End":"00:30.330","Text":"To be precise,"},{"Start":"00:30.330 ","End":"00:33.960","Text":"row take away 3 times row 1,"},{"Start":"00:33.960 ","End":"00:36.000","Text":"and we put that into row 2,"},{"Start":"00:36.000 ","End":"00:39.750","Text":"and then we\u0027ll get this determinant,"},{"Start":"00:39.750 ","End":"00:41.640","Text":"which is obviously,"},{"Start":"00:41.640 ","End":"00:47.270","Text":"what we wanted to do is expand along the last column."},{"Start":"00:47.270 ","End":"00:49.564","Text":"If we do that,"},{"Start":"00:49.564 ","End":"00:51.510","Text":"because it\u0027s mostly 0s,"},{"Start":"00:51.510 ","End":"00:54.830","Text":"we\u0027ll only get the element 1 contributing and now"},{"Start":"00:54.830 ","End":"00:59.610","Text":"think of the checkerboard pattern plus minus, plus, minus."},{"Start":"00:59.870 ","End":"01:04.910","Text":"We get a minus with the 1 and with the minor,"},{"Start":"01:04.910 ","End":"01:08.305","Text":"which is this determinant 3-by-3."},{"Start":"01:08.305 ","End":"01:10.355","Text":"Then I look at this,"},{"Start":"01:10.355 ","End":"01:12.155","Text":"stare at it a while,"},{"Start":"01:12.155 ","End":"01:15.735","Text":"and then I see that I have a 6 and a minus 6."},{"Start":"01:15.735 ","End":"01:21.920","Text":"If I add these 2 rows into either 1 of them,"},{"Start":"01:21.920 ","End":"01:26.585","Text":"let\u0027s say I add the second row to the third row and put it in the third row,"},{"Start":"01:26.585 ","End":"01:30.535","Text":"then we\u0027ll get a 0 here."},{"Start":"01:30.535 ","End":"01:33.195","Text":"In fact, this is what we get."},{"Start":"01:33.195 ","End":"01:39.740","Text":"At this point, the obvious thing to do is to expand along the first column, mostly 0s."},{"Start":"01:39.740 ","End":"01:41.360","Text":"The only non-zero is this."},{"Start":"01:41.360 ","End":"01:43.450","Text":"We\u0027re only going to get 1 term."},{"Start":"01:43.450 ","End":"01:46.700","Text":"Now, this 6 on the checkerboard is plus minus,"},{"Start":"01:46.700 ","End":"01:50.015","Text":"so it\u0027s minus 6 times the minor."},{"Start":"01:50.015 ","End":"01:52.280","Text":"It\u0027s minus 1 from before,"},{"Start":"01:52.280 ","End":"01:56.485","Text":"minus 6, and then this minor."},{"Start":"01:56.485 ","End":"02:01.530","Text":"Now, the minus 1 with the minus 6 gives me this 6."},{"Start":"02:01.530 ","End":"02:09.090","Text":"This diagonal minus 9 takeaway minus 10 is the 1 and 6 times 1 is 6,"},{"Start":"02:09.090 ","End":"02:13.870","Text":"and that\u0027s the answer. We\u0027re done."}],"ID":9890},{"Watched":false,"Name":"Exercise 5","Duration":"2m 24s","ChapterTopicVideoID":9601,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9601.jpeg","UploadDate":"2017-07-26T08:38:03.3370000","DurationForVideoObject":"PT2M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"This exercise has 3 parts."},{"Start":"00:02.730 ","End":"00:04.260","Text":"This is the first 1."},{"Start":"00:04.260 ","End":"00:06.240","Text":"Each of them were given a determinant,"},{"Start":"00:06.240 ","End":"00:10.440","Text":"and we have to prove without actually computing the determinant,"},{"Start":"00:10.440 ","End":"00:13.530","Text":"that they\u0027re all 0."},{"Start":"00:13.530 ","End":"00:15.315","Text":"Now in this 1,"},{"Start":"00:15.315 ","End":"00:17.130","Text":"if we just look at it for a second,"},{"Start":"00:17.130 ","End":"00:21.930","Text":"you\u0027ll see that the middle column consists of all 0s."},{"Start":"00:21.930 ","End":"00:27.075","Text":"Middle column is also called the C2 at the second column."},{"Start":"00:27.075 ","End":"00:31.860","Text":"Whenever we have a row or a column that\u0027s entirely 0,"},{"Start":"00:31.860 ","End":"00:34.650","Text":"then the determinant is 0."},{"Start":"00:34.650 ","End":"00:35.910","Text":"It\u0027s 1 of the properties,"},{"Start":"00:35.910 ","End":"00:39.300","Text":"but you can easily see why if you expand along the second column,"},{"Start":"00:39.300 ","End":"00:40.980","Text":"you\u0027ll see that everything is 0."},{"Start":"00:40.980 ","End":"00:42.865","Text":"Okay, next 1."},{"Start":"00:42.865 ","End":"00:45.800","Text":"Now here is not immediately obvious."},{"Start":"00:45.800 ","End":"00:49.470","Text":"But notice that 1 plus 4 is 5,"},{"Start":"00:49.470 ","End":"00:51.600","Text":"and 2 plus 5 is 7, and 3,"},{"Start":"00:51.600 ","End":"00:53.100","Text":"plus 6 is 9,"},{"Start":"00:53.100 ","End":"00:58.820","Text":"which means that the third row is the sum of the first second row."},{"Start":"00:58.820 ","End":"01:03.725","Text":"Now in general, when 1 row is the linear combination of the other rows,"},{"Start":"01:03.725 ","End":"01:09.160","Text":"the determinant is 0, and the sum of 2 is certainly a linear combination."},{"Start":"01:09.160 ","End":"01:15.000","Text":"We can immediately conclude that this determinant is 0."},{"Start":"01:15.000 ","End":"01:17.265","Text":"On to the third."},{"Start":"01:17.265 ","End":"01:20.450","Text":"In this 1, if you just look at it for a few seconds,"},{"Start":"01:20.450 ","End":"01:25.580","Text":"you\u0027ll see that if I take this top row and add 1 to all of them,"},{"Start":"01:25.580 ","End":"01:27.065","Text":"I get the second row."},{"Start":"01:27.065 ","End":"01:30.890","Text":"Similarly, if I add 1 to the second row,"},{"Start":"01:30.890 ","End":"01:33.510","Text":"I\u0027ll get the third row."},{"Start":"01:33.610 ","End":"01:39.110","Text":"If I subtract, let\u0027s say the second row from the third row,"},{"Start":"01:39.110 ","End":"01:41.110","Text":"then I\u0027ll get all 1s."},{"Start":"01:41.110 ","End":"01:46.310","Text":"Similarly, if I subtract the first row from the second row,"},{"Start":"01:46.310 ","End":"01:52.515","Text":"then I\u0027m also going to get all 1s here in the second row."},{"Start":"01:52.515 ","End":"01:57.725","Text":"Now the thing to notice is that I have 2 identical rows."},{"Start":"01:57.725 ","End":"02:02.580","Text":"I could write that 2 equals 3."},{"Start":"02:02.580 ","End":"02:06.230","Text":"When this happens by the property of determinants,"},{"Start":"02:06.230 ","End":"02:09.830","Text":"we get that the determinant is 0."},{"Start":"02:09.830 ","End":"02:13.310","Text":"I just should have emphasized earlier that of course,"},{"Start":"02:13.310 ","End":"02:16.385","Text":"row operations like the ones we did here,"},{"Start":"02:16.385 ","End":"02:19.770","Text":"don\u0027t change the value of the determinant."},{"Start":"02:19.910 ","End":"02:24.640","Text":"That\u0027s the third and last part we\u0027re done."}],"ID":9891},{"Watched":false,"Name":"Exercise 6","Duration":"6m 32s","ChapterTopicVideoID":9602,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9602.jpeg","UploadDate":"2017-07-26T08:38:21.3800000","DurationForVideoObject":"PT6M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"This exercise has 4 parts,"},{"Start":"00:03.210 ","End":"00:06.540","Text":"and in each of them we have to prove that"},{"Start":"00:06.540 ","End":"00:11.925","Text":"the given determinant is 0 without actually doing the computation."},{"Start":"00:11.925 ","End":"00:14.190","Text":"Using the properties of determinants."},{"Start":"00:14.190 ","End":"00:16.440","Text":"Here\u0027s the first one."},{"Start":"00:16.440 ","End":"00:18.740","Text":"If you look at it a few moments,"},{"Start":"00:18.740 ","End":"00:26.235","Text":"you\u0027ll see that if we add the first and second rows,"},{"Start":"00:26.235 ","End":"00:27.795","Text":"we get the same thing."},{"Start":"00:27.795 ","End":"00:31.040","Text":"In each case we get x plus y plus z, this and this,"},{"Start":"00:31.040 ","End":"00:32.435","Text":"x plus y plus z,"},{"Start":"00:32.435 ","End":"00:34.850","Text":"this and this, x plus y plus z."},{"Start":"00:34.850 ","End":"00:37.670","Text":"Let\u0027s add them and we\u0027ll put the result,"},{"Start":"00:37.670 ","End":"00:40.370","Text":"say in the second row."},{"Start":"00:40.370 ","End":"00:43.715","Text":"This is the formal way of saying what I just said."},{"Start":"00:43.715 ","End":"00:45.670","Text":"If we do that,"},{"Start":"00:45.670 ","End":"00:47.850","Text":"then we get this."},{"Start":"00:47.850 ","End":"00:49.865","Text":"Notice that the middle row,"},{"Start":"00:49.865 ","End":"00:52.955","Text":"all the entries are x plus y plus z."},{"Start":"00:52.955 ","End":"00:57.469","Text":"Of course, this row operation doesn\u0027t change the determinants."},{"Start":"00:57.469 ","End":"01:02.164","Text":"You can add a multiple of 1 row to another and it won\u0027t change."},{"Start":"01:02.164 ","End":"01:04.625","Text":"What we can do here though,"},{"Start":"01:04.625 ","End":"01:09.814","Text":"is take this x plus y plus z out of the brackets,"},{"Start":"01:09.814 ","End":"01:11.345","Text":"take a factor out."},{"Start":"01:11.345 ","End":"01:14.150","Text":"When you do this to any row or column,"},{"Start":"01:14.150 ","End":"01:16.870","Text":"it comes outside the determinant."},{"Start":"01:16.870 ","End":"01:20.900","Text":"This is what we get, first and last row unchanged and in the second row,"},{"Start":"01:20.900 ","End":"01:23.195","Text":"I took out the common factor."},{"Start":"01:23.195 ","End":"01:25.399","Text":"Now if we look at this determinant,"},{"Start":"01:25.399 ","End":"01:27.710","Text":"it has 2 rows the same,"},{"Start":"01:27.710 ","End":"01:29.245","Text":"the second and third."},{"Start":"01:29.245 ","End":"01:31.565","Text":"We\u0027ve seen this kind of thing before."},{"Start":"01:31.565 ","End":"01:34.190","Text":"This means that this determinant is 0,"},{"Start":"01:34.190 ","End":"01:38.225","Text":"so multiplied by anything it\u0027s still 0."},{"Start":"01:38.225 ","End":"01:40.825","Text":"Let\u0027s move on to the next."},{"Start":"01:40.825 ","End":"01:45.040","Text":"Here it is. Let\u0027s see if we can find the trick here."},{"Start":"01:45.040 ","End":"01:49.840","Text":"Well, the thing that most ensaults is that we have an a, b,"},{"Start":"01:49.840 ","End":"01:51.760","Text":"c here, also a b,"},{"Start":"01:51.760 ","End":"01:53.130","Text":"c here, and a,"},{"Start":"01:53.130 ","End":"01:54.695","Text":"b, c here."},{"Start":"01:54.695 ","End":"01:58.900","Text":"We can actually get rid of these last 2 a, b,"},{"Start":"01:58.900 ","End":"02:06.350","Text":"c columns by just subtracting the first column from both the second and from the third."},{"Start":"02:06.350 ","End":"02:09.160","Text":"This is what it is in formal terms."},{"Start":"02:09.160 ","End":"02:13.560","Text":"This bit says, subtract the first from the second."},{"Start":"02:13.560 ","End":"02:18.125","Text":"Of course, we put it in the second and here we take the third,"},{"Start":"02:18.125 ","End":"02:23.150","Text":"subtract the first, and it stays in the third column."},{"Start":"02:23.640 ","End":"02:27.515","Text":"Now we get this which looks much simpler."},{"Start":"02:27.515 ","End":"02:32.470","Text":"Now, previously in the exercise we took"},{"Start":"02:32.470 ","End":"02:37.240","Text":"a common factor out of 1 of the rows and the same thing works with columns."},{"Start":"02:37.240 ","End":"02:42.550","Text":"I see this column and I can see that I can take x as a factor out of this."},{"Start":"02:42.550 ","End":"02:45.580","Text":"I could take x out and it would be 1, 1, 1;"},{"Start":"02:45.580 ","End":"02:47.335","Text":"but let\u0027s do 2 steps in 1."},{"Start":"02:47.335 ","End":"02:50.905","Text":"I can also take y out of this column."},{"Start":"02:50.905 ","End":"02:53.915","Text":"I take x and y out."},{"Start":"02:53.915 ","End":"02:56.115","Text":"Then we get x, y."},{"Start":"02:56.115 ","End":"03:00.090","Text":"These 2 columns have been replaced by all 1s."},{"Start":"03:00.090 ","End":"03:02.520","Text":"Now, of course, this is familiar,"},{"Start":"03:02.520 ","End":"03:06.450","Text":"we have 2 equal columns,"},{"Start":"03:06.450 ","End":"03:08.750","Text":"it could have been rows or columns,"},{"Start":"03:08.750 ","End":"03:10.460","Text":"and we have 2 identical 1s,"},{"Start":"03:10.460 ","End":"03:12.950","Text":"then the determinant is 0."},{"Start":"03:12.950 ","End":"03:17.425","Text":"Multiplying by x, y still leaves it 0."},{"Start":"03:17.425 ","End":"03:20.370","Text":"On to the next."},{"Start":"03:20.370 ","End":"03:24.575","Text":"Here it is, and it\u0027s trigonometry,"},{"Start":"03:24.575 ","End":"03:29.800","Text":"the sine squareds everywhere and the cosine squareds."},{"Start":"03:29.800 ","End":"03:33.050","Text":"The formula that comes to mind is"},{"Start":"03:33.050 ","End":"03:37.490","Text":"the trigonometrical identity is that sine squared of anything,"},{"Start":"03:37.490 ","End":"03:44.020","Text":"say Alpha plus cosine squared of Alpha is equal to 1."},{"Start":"03:44.020 ","End":"03:48.610","Text":"This would work if alpha was x or y or z."},{"Start":"03:48.610 ","End":"03:55.970","Text":"In fact, let\u0027s do this by adding the first column to the second column."},{"Start":"03:55.970 ","End":"03:59.510","Text":"We\u0027ll put the sum, let\u0027s say in the second column,"},{"Start":"03:59.510 ","End":"04:00.620","Text":"it doesn\u0027t really matter,"},{"Start":"04:00.620 ","End":"04:03.285","Text":"we could have put it in the first,"},{"Start":"04:03.285 ","End":"04:04.805","Text":"we put it in the second."},{"Start":"04:04.805 ","End":"04:08.225","Text":"If we do that, then here we\u0027ll get"},{"Start":"04:08.225 ","End":"04:13.055","Text":"all ones because sine squared x plus cosine squared x is 1."},{"Start":"04:13.055 ","End":"04:15.295","Text":"Similarly for y and for z."},{"Start":"04:15.295 ","End":"04:19.220","Text":"Well, I wrote it out as the middle step just so you can see,"},{"Start":"04:19.220 ","End":"04:24.445","Text":"this is what we get and each of these is equal to 1 because of this."},{"Start":"04:24.445 ","End":"04:27.020","Text":"Like so, and this is already familiar."},{"Start":"04:27.020 ","End":"04:28.984","Text":"We have 2 identical columns."},{"Start":"04:28.984 ","End":"04:35.620","Text":"Whenever that happens, the determinant is 0."},{"Start":"04:35.620 ","End":"04:37.420","Text":"That\u0027s number 3."},{"Start":"04:37.420 ","End":"04:42.220","Text":"We have the fourth part of this exercise."},{"Start":"04:42.220 ","End":"04:44.960","Text":"Here it is. Don\u0027t be alarmed."},{"Start":"04:44.960 ","End":"04:48.545","Text":"Yes, it\u0027s a 7-by-7 determinant,"},{"Start":"04:48.545 ","End":"04:50.270","Text":"but there is a trick."},{"Start":"04:50.270 ","End":"04:56.435","Text":"It\u0027s not something you\u0027d be expected to find on your own in an exam,"},{"Start":"04:56.435 ","End":"05:03.230","Text":"it\u0027s more for educational purposes to illustrate some examples."},{"Start":"05:03.230 ","End":"05:05.360","Text":"The trick is this."},{"Start":"05:05.360 ","End":"05:12.405","Text":"If you add up the entries on any given row, we get 0."},{"Start":"05:12.405 ","End":"05:17.060","Text":"For example, 3 minus 1 is 2,"},{"Start":"05:17.060 ","End":"05:21.230","Text":"6, 11, still 11."},{"Start":"05:21.230 ","End":"05:24.095","Text":"12 minus 12 is 0."},{"Start":"05:24.095 ","End":"05:26.705","Text":"Let\u0027s take 1 more say this one,"},{"Start":"05:26.705 ","End":"05:29.645","Text":"3 and 5 is 8, minus 2 is 6."},{"Start":"05:29.645 ","End":"05:33.140","Text":"6 minus 4 is 2,"},{"Start":"05:33.140 ","End":"05:36.390","Text":"3 minus 3 is 0."},{"Start":"05:37.100 ","End":"05:44.510","Text":"Once we\u0027ve noticed that the sum across all the rows across each one is 0,"},{"Start":"05:44.510 ","End":"05:47.930","Text":"what we can do is say add"},{"Start":"05:47.930 ","End":"05:55.385","Text":"the first 6 rows to the 7th row and put it in the 7th row and that will make it 0."},{"Start":"05:55.385 ","End":"05:57.200","Text":"Here\u0027s how we write that,"},{"Start":"05:57.200 ","End":"06:03.080","Text":"the column 7 plus the first 6 columns into the 7th column."},{"Start":"06:03.080 ","End":"06:05.200","Text":"If we do all that,"},{"Start":"06:05.200 ","End":"06:11.625","Text":"then these columns are unchanged but the last one is now all 0s."},{"Start":"06:11.625 ","End":"06:15.290","Text":"We haven\u0027t changed the value of the determinant by adding"},{"Start":"06:15.290 ","End":"06:20.010","Text":"multiples of some other rows to a given row."},{"Start":"06:20.750 ","End":"06:24.619","Text":"We know that if we have a row or column with 0,"},{"Start":"06:24.619 ","End":"06:26.300","Text":"then the determinant is 0,"},{"Start":"06:26.300 ","End":"06:28.820","Text":"so the answer is just 0."},{"Start":"06:28.820 ","End":"06:33.360","Text":"Now we\u0027re done with all 4 parts of this exercise."}],"ID":9892},{"Watched":false,"Name":"Exercise 8","Duration":"3m 30s","ChapterTopicVideoID":9583,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9583.jpeg","UploadDate":"2017-07-26T08:32:59.0300000","DurationForVideoObject":"PT3M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"This exercise is similar to the previous 1."},{"Start":"00:03.030 ","End":"00:05.925","Text":"We\u0027re given a 3-by-3 determinant."},{"Start":"00:05.925 ","End":"00:08.640","Text":"If you look at just the first 9 letters of the alphabet,"},{"Start":"00:08.640 ","End":"00:09.690","Text":"a, b, c, d, e,"},{"Start":"00:09.690 ","End":"00:11.145","Text":"f, g, h, i,"},{"Start":"00:11.145 ","End":"00:13.995","Text":"and we\u0027re given that this is equal to 4."},{"Start":"00:13.995 ","End":"00:18.480","Text":"Using this, we have to figure out what is this determinant,"},{"Start":"00:18.480 ","End":"00:22.450","Text":"which also uses the letters from a through i."},{"Start":"00:22.450 ","End":"00:26.070","Text":"We\u0027re going to do a series of operations each time getting closer and"},{"Start":"00:26.070 ","End":"00:29.415","Text":"closer to the original determinant."},{"Start":"00:29.415 ","End":"00:31.305","Text":"Let\u0027s get started,"},{"Start":"00:31.305 ","End":"00:34.855","Text":"what I\u0027d like to do first is look at the second column,"},{"Start":"00:34.855 ","End":"00:40.445","Text":"take the 2 that\u0027s here outside the determinant."},{"Start":"00:40.445 ","End":"00:46.430","Text":"Here it is, the 2 that was here and now we have in the middle just d, e,"},{"Start":"00:46.430 ","End":"00:53.310","Text":"f and what I\u0027m going to do is add 3 times"},{"Start":"00:53.310 ","End":"01:00.330","Text":"this column to the first column and that\u0027ll get rid of all these minus 3d,"},{"Start":"01:00.330 ","End":"01:02.820","Text":"minus 3, minus 3f."},{"Start":"01:02.820 ","End":"01:07.160","Text":"I indicate what I\u0027m going to do and the column c1 is"},{"Start":"01:07.160 ","End":"01:12.665","Text":"replaced by whatever it was plus 3 times the second column."},{"Start":"01:12.665 ","End":"01:15.140","Text":"We get this, which is like this,"},{"Start":"01:15.140 ","End":"01:18.020","Text":"but with all this bit knocked off."},{"Start":"01:18.020 ","End":"01:24.115","Text":"Now I see that there\u0027s just like we had here, a 2."},{"Start":"01:24.115 ","End":"01:28.820","Text":"I can take that out of the first column and bring it outside"},{"Start":"01:28.820 ","End":"01:36.390","Text":"the determinant and that will give us this and this 2 is this 2,"},{"Start":"01:36.390 ","End":"01:38.295","Text":"we already had a 2 from before,"},{"Start":"01:38.295 ","End":"01:41.470","Text":"so we have a 2 times 2 now."},{"Start":"01:41.570 ","End":"01:43.710","Text":"Now notice I have a,"},{"Start":"01:43.710 ","End":"01:45.000","Text":"b, c here,"},{"Start":"01:45.000 ","End":"01:47.520","Text":"and I have also a, b, c here,"},{"Start":"01:47.520 ","End":"01:50.595","Text":"each of them with a 4 in front."},{"Start":"01:50.595 ","End":"01:55.460","Text":"If I subtract 4 times the first term from the last,"},{"Start":"01:55.460 ","End":"02:01.470","Text":"and I\u0027ll just write that what we\u0027re doing is subtracting"},{"Start":"02:01.470 ","End":"02:09.080","Text":"4 times the first column"},{"Start":"02:09.080 ","End":"02:16.145","Text":"from the third column and leave that in the third column and this is what we get."},{"Start":"02:16.145 ","End":"02:18.335","Text":"Once in a while I want to remind you that"},{"Start":"02:18.335 ","End":"02:20.600","Text":"this kind of operation where we add a multiple of"},{"Start":"02:20.600 ","End":"02:24.829","Text":"1 column or row to another doesn\u0027t change the value of the determinant,"},{"Start":"02:24.829 ","End":"02:26.890","Text":"which is why I can do it."},{"Start":"02:26.890 ","End":"02:29.750","Text":"Now let\u0027s see where we are."},{"Start":"02:29.750 ","End":"02:32.255","Text":"Well, once again,"},{"Start":"02:32.255 ","End":"02:34.610","Text":"just like in the previous exercise,"},{"Start":"02:34.610 ","End":"02:36.919","Text":"we have a, b, c as a column,"},{"Start":"02:36.919 ","End":"02:38.430","Text":"and we want it as a row,"},{"Start":"02:38.430 ","End":"02:46.505","Text":"so the natural thing to do is to transpose this matrix that\u0027s inside the determinant."},{"Start":"02:46.505 ","End":"02:51.290","Text":"Remember that when you take the transpose of a matrix,"},{"Start":"02:51.290 ","End":"02:54.720","Text":"it has the same determinant as the original."},{"Start":"02:54.940 ","End":"02:57.755","Text":"Now, as for the transpose,"},{"Start":"02:57.755 ","End":"03:01.130","Text":"we change columns to rows or vice versa."},{"Start":"03:01.130 ","End":"03:04.495","Text":"This first column becomes the first row,"},{"Start":"03:04.495 ","End":"03:08.420","Text":"the second column becomes the second row,"},{"Start":"03:08.420 ","End":"03:12.515","Text":"and the third column becomes the third row."},{"Start":"03:12.515 ","End":"03:16.950","Text":"Now this is exactly the original determinant that was given a,"},{"Start":"03:16.950 ","End":"03:18.240","Text":"b, c, d, e, f, g, h, i, j,"},{"Start":"03:18.240 ","End":"03:20.565","Text":"we know that this is equal to 4."},{"Start":"03:20.565 ","End":"03:22.865","Text":"With this 2 times 2,"},{"Start":"03:22.865 ","End":"03:25.790","Text":"we have 4 times 4,"},{"Start":"03:25.790 ","End":"03:30.210","Text":"which is 16 and that\u0027s the answer."}],"ID":9893},{"Watched":false,"Name":"Exercise 7","Duration":"3m 35s","ChapterTopicVideoID":9603,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9603.jpeg","UploadDate":"2017-07-26T08:38:30.7330000","DurationForVideoObject":"PT3M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.000","Text":"In this exercise, we\u0027re given that the determinant of, well,"},{"Start":"00:06.000 ","End":"00:08.790","Text":"it\u0027s just the consecutive letters a, b, c, d,"},{"Start":"00:08.790 ","End":"00:10.305","Text":"e, f, g, h, i,"},{"Start":"00:10.305 ","End":"00:12.285","Text":"given that this is 4."},{"Start":"00:12.285 ","End":"00:22.610","Text":"Using this, we have to figure out the following determinant. I won\u0027t read it out."},{"Start":"00:22.610 ","End":"00:27.070","Text":"You can see what it is also involves the same letters from a through i."},{"Start":"00:27.070 ","End":"00:29.460","Text":"What we\u0027ll do is we\u0027ll start from this and"},{"Start":"00:29.460 ","End":"00:34.510","Text":"step-by-step modify it until we end up with this."},{"Start":"00:35.870 ","End":"00:38.720","Text":"Well, just to begin."},{"Start":"00:38.720 ","End":"00:40.340","Text":"Now, looking at this,"},{"Start":"00:40.340 ","End":"00:42.530","Text":"I see here there\u0027s a d,"},{"Start":"00:42.530 ","End":"00:45.725","Text":"e, f, but there\u0027s a 2 in front of each."},{"Start":"00:45.725 ","End":"00:48.325","Text":"There\u0027s also a d, e, f here,"},{"Start":"00:48.325 ","End":"00:50.135","Text":"maybe not a column but a row,"},{"Start":"00:50.135 ","End":"00:51.905","Text":"and there\u0027s also a d, e, f here."},{"Start":"00:51.905 ","End":"00:56.160","Text":"I suggest let\u0027s divide by 2."},{"Start":"00:56.160 ","End":"00:58.970","Text":"It meaning let\u0027s take the 2 outside of the determinant."},{"Start":"00:58.970 ","End":"01:01.860","Text":"We can do it for any row or column."},{"Start":"01:01.930 ","End":"01:05.510","Text":"Now this 2 is in front and here we have just d,"},{"Start":"01:05.510 ","End":"01:08.105","Text":"e, f. Now,"},{"Start":"01:08.105 ","End":"01:13.685","Text":"if I subtract this last column from the middle column,"},{"Start":"01:13.685 ","End":"01:14.960","Text":"I\u0027ll get rid of the d, e,"},{"Start":"01:14.960 ","End":"01:20.085","Text":"f and I\u0027ll have just the letters from a through i scrambled still."},{"Start":"01:20.085 ","End":"01:22.220","Text":"Wait a minute."},{"Start":"01:22.220 ","End":"01:24.455","Text":"Let\u0027s get some space."},{"Start":"01:24.455 ","End":"01:28.610","Text":"This is the way we write that we\u0027re going to subtract"},{"Start":"01:28.610 ","End":"01:35.560","Text":"the 3rd column from the 2nd column and put the result in the 2nd column."},{"Start":"01:35.560 ","End":"01:37.550","Text":"This is what we get,"},{"Start":"01:37.550 ","End":"01:39.815","Text":"so it\u0027s beginning to take shape."},{"Start":"01:39.815 ","End":"01:46.365","Text":"Now, in our original determinant,"},{"Start":"01:46.365 ","End":"01:47.950","Text":"there wasn\u0027t a, b, c,"},{"Start":"01:47.950 ","End":"01:49.640","Text":"but it was the first row,"},{"Start":"01:49.640 ","End":"01:51.200","Text":"not the first column."},{"Start":"01:51.200 ","End":"01:58.865","Text":"What we can do is to take the transpose of the matrix inside the determinant,"},{"Start":"01:58.865 ","End":"02:02.570","Text":"meaning change rows to columns and vice versa."},{"Start":"02:02.570 ","End":"02:08.160","Text":"Remember the transpose is written with a T as a superscript,"},{"Start":"02:08.160 ","End":"02:13.010","Text":"and the thing is that it has the same determinant as the original."},{"Start":"02:13.010 ","End":"02:19.655","Text":"What we\u0027ll get if we do that is,"},{"Start":"02:19.655 ","End":"02:21.770","Text":"this is the transpose look,"},{"Start":"02:21.770 ","End":"02:30.500","Text":"the first column here becomes the first row and the second column becomes the second row,"},{"Start":"02:30.500 ","End":"02:34.595","Text":"and the 3rd column becomes the third row."},{"Start":"02:34.595 ","End":"02:39.800","Text":"Now we\u0027re very close to what we originally wanted,"},{"Start":"02:39.800 ","End":"02:42.540","Text":"but it starts out, a, b,"},{"Start":"02:42.540 ","End":"02:44.310","Text":"c, but we want d, e,"},{"Start":"02:44.310 ","End":"02:46.665","Text":"f here and g, h, i here."},{"Start":"02:46.665 ","End":"02:50.645","Text":"It looks like we want to switch these 2 rows around,"},{"Start":"02:50.645 ","End":"02:56.140","Text":"which we indicate with a double-arrow row 2 switch with row 3."},{"Start":"02:56.140 ","End":"02:58.010","Text":"Here I have done that,"},{"Start":"02:58.010 ","End":"03:05.510","Text":"but the determinant changes sign when you switch 2 rows,"},{"Start":"03:05.510 ","End":"03:11.765","Text":"which is why I put the minus here."},{"Start":"03:11.765 ","End":"03:21.535","Text":"Now, here we have the original determinant and we were given that this is equal to 4,"},{"Start":"03:21.535 ","End":"03:24.155","Text":"I guess I should have written an extra step,"},{"Start":"03:24.155 ","End":"03:27.170","Text":"which is equal to minus the 2 from here,"},{"Start":"03:27.170 ","End":"03:32.600","Text":"and then the 4 from the given and minus 2 times 4 is minus 8,"},{"Start":"03:32.600 ","End":"03:35.430","Text":"which is our answer."}],"ID":9894},{"Watched":false,"Name":"Exercise 9","Duration":"5m 13s","ChapterTopicVideoID":9584,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9584.jpeg","UploadDate":"2017-07-26T08:33:15.4870000","DurationForVideoObject":"PT5M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"This exercise is a bit like the previous one where we were also"},{"Start":"00:03.600 ","End":"00:07.605","Text":"given this determinant 3 by 3,"},{"Start":"00:07.605 ","End":"00:09.705","Text":"a, b, c, d, e, f, g, h, i."},{"Start":"00:09.705 ","End":"00:11.580","Text":"We\u0027re given that it\u0027s equal to 4,"},{"Start":"00:11.580 ","End":"00:14.490","Text":"and we have to evaluate another determinant."},{"Start":"00:14.490 ","End":"00:17.475","Text":"The previous one, it was also a 3 by 3."},{"Start":"00:17.475 ","End":"00:19.800","Text":"Here we have a 4 by 4,"},{"Start":"00:19.800 ","End":"00:26.190","Text":"but if you notice this column or this row are just 1s and 0s,"},{"Start":"00:26.190 ","End":"00:29.115","Text":"so we could easily get rid of that."},{"Start":"00:29.115 ","End":"00:31.050","Text":"What we can do is, let\u0027s say,"},{"Start":"00:31.050 ","End":"00:35.995","Text":"we expand along the fourth row."},{"Start":"00:35.995 ","End":"00:40.060","Text":"Now, the only one that\u0027s non-zero is this,"},{"Start":"00:40.060 ","End":"00:41.600","Text":"these 3 don\u0027t matter,"},{"Start":"00:41.600 ","End":"00:45.120","Text":"so we just need the entry 1."},{"Start":"00:47.060 ","End":"00:49.730","Text":"What we do is we take the entry,"},{"Start":"00:49.730 ","End":"00:52.130","Text":"multiply it by the checkerboard sign,"},{"Start":"00:52.130 ","End":"00:55.445","Text":"remember plus, minus, plus, minus."},{"Start":"00:55.445 ","End":"00:59.105","Text":"Then we also have to multiply by the minor."},{"Start":"00:59.105 ","End":"01:04.770","Text":"When I cross out the row and column what\u0027s left? Here we are."},{"Start":"01:04.770 ","End":"01:09.080","Text":"Once again, the minus from the plus, minus, plus, minus."},{"Start":"01:09.080 ","End":"01:11.390","Text":"The one is the entry here,"},{"Start":"01:11.390 ","End":"01:15.095","Text":"and this 3 by 3 determinant is the minor."},{"Start":"01:15.095 ","End":"01:19.550","Text":"Now it\u0027s at least the right size and we also have the letters from a through i,"},{"Start":"01:19.550 ","End":"01:21.830","Text":"but a bit scrambled."},{"Start":"01:21.830 ","End":"01:23.920","Text":"Taking a look at this,"},{"Start":"01:23.920 ","End":"01:27.720","Text":"I noticed the middle column,"},{"Start":"01:27.720 ","End":"01:31.400","Text":"and I would like to take these 3,"},{"Start":"01:31.400 ","End":"01:34.205","Text":"which is common to all the elements of the middle column,"},{"Start":"01:34.205 ","End":"01:38.530","Text":"I can pull the 3 outside the determinant."},{"Start":"01:38.530 ","End":"01:42.600","Text":"Here is the 3, so that\u0027s that."},{"Start":"01:42.600 ","End":"01:45.135","Text":"Now let\u0027s see what else we can do."},{"Start":"01:45.135 ","End":"01:50.010","Text":"I see that there\u0027s this a,"},{"Start":"01:50.010 ","End":"01:51.720","Text":"b, c, there is also an a,"},{"Start":"01:51.720 ","End":"01:53.250","Text":"b, c here."},{"Start":"01:53.250 ","End":"01:57.470","Text":"If I subtract the second column from the third column,"},{"Start":"01:57.470 ","End":"01:58.910","Text":"I can get rid of this,"},{"Start":"01:58.910 ","End":"02:01.315","Text":"knock this a, b, c out."},{"Start":"02:01.315 ","End":"02:03.125","Text":"This is how I write,"},{"Start":"02:03.125 ","End":"02:07.960","Text":"subtracting the second column from the third and putting it into the third."},{"Start":"02:07.960 ","End":"02:11.205","Text":"Notice how this a, b, c has disappeared."},{"Start":"02:11.205 ","End":"02:13.205","Text":"Of course, this kind of operation,"},{"Start":"02:13.205 ","End":"02:18.860","Text":"subtracting a multiple of one row or column from another doesn\u0027t change the determinant."},{"Start":"02:18.860 ","End":"02:22.065","Text":"So we\u0027re still with the minus 3 times this."},{"Start":"02:22.065 ","End":"02:25.640","Text":"Now let\u0027s see what we can do next."},{"Start":"02:25.940 ","End":"02:29.330","Text":"Well, I could do the trick with the 3 again,"},{"Start":"02:29.330 ","End":"02:31.070","Text":"because look, I have a 3 here,"},{"Start":"02:31.070 ","End":"02:33.410","Text":"here and here in the last column,"},{"Start":"02:33.410 ","End":"02:36.200","Text":"I can pull this out front."},{"Start":"02:36.200 ","End":"02:40.255","Text":"These 3s disappear and it appears here."},{"Start":"02:40.255 ","End":"02:43.705","Text":"It\u0027s getting tidier and tidier."},{"Start":"02:43.705 ","End":"02:46.270","Text":"Now let\u0027s see what else we can do."},{"Start":"02:46.270 ","End":"02:49.710","Text":"Well, I see this 3d, 3e, 3f,"},{"Start":"02:49.710 ","End":"02:50.970","Text":"which I\u0027d like to get rid of."},{"Start":"02:50.970 ","End":"02:52.170","Text":"Here I see, d, e,"},{"Start":"02:52.170 ","End":"02:58.280","Text":"f. If I subtract 3 times the last column from the first column,"},{"Start":"02:58.280 ","End":"03:00.025","Text":"that should do it."},{"Start":"03:00.025 ","End":"03:07.400","Text":"This is the notation for subtracting 3 times the last column from the first."},{"Start":"03:07.400 ","End":"03:11.180","Text":"If we do it, we end up with this."},{"Start":"03:11.180 ","End":"03:13.370","Text":"Notice that these have just disappeared,"},{"Start":"03:13.370 ","End":"03:17.180","Text":"these 3s in the first column."},{"Start":"03:17.180 ","End":"03:24.640","Text":"Now, it\u0027s time for a transpose because we have an a,"},{"Start":"03:24.640 ","End":"03:29.980","Text":"b, c, but here it\u0027s a column and we want it as a row."},{"Start":"03:29.980 ","End":"03:34.835","Text":"Remember, taking a transpose of a matrix doesn\u0027t change its determinant."},{"Start":"03:34.835 ","End":"03:39.245","Text":"What we get is this, let me explain."},{"Start":"03:39.245 ","End":"03:44.015","Text":"The first column here becomes the first row,"},{"Start":"03:44.015 ","End":"03:48.305","Text":"the second column becomes the second row,"},{"Start":"03:48.305 ","End":"03:51.920","Text":"and the third column becomes the third row."},{"Start":"03:51.920 ","End":"03:53.950","Text":"That\u0027s the transpose."},{"Start":"03:53.950 ","End":"03:57.710","Text":"There\u0027s still more work to do because if you recall,"},{"Start":"03:57.710 ","End":"04:00.300","Text":"what we want to arrive at his a, b,"},{"Start":"04:00.300 ","End":"04:02.150","Text":"c on the top and then d,"},{"Start":"04:02.150 ","End":"04:03.230","Text":"e, f, g, h, i."},{"Start":"04:03.230 ","End":"04:05.015","Text":"The rows are scrambled."},{"Start":"04:05.015 ","End":"04:06.770","Text":"Well, let\u0027s do it one at a time."},{"Start":"04:06.770 ","End":"04:08.010","Text":"Let\u0027s first of all get a, b,"},{"Start":"04:08.010 ","End":"04:12.365","Text":"c to the top by switching the first and second rows."},{"Start":"04:12.365 ","End":"04:16.805","Text":"But remember switching rows changes the determinant."},{"Start":"04:16.805 ","End":"04:20.659","Text":"If we switch any two rows like row 2 with row 1,"},{"Start":"04:20.659 ","End":"04:26.435","Text":"then we have to make the determinant negative. Here we are."},{"Start":"04:26.435 ","End":"04:27.830","Text":"The last row is the same,"},{"Start":"04:27.830 ","End":"04:29.180","Text":"but we swapped these two,"},{"Start":"04:29.180 ","End":"04:31.490","Text":"and now a, b, c is in the right position."},{"Start":"04:31.490 ","End":"04:34.220","Text":"Notice that I\u0027ve changed this minus to a plus."},{"Start":"04:34.220 ","End":"04:40.490","Text":"I deliberately wrote the plus to emphasize exchanging rows changes sign."},{"Start":"04:40.490 ","End":"04:44.375","Text":"Now we do the same trick again with the second and third rows."},{"Start":"04:44.375 ","End":"04:48.545","Text":"I indicated like this and I\u0027m going to exchange these two."},{"Start":"04:48.545 ","End":"04:53.070","Text":"We\u0027re going to then exchange these two and put a minus."},{"Start":"04:53.070 ","End":"04:55.740","Text":"Here\u0027s that minus again."},{"Start":"04:55.740 ","End":"04:57.810","Text":"Now we have a, b, c, d, e,"},{"Start":"04:57.810 ","End":"05:00.255","Text":"f, g, h, i just at the beginning."},{"Start":"05:00.255 ","End":"05:03.285","Text":"Remember this was equal to 4."},{"Start":"05:03.285 ","End":"05:08.235","Text":"We now have minus 3 times 3 is minus 9 times 4."},{"Start":"05:08.235 ","End":"05:11.520","Text":"The answer is minus 36,"},{"Start":"05:11.520 ","End":"05:14.080","Text":"and we are done."}],"ID":9895},{"Watched":false,"Name":"Exercise 10","Duration":"3m 19s","ChapterTopicVideoID":9585,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9585.jpeg","UploadDate":"2017-07-26T08:33:26.7500000","DurationForVideoObject":"PT3M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.160","Text":"In this exercise, you have to evaluate the following determinant."},{"Start":"00:05.160 ","End":"00:12.045","Text":"What I\u0027m going to do first are some row operations to get 0s below the 1 here."},{"Start":"00:12.045 ","End":"00:16.380","Text":"I\u0027m going to subtract the first row from both the second and the third."},{"Start":"00:16.380 ","End":"00:21.765","Text":"This is how I express this more precisely with the notation."},{"Start":"00:21.765 ","End":"00:23.415","Text":"After we do that,"},{"Start":"00:23.415 ","End":"00:25.030","Text":"we get this determinant,"},{"Start":"00:25.030 ","End":"00:27.435","Text":"you can see b minus a is here,"},{"Start":"00:27.435 ","End":"00:28.680","Text":"c minus a is here,"},{"Start":"00:28.680 ","End":"00:32.410","Text":"b squared minus a squared, and so on."},{"Start":"00:32.570 ","End":"00:36.510","Text":"Now what we\u0027re going to do is"},{"Start":"00:36.510 ","End":"00:42.370","Text":"factorize these 2 expressions using the difference of squares formula."},{"Start":"00:42.370 ","End":"00:48.500","Text":"I\u0027m talking about the formula that says that x squared minus y squared is equal to"},{"Start":"00:48.500 ","End":"00:55.040","Text":"x minus y times x plus y called the difference of squares formula."},{"Start":"00:55.040 ","End":"00:57.895","Text":"We\u0027ll apply it here and here."},{"Start":"00:57.895 ","End":"01:02.210","Text":"After we do that, these 2 elements change to this,"},{"Start":"01:02.210 ","End":"01:10.375","Text":"and the next thing we\u0027re going to do is to take out a common factor."},{"Start":"01:10.375 ","End":"01:17.680","Text":"See the second row has b minus a in all 3 entries really,"},{"Start":"01:17.680 ","End":"01:24.275","Text":"because this is also multiple of b minus a. I can take c minus a out of this."},{"Start":"01:24.275 ","End":"01:27.045","Text":"Highlight that have a b minus a here,"},{"Start":"01:27.045 ","End":"01:29.010","Text":"have a b minus a here,"},{"Start":"01:29.010 ","End":"01:30.750","Text":"and somewhere hidden in the 0,"},{"Start":"01:30.750 ","End":"01:34.240","Text":"there\u0027s a b minus a times 0."},{"Start":"01:34.460 ","End":"01:37.130","Text":"I have a c minus a here."},{"Start":"01:37.130 ","End":"01:38.740","Text":"I have a c minus a here,"},{"Start":"01:38.740 ","End":"01:43.830","Text":"and there\u0027s a hidden c minus a times 0 here,"},{"Start":"01:44.450 ","End":"01:47.640","Text":"and here\u0027s the b minus a,"},{"Start":"01:47.640 ","End":"01:49.334","Text":"we pulled it out front."},{"Start":"01:49.334 ","End":"01:51.945","Text":"The 0 stays 0,"},{"Start":"01:51.945 ","End":"01:54.995","Text":"and the c minus a,"},{"Start":"01:54.995 ","End":"01:58.410","Text":"which we took from the bottom row."},{"Start":"01:59.570 ","End":"02:03.060","Text":"Now I look at this and I say,"},{"Start":"02:03.060 ","End":"02:05.310","Text":"I want to get rid of this 1."},{"Start":"02:05.310 ","End":"02:09.940","Text":"If I just subtract the second row from the third,"},{"Start":"02:09.940 ","End":"02:14.495","Text":"and I\u0027ll get an upper diagonal matrix."},{"Start":"02:14.495 ","End":"02:17.850","Text":"Let me just write precisely what I\u0027m doing."},{"Start":"02:17.850 ","End":"02:20.290","Text":"I\u0027m subtracting row 2 from row 3,"},{"Start":"02:20.290 ","End":"02:23.720","Text":"this is how it\u0027s expressed formally."},{"Start":"02:23.720 ","End":"02:28.140","Text":"The 1 minus 1 is 0,"},{"Start":"02:28.140 ","End":"02:34.120","Text":"and this entry comes from just subtracting c plus a."},{"Start":"02:34.120 ","End":"02:38.900","Text":"If I subtract b plus a,"},{"Start":"02:38.900 ","End":"02:41.250","Text":"the a minus a cancels,"},{"Start":"02:41.250 ","End":"02:43.930","Text":"and I\u0027m left with c minus b,"},{"Start":"02:43.930 ","End":"02:46.190","Text":"which is what I have here."},{"Start":"02:46.190 ","End":"02:55.520","Text":"As I said, we\u0027re going to get an upper diagonal matrix determinant,"},{"Start":"02:56.150 ","End":"02:58.745","Text":"and when we have this,"},{"Start":"02:58.745 ","End":"03:04.495","Text":"and all we have to do is multiply the elements on the diagonal,"},{"Start":"03:04.495 ","End":"03:09.925","Text":"1 times 1 times c minus b,"},{"Start":"03:09.925 ","End":"03:12.680","Text":"which is just c minus b,"},{"Start":"03:12.680 ","End":"03:15.035","Text":"and this is what we had before."},{"Start":"03:15.035 ","End":"03:19.290","Text":"This is our answer and we\u0027re done."}],"ID":9896},{"Watched":false,"Name":"Exercise 11","Duration":"5m 7s","ChapterTopicVideoID":9586,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9586.jpeg","UploadDate":"2017-07-26T08:33:48.8770000","DurationForVideoObject":"PT5M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"In this exercise, we have a determinant to evaluate,"},{"Start":"00:04.140 ","End":"00:05.970","Text":"its 4 by 4."},{"Start":"00:05.970 ","End":"00:12.210","Text":"The way we\u0027ll tackle it will be to do some row operations with the aim"},{"Start":"00:12.210 ","End":"00:18.600","Text":"of bringing it to row-echelon or triangular form."},{"Start":"00:18.600 ","End":"00:25.200","Text":"The first thing to do would be to subtract the top row from each of the other 3 rows,"},{"Start":"00:25.200 ","End":"00:27.690","Text":"and that will give us zeros here."},{"Start":"00:27.690 ","End":"00:29.670","Text":"In the proper notation,"},{"Start":"00:29.670 ","End":"00:31.845","Text":"these are the 3 operations we\u0027ll do."},{"Start":"00:31.845 ","End":"00:33.720","Text":"This first 1, for example,"},{"Start":"00:33.720 ","End":"00:36.285","Text":"says we subtract the first from the second and"},{"Start":"00:36.285 ","End":"00:39.640","Text":"we put the result in the second row, and so on."},{"Start":"00:39.640 ","End":"00:43.845","Text":"We get this determinant."},{"Start":"00:43.845 ","End":"00:47.060","Text":"The value of the determinant is not changed when"},{"Start":"00:47.060 ","End":"00:50.585","Text":"you add or subtract multiples of 1 row from another."},{"Start":"00:50.585 ","End":"00:53.520","Text":"We\u0027re at the same determinant,"},{"Start":"00:53.520 ","End":"00:55.005","Text":"and now what we\u0027re going to do,"},{"Start":"00:55.005 ","End":"01:02.405","Text":"is it looks a bit of a mess so we\u0027ll simplify it by using some rules from algebra."},{"Start":"01:02.405 ","End":"01:06.470","Text":"There\u0027s a difference of squares law and a difference of cubes law"},{"Start":"01:06.470 ","End":"01:08.645","Text":"or maybe rule or formula."},{"Start":"01:08.645 ","End":"01:12.905","Text":"Anyway, these are the 2 that I\u0027m talking about in terms of A and B."},{"Start":"01:12.905 ","End":"01:20.125","Text":"We\u0027ll apply them once with y and x and once with z and x and then with t and x."},{"Start":"01:20.125 ","End":"01:22.485","Text":"This 1 for these,"},{"Start":"01:22.485 ","End":"01:25.390","Text":"and this rule for the difference of cubes."},{"Start":"01:25.390 ","End":"01:28.235","Text":"This is what we get after factoring."},{"Start":"01:28.235 ","End":"01:31.610","Text":"Now there\u0027s another thing we can do with determinants is if we"},{"Start":"01:31.610 ","End":"01:34.850","Text":"can take a factor out of a given row or column,"},{"Start":"01:34.850 ","End":"01:36.500","Text":"we can bring it to the front."},{"Start":"01:36.500 ","End":"01:39.170","Text":"So what I\u0027m going to do is for the second row,"},{"Start":"01:39.170 ","End":"01:44.500","Text":"I\u0027m going to take out y minus x from here and from here, and from here."},{"Start":"01:44.500 ","End":"01:47.750","Text":"Here you can take y minus x out of 0."},{"Start":"01:47.750 ","End":"01:50.105","Text":"I mean just gives you 0"},{"Start":"01:50.105 ","End":"01:52.130","Text":"but wait, before we do that,"},{"Start":"01:52.130 ","End":"01:53.450","Text":"I\u0027m going to do 3 things at once."},{"Start":"01:53.450 ","End":"01:56.255","Text":"I\u0027m also going to do the third row,"},{"Start":"01:56.255 ","End":"02:02.750","Text":"where I can take z minus x out as a factor from all of them."},{"Start":"02:02.750 ","End":"02:09.900","Text":"The last row, where I can take out t minus x to the front."},{"Start":"02:09.980 ","End":"02:14.200","Text":"If I do all those operations,"},{"Start":"02:14.200 ","End":"02:20.170","Text":"then here\u0027s my fourth row with the t minus x here,"},{"Start":"02:20.170 ","End":"02:24.495","Text":"z minus x from the third row here."},{"Start":"02:24.495 ","End":"02:28.990","Text":"The y minus x from the second row is here."},{"Start":"02:28.990 ","End":"02:31.870","Text":"Now we\u0027ll continue subtracting rows."},{"Start":"02:31.870 ","End":"02:35.935","Text":"I can get 0 here and here also,"},{"Start":"02:35.935 ","End":"02:40.224","Text":"if I subtract the second from the third and the fourth rows,"},{"Start":"02:40.224 ","End":"02:43.420","Text":"and here is the proper notation for what I just said."},{"Start":"02:43.420 ","End":"02:47.650","Text":"Although it\u0027s easier to just say subtract the second row from the"},{"Start":"02:47.650 ","End":"02:52.570","Text":"third and the fourth here. Here\u0027s what we get."},{"Start":"02:52.570 ","End":"02:53.650","Text":"There are zeros here,"},{"Start":"02:53.650 ","End":"02:55.195","Text":"but this looks a bit of a mess;"},{"Start":"02:55.195 ","End":"02:59.260","Text":"but actually it\u0027s not so bad because this z minus y"},{"Start":"02:59.260 ","End":"03:03.505","Text":"is actually a factor of this and t minus y is a factor of this."},{"Start":"03:03.505 ","End":"03:08.155","Text":"Because again, I can use the difference of squares rule on this."},{"Start":"03:08.155 ","End":"03:11.650","Text":"So z squared minus y squared is z minus y, z plus y."},{"Start":"03:11.650 ","End":"03:14.785","Text":"Here I just take x out and I get z minus y,"},{"Start":"03:14.785 ","End":"03:18.475","Text":"similarly in the bottom row."},{"Start":"03:18.475 ","End":"03:26.290","Text":"So what I\u0027ll do now is take z minus y out of the third row."},{"Start":"03:26.290 ","End":"03:30.160","Text":"This and this is part of the same term."},{"Start":"03:30.160 ","End":"03:32.825","Text":"Then z minus y from here and of course,"},{"Start":"03:32.825 ","End":"03:36.410","Text":"this I can look upon a 0 times z minus y."},{"Start":"03:36.410 ","End":"03:44.645","Text":"Similarly with t minus y in the fourth and last row,"},{"Start":"03:44.645 ","End":"03:46.400","Text":"just cleared some space."},{"Start":"03:46.400 ","End":"03:50.840","Text":"Anyway, what we get is this where these 3 factors are here,"},{"Start":"03:50.840 ","End":"03:57.510","Text":"and this is the z minus y from the third row and the t minus y from the fourth row."},{"Start":"03:57.510 ","End":"04:01.010","Text":"We just have 1 more row operation to do."},{"Start":"04:01.010 ","End":"04:08.225","Text":"I\u0027m going to get a 0 here by subtracting the third row from the fourth row."},{"Start":"04:08.225 ","End":"04:10.910","Text":"This is the precise notation for doing that."},{"Start":"04:10.910 ","End":"04:13.760","Text":"After we do that and we get a 0 here,"},{"Start":"04:13.760 ","End":"04:16.610","Text":"then we will have an upper triangular,"},{"Start":"04:16.610 ","End":"04:19.055","Text":"which is a special kind of row echelon,"},{"Start":"04:19.055 ","End":"04:21.425","Text":"and it\u0027s easy to compute the determinant of."},{"Start":"04:21.425 ","End":"04:23.480","Text":"So if we do that,"},{"Start":"04:23.480 ","End":"04:27.855","Text":"we will get the following"},{"Start":"04:27.855 ","End":"04:34.460","Text":"which is in upper triangular form because here\u0027s the diagonal,"},{"Start":"04:34.460 ","End":"04:36.475","Text":"perhaps I\u0027ll highlight it."},{"Start":"04:36.475 ","End":"04:38.060","Text":"That\u0027s the diagonal,"},{"Start":"04:38.060 ","End":"04:41.800","Text":"below the diagonal only zeros."},{"Start":"04:42.410 ","End":"04:45.715","Text":"It\u0027s what we call the upper triangular."},{"Start":"04:45.715 ","End":"04:47.825","Text":"When we have an upper triangular,"},{"Start":"04:47.825 ","End":"04:51.170","Text":"then the formula for the determinant is just the product of"},{"Start":"04:51.170 ","End":"04:54.695","Text":"the entries on the main diagonal."},{"Start":"04:54.695 ","End":"04:58.795","Text":"The product of these is t minus z."},{"Start":"04:58.795 ","End":"05:03.335","Text":"Here is that t minus z together with all these others."},{"Start":"05:03.335 ","End":"05:05.660","Text":"This is our final answer."},{"Start":"05:05.660 ","End":"05:07.980","Text":"So we are done."}],"ID":9897},{"Watched":false,"Name":"Exercise 12","Duration":"3m 48s","ChapterTopicVideoID":9587,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9587.jpeg","UploadDate":"2017-07-26T08:34:04.0270000","DurationForVideoObject":"PT3M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, we have to evaluate the following determinant."},{"Start":"00:03.720 ","End":"00:07.575","Text":"It\u0027s a square matrix,"},{"Start":"00:07.575 ","End":"00:11.475","Text":"1, 2, 3, 4, 5 by 5."},{"Start":"00:11.475 ","End":"00:16.950","Text":"Everything\u0027s 1 except on the secondary diagonal,"},{"Start":"00:16.950 ","End":"00:21.240","Text":"the counter diagonal where these ks all elongate."},{"Start":"00:21.240 ","End":"00:25.390","Text":"It came out a bit crooked, never mind that."},{"Start":"00:26.510 ","End":"00:29.325","Text":"It\u0027s not intuitively obvious,"},{"Start":"00:29.325 ","End":"00:33.450","Text":"but if we subtract"},{"Start":"00:33.450 ","End":"00:41.925","Text":"the top row from all the other rows we\u0027ll get mostly 0s because this is mostly 1s."},{"Start":"00:41.925 ","End":"00:49.180","Text":"To be precise, these are the 4 row operations we\u0027ll be doing."},{"Start":"00:49.220 ","End":"00:55.625","Text":"In each case, we\u0027ll be subtracting the top 1 from whatever 1 it is."},{"Start":"00:55.625 ","End":"00:57.560","Text":"After we do this,"},{"Start":"00:57.560 ","End":"01:01.035","Text":"we will get the following,"},{"Start":"01:01.035 ","End":"01:04.770","Text":"I thought you see most of the entries are 0s,"},{"Start":"01:04.770 ","End":"01:09.650","Text":"but we have a lot of rows with a pair of k minus 1,"},{"Start":"01:09.650 ","End":"01:12.470","Text":"and a 1 minus k. Notice that the sum of"},{"Start":"01:12.470 ","End":"01:16.045","Text":"these 2 is 0 because they\u0027re opposites of each other."},{"Start":"01:16.045 ","End":"01:19.640","Text":"Here\u0027s the trick we\u0027re going to use next,"},{"Start":"01:19.640 ","End":"01:27.875","Text":"we\u0027re going to add up all the columns and put that total in the last column."},{"Start":"01:27.875 ","End":"01:30.605","Text":"Well, maybe it\u0027s easier to look at it as saying,"},{"Start":"01:30.605 ","End":"01:36.260","Text":"add the sum of these 4 columns to the 5th column."},{"Start":"01:36.260 ","End":"01:40.555","Text":"If we write that with row/column notation,"},{"Start":"01:40.555 ","End":"01:45.330","Text":"then we take the 5th column and add to it the 1st,"},{"Start":"01:45.330 ","End":"01:46.830","Text":"2nd, 3rd, and 4th columns,"},{"Start":"01:46.830 ","End":"01:48.720","Text":"and that stays in the 5th."},{"Start":"01:48.720 ","End":"01:55.950","Text":"If we do that, we get this and notice that all these 4 entries have become"},{"Start":"01:55.950 ","End":"02:04.280","Text":"0 because the k minus 1 cancels the 1 minus k. Here we have accumulated 1,"},{"Start":"02:04.280 ","End":"02:07.790","Text":"2, 3, 4, so we\u0027ve got k plus 4."},{"Start":"02:07.790 ","End":"02:14.180","Text":"Now, this may look like a triangular matrix,"},{"Start":"02:14.180 ","End":"02:15.320","Text":"but it isn\u0027t,"},{"Start":"02:15.320 ","End":"02:17.225","Text":"because to be triangular,"},{"Start":"02:17.225 ","End":"02:24.750","Text":"it has to be triangular above the main or primary principal diagonal."},{"Start":"02:24.750 ","End":"02:29.375","Text":"We have 0 above the secondary,"},{"Start":"02:29.375 ","End":"02:32.105","Text":"not the main diagonal,"},{"Start":"02:32.105 ","End":"02:34.040","Text":"so be careful there."},{"Start":"02:34.040 ","End":"02:39.300","Text":"This is not an upper triangular matrix"},{"Start":"02:39.300 ","End":"02:42.415","Text":"but we can easily get it that way."},{"Start":"02:42.415 ","End":"02:45.380","Text":"Because of this backwards nature,"},{"Start":"02:45.380 ","End":"02:48.530","Text":"if we swap some rows around, for example,"},{"Start":"02:48.530 ","End":"02:54.890","Text":"if I put this row here and vice versa,"},{"Start":"02:54.890 ","End":"02:58.010","Text":"switch these 2 around and then switch these 2 around,"},{"Start":"02:58.010 ","End":"03:00.290","Text":"then we\u0027ll get it the right way."},{"Start":"03:00.290 ","End":"03:05.915","Text":"Just to be precise, we swap row 1 with row 5 and row 2 with row 4."},{"Start":"03:05.915 ","End":"03:08.645","Text":"After we do that,"},{"Start":"03:08.645 ","End":"03:12.980","Text":"then this goes down here and this one goes up here,"},{"Start":"03:12.980 ","End":"03:16.895","Text":"and we do get the diagonal the right way."},{"Start":"03:16.895 ","End":"03:27.000","Text":"Now, we really do have upper triangular matrix."},{"Start":"03:27.000 ","End":"03:37.420","Text":"We get its determinant by just multiplying the terms along the main principle diagonal."},{"Start":"03:37.420 ","End":"03:41.745","Text":"That is just this because we have k minus 1."},{"Start":"03:41.745 ","End":"03:43.485","Text":"There\u0027s 4 factors of those."},{"Start":"03:43.485 ","End":"03:48.930","Text":"K minus 1 to the 4th times k plus 4, and we\u0027re done."}],"ID":9898},{"Watched":false,"Name":"Exercise 13","Duration":"5m 34s","ChapterTopicVideoID":9588,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9588.jpeg","UploadDate":"2017-07-26T08:34:23.9630000","DurationForVideoObject":"PT5M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we have to compute the determinant of a matrix,"},{"Start":"00:03.630 ","End":"00:07.230","Text":"but we\u0027re not given the picture or the diagram of the matrix,"},{"Start":"00:07.230 ","End":"00:10.290","Text":"we\u0027re given it in case form."},{"Start":"00:10.290 ","End":"00:13.965","Text":"The matrix, A is a_ij."},{"Start":"00:13.965 ","End":"00:23.265","Text":"As usual, I should mention that i is the row number and j is the column number,"},{"Start":"00:23.265 ","End":"00:29.355","Text":"so that for example a_ 23 would be row 2, column 3."},{"Start":"00:29.355 ","End":"00:32.175","Text":"Now we\u0027re giving it in cases."},{"Start":"00:32.175 ","End":"00:34.230","Text":"The first task is"},{"Start":"00:34.230 ","End":"00:40.860","Text":"to draw a diagram of what the matrix looks like and then to find the determinant,"},{"Start":"00:40.860 ","End":"00:43.200","Text":"but since finding the determinant is more important,"},{"Start":"00:43.200 ","End":"00:45.555","Text":"I\u0027m going to do the second part first,"},{"Start":"00:45.555 ","End":"00:50.955","Text":"and I\u0027ll just show you what the matrix looks like. Here it is."},{"Start":"00:50.955 ","End":"00:55.330","Text":"Take a moment to see if you can see the pattern."},{"Start":"00:55.790 ","End":"01:00.405","Text":"Well, you can see it\u0027s basically from 1 to n everywhere,"},{"Start":"01:00.405 ","End":"01:02.955","Text":"except that there are some signs,"},{"Start":"01:02.955 ","End":"01:04.980","Text":"minuses some places,"},{"Start":"01:04.980 ","End":"01:07.560","Text":"and on the diagonal we have a lot of 0s."},{"Start":"01:07.560 ","End":"01:10.410","Text":"I\u0027ll explain this at the end how we got to that."},{"Start":"01:10.410 ","End":"01:13.620","Text":"Let\u0027s now concentrate on finding the determinant."},{"Start":"01:13.620 ","End":"01:17.490","Text":"As usual, we\u0027ll do row or column operations,"},{"Start":"01:17.490 ","End":"01:21.525","Text":"in this case row operations to try and bring it into echelon form."},{"Start":"01:21.525 ","End":"01:24.510","Text":"I have a 1 here and I have minus 1s everywhere,"},{"Start":"01:24.510 ","End":"01:33.210","Text":"so it seems fairly straightforward to just add the first row to each of the other rows."},{"Start":"01:33.210 ","End":"01:35.835","Text":"I\u0027m using the notation."},{"Start":"01:35.835 ","End":"01:41.280","Text":"For example, we take row 2 add R_1 to it,"},{"Start":"01:41.280 ","End":"01:43.410","Text":"row 1 and the value in row 2,"},{"Start":"01:43.410 ","End":"01:45.240","Text":"and similarly for row 3."},{"Start":"01:45.240 ","End":"01:46.950","Text":"Finally, the last row,"},{"Start":"01:46.950 ","End":"01:51.780","Text":"we add row 1 to it and leave it where it is."},{"Start":"01:51.780 ","End":"01:55.560","Text":"We\u0027re just adding row 1 to all the other rows."},{"Start":"01:55.560 ","End":"01:57.990","Text":"Now after we do that, well,"},{"Start":"01:57.990 ","End":"02:01.800","Text":"the first column is 1 with all 0s as expected."},{"Start":"02:01.800 ","End":"02:03.495","Text":"The second column,"},{"Start":"02:03.495 ","End":"02:09.675","Text":"this 2 make this a 2 but all the others cancel out and we get 0s."},{"Start":"02:09.675 ","End":"02:14.340","Text":"Similarly, we have 3 and then 6 and then another 3."},{"Start":"02:14.340 ","End":"02:17.475","Text":"Then all these cancel out and become 0s."},{"Start":"02:17.475 ","End":"02:21.930","Text":"You can see that it\u0027s already in triangular form"},{"Start":"02:21.930 ","End":"02:27.615","Text":"because below the main diagonal everything is 0."},{"Start":"02:27.615 ","End":"02:32.340","Text":"We have an upper triangular matrix, and in that case,"},{"Start":"02:32.340 ","End":"02:35.730","Text":"we know that the determinant is just the product of"},{"Start":"02:35.730 ","End":"02:39.420","Text":"the elements along the main diagonal,"},{"Start":"02:39.420 ","End":"02:43.650","Text":"which is just 1 times 2 times 3 times 4, and so on up to n,"},{"Start":"02:43.650 ","End":"02:46.545","Text":"which is n factorial,"},{"Start":"02:46.545 ","End":"02:48.525","Text":"and that\u0027s the answer."},{"Start":"02:48.525 ","End":"02:56.250","Text":"But now we have to get back and see if we can interpret this definition by cases,"},{"Start":"02:56.250 ","End":"03:00.735","Text":"that\u0027s how we get this depiction of the matrix."},{"Start":"03:00.735 ","End":"03:02.970","Text":"There several ways you could do it."},{"Start":"03:02.970 ","End":"03:06.900","Text":"We could just start in order and to see what this means."},{"Start":"03:06.900 ","End":"03:09.150","Text":"Well, when i and j are both 1,"},{"Start":"03:09.150 ","End":"03:10.770","Text":"which means just row 1,"},{"Start":"03:10.770 ","End":"03:12.495","Text":"column 1, it\u0027s a 1."},{"Start":"03:12.495 ","End":"03:15.285","Text":"That\u0027s dictated explicitly."},{"Start":"03:15.285 ","End":"03:19.425","Text":"Next, I see that if i equals j,"},{"Start":"03:19.425 ","End":"03:21.750","Text":"the row and the column are the same,"},{"Start":"03:21.750 ","End":"03:25.020","Text":"but not 1, then it\u0027s 0."},{"Start":"03:25.020 ","End":"03:30.180","Text":"That will give us all these 0s along the diagonal,"},{"Start":"03:30.180 ","End":"03:32.805","Text":"so we have a 1 and all 0s."},{"Start":"03:32.805 ","End":"03:38.280","Text":"Now, this takes care of all the cases where i equals j."},{"Start":"03:38.280 ","End":"03:42.660","Text":"Notice that if I pick an element here above the diagonal,"},{"Start":"03:42.660 ","End":"03:44.220","Text":"the column is bigger than the row,"},{"Start":"03:44.220 ","End":"03:45.600","Text":"like this would be, I don\u0027t know,"},{"Start":"03:45.600 ","End":"03:47.625","Text":"column 4, row 1."},{"Start":"03:47.625 ","End":"03:52.785","Text":"Here above the diagonal we\u0027ll have i less than j,"},{"Start":"03:52.785 ","End":"03:55.260","Text":"in which case we\u0027ll be reading from here,"},{"Start":"03:55.260 ","End":"03:59.655","Text":"so this will be above the main diagonal."},{"Start":"03:59.655 ","End":"04:02.610","Text":"If we go here, for example,"},{"Start":"04:02.610 ","End":"04:08.715","Text":"this entry will have row is 4 and the column is 2,"},{"Start":"04:08.715 ","End":"04:12.075","Text":"so the row is bigger than the column."},{"Start":"04:12.075 ","End":"04:20.790","Text":"So i is bigger than j for the lower part,"},{"Start":"04:20.790 ","End":"04:23.830","Text":"below the main diagonal."},{"Start":"04:23.900 ","End":"04:29.190","Text":"We\u0027ll just write that that\u0027s below the diagonal."},{"Start":"04:29.190 ","End":"04:32.055","Text":"Now we can take a look at what this means."},{"Start":"04:32.055 ","End":"04:34.335","Text":"Above the diagonal,"},{"Start":"04:34.335 ","End":"04:38.070","Text":"the entry is just j,"},{"Start":"04:38.070 ","End":"04:40.470","Text":"which is the column number."},{"Start":"04:40.470 ","End":"04:43.200","Text":"For example, if j is 3,"},{"Start":"04:43.200 ","End":"04:44.640","Text":"which is column 3,"},{"Start":"04:44.640 ","End":"04:47.310","Text":"then it\u0027s just 3 and that gives us these."},{"Start":"04:47.310 ","End":"04:50.265","Text":"In column 4, it\u0027s 4."},{"Start":"04:50.265 ","End":"04:52.230","Text":"In column 2, it\u0027s 2."},{"Start":"04:52.230 ","End":"04:53.625","Text":"In column n,"},{"Start":"04:53.625 ","End":"04:55.860","Text":"its n, and so on."},{"Start":"04:55.860 ","End":"04:58.065","Text":"That\u0027s for the above the diagonal."},{"Start":"04:58.065 ","End":"04:59.655","Text":"Below the diagonal,"},{"Start":"04:59.655 ","End":"05:01.155","Text":"it\u0027s very similar,"},{"Start":"05:01.155 ","End":"05:02.640","Text":"if not the column number,"},{"Start":"05:02.640 ","End":"05:05.065","Text":"it\u0027s minus the column number."},{"Start":"05:05.065 ","End":"05:08.885","Text":"For example, in the third column where j is 3,"},{"Start":"05:08.885 ","End":"05:12.350","Text":"we take minus j, minus 3."},{"Start":"05:12.350 ","End":"05:13.550","Text":"In column 2,"},{"Start":"05:13.550 ","End":"05:14.840","Text":"we have a minus 2,"},{"Start":"05:14.840 ","End":"05:16.100","Text":"in column 1,"},{"Start":"05:16.100 ","End":"05:19.380","Text":"minus 1, and so on."},{"Start":"05:19.380 ","End":"05:24.165","Text":"Column n minus 1 minus n minus 1."},{"Start":"05:24.165 ","End":"05:27.045","Text":"That basically describes it."},{"Start":"05:27.045 ","End":"05:29.670","Text":"Anyway, that\u0027s the less important part."},{"Start":"05:29.670 ","End":"05:33.030","Text":"Finding the determinant was the important part and we\u0027ve done that,"},{"Start":"05:33.030 ","End":"05:35.680","Text":"and so we\u0027ve finished."}],"ID":9899},{"Watched":false,"Name":"Exercise 14","Duration":"5m 45s","ChapterTopicVideoID":9589,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9589.jpeg","UploadDate":"2017-07-26T08:34:38.4570000","DurationForVideoObject":"PT5M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this exercise, we have to evaluate,"},{"Start":"00:04.560 ","End":"00:07.740","Text":"compute the determinant of matrix A,"},{"Start":"00:07.740 ","End":"00:12.195","Text":"where A is an n by n matrix given in analytic form."},{"Start":"00:12.195 ","End":"00:18.780","Text":"The elements aij are given in cases depending on i and j."},{"Start":"00:18.780 ","End":"00:22.155","Text":"Let\u0027s skip this part we\u0027ll do that at the end."},{"Start":"00:22.155 ","End":"00:30.635","Text":"I\u0027ll show you what the matrix looks like and afterwards we will see why. Here it is."},{"Start":"00:30.635 ","End":"00:32.780","Text":"If you look at it in a moment,"},{"Start":"00:32.780 ","End":"00:36.725","Text":"you\u0027ll see that mostly it\u0027s 0s and the places where it\u0027s not 0"},{"Start":"00:36.725 ","End":"00:41.520","Text":"is here just below the main diagonal we have all the numbers from 1,"},{"Start":"00:41.520 ","End":"00:47.670","Text":"2 and minus 1 and then we have an odd 1 out an n here."},{"Start":"00:47.670 ","End":"00:56.630","Text":"The way we tackle this the n here is just to expand by the top row."},{"Start":"00:56.630 ","End":"01:00.025","Text":"Now this top row only has"},{"Start":"01:00.025 ","End":"01:06.930","Text":"1 non-zero element in the expansion we just take care of this term."},{"Start":"01:06.930 ","End":"01:16.380","Text":"We find the minor by crossing out the row we\u0027ve done already and the column."},{"Start":"01:16.520 ","End":"01:21.170","Text":"Then we have to take a sign plus or minus times"},{"Start":"01:21.170 ","End":"01:25.775","Text":"the element times the determinant of what\u0027s left."},{"Start":"01:25.775 ","End":"01:32.880","Text":"The only technical problem which is not really a problem a minor 1 is the sign."},{"Start":"01:32.880 ","End":"01:35.450","Text":"We start off with the checkerboard pattern plus,"},{"Start":"01:35.450 ","End":"01:38.945","Text":"minus, plus, and so on."},{"Start":"01:38.945 ","End":"01:42.140","Text":"The question is, is it a plus or is it"},{"Start":"01:42.140 ","End":"01:45.770","Text":"a minus when we get to the nth because we don\u0027t know what n is?"},{"Start":"01:45.770 ","End":"01:52.990","Text":"We want something that\u0027s going to be plus when n is,"},{"Start":"01:52.990 ","End":"01:55.045","Text":"well, let\u0027s see odd or even."},{"Start":"01:55.045 ","End":"01:58.330","Text":"It starts off that when n is 1, you want to plus."},{"Start":"01:58.330 ","End":"02:03.400","Text":"In general when n is odd you want to plus and it\u0027s going to be a minus if n"},{"Start":"02:03.400 ","End":"02:09.220","Text":"is even like the second and the fourth 1 would be a minus."},{"Start":"02:09.220 ","End":"02:11.710","Text":"This is the thing we\u0027ve seen before."},{"Start":"02:11.710 ","End":"02:17.040","Text":"I could say plus 1 or minus 1 when we multiply."},{"Start":"02:17.040 ","End":"02:20.620","Text":"The trick is to take minus 1 to the power of"},{"Start":"02:20.620 ","End":"02:26.125","Text":"n and possibly adjust it by adding or subtracting 1 if we\u0027re off by 1."},{"Start":"02:26.125 ","End":"02:28.750","Text":"Let\u0027s see, will minus 1 to the n do it."},{"Start":"02:28.750 ","End":"02:30.665","Text":"Well, if n is odd,"},{"Start":"02:30.665 ","End":"02:37.485","Text":"actually we get minus 1 and when n is even we get plus 1 so it\u0027s the wrong way around."},{"Start":"02:37.485 ","End":"02:40.550","Text":"When that happens, we just add or subtract 1."},{"Start":"02:40.550 ","End":"02:42.430","Text":"Let\u0027s say I subtract 1."},{"Start":"02:42.430 ","End":"02:48.945","Text":"You can check if n is 1 we get minus 1 to the 0 which is plus 1."},{"Start":"02:48.945 ","End":"02:58.470","Text":"If n is 2 we have minus 1 to the power of 1 is minus 1 to the minus. This works."},{"Start":"02:59.420 ","End":"03:01.655","Text":"Here is the sign."},{"Start":"03:01.655 ","End":"03:07.640","Text":"The minus 1 to the n minus 1 here is the element that\u0027s this n here."},{"Start":"03:07.640 ","End":"03:14.250","Text":"Then we have the determinant of what\u0027s left this is called the minor."},{"Start":"03:14.450 ","End":"03:19.880","Text":"Because it\u0027s a triangular and it\u0027s even a diagonal matrix,"},{"Start":"03:19.880 ","End":"03:21.890","Text":"only entries on the main diagonal"},{"Start":"03:21.890 ","End":"03:25.610","Text":"the determinant is what you get when you multiply all the elements of"},{"Start":"03:25.610 ","End":"03:33.185","Text":"the diagonal which is 1 times 2 times 3 and so on up to n minus 1,"},{"Start":"03:33.185 ","End":"03:36.800","Text":"but notice that we also have this n here which we could throw at"},{"Start":"03:36.800 ","End":"03:42.500","Text":"the end and then we have the product of all the numbers from 1 to n which is n factorial."},{"Start":"03:42.500 ","End":"03:48.260","Text":"This is our final answer but we still have to go back so that I explain how we"},{"Start":"03:48.260 ","End":"03:54.035","Text":"got the matrix that I displayed in a more explicit way."},{"Start":"03:54.035 ","End":"03:56.750","Text":"This is a unique case."},{"Start":"03:56.750 ","End":"04:03.185","Text":"It\u0027s specifically on the first row on the nth which is last column"},{"Start":"04:03.185 ","End":"04:10.190","Text":"we have an n. That explains this 1 and that\u0027s 1 unique case so we\u0027re done with that."},{"Start":"04:10.190 ","End":"04:12.575","Text":"Now let\u0027s look at this top row."},{"Start":"04:12.575 ","End":"04:18.905","Text":"What this says is that when i is j plus 1, we have a j."},{"Start":"04:18.905 ","End":"04:22.610","Text":"Let\u0027s just take different values of j."},{"Start":"04:22.610 ","End":"04:25.085","Text":"If j equals 1,"},{"Start":"04:25.085 ","End":"04:29.175","Text":"for example, then i is 2."},{"Start":"04:29.175 ","End":"04:34.890","Text":"It says that a"},{"Start":"04:34.890 ","End":"04:41.190","Text":"2,1 is equal to 1 and a 2,1 is this."},{"Start":"04:41.190 ","End":"04:45.075","Text":"If we let j equals 2,"},{"Start":"04:45.075 ","End":"04:51.530","Text":"then we\u0027ll get i equals 3 and this will say that a 3,"},{"Start":"04:51.530 ","End":"04:54.560","Text":"2 is 2 but a 3,"},{"Start":"04:54.560 ","End":"04:57.695","Text":"2 third row second column is this."},{"Start":"04:57.695 ","End":"05:00.080","Text":"Then we\u0027ll get that a 4,"},{"Start":"05:00.080 ","End":"05:06.290","Text":"3 is 3 and so on until we get"},{"Start":"05:06.290 ","End":"05:13.605","Text":"to a n minus 1 is n minus 1."},{"Start":"05:13.605 ","End":"05:18.465","Text":"The a 4, 3 equals 3 is this and this nth row,"},{"Start":"05:18.465 ","End":"05:21.980","Text":"n minus 1th in a column is this."},{"Start":"05:21.980 ","End":"05:29.230","Text":"That gives us this sub diagonal the diagonal that\u0027s below the main diagonal."},{"Start":"05:29.230 ","End":"05:35.615","Text":"The special 1 and everything else what says here otherwise filled in with 0s."},{"Start":"05:35.615 ","End":"05:39.200","Text":"That explains how we got to this and the rest we\u0027ve"},{"Start":"05:39.200 ","End":"05:46.270","Text":"already done we\u0027ve shown what the determinant is. That\u0027s it."}],"ID":9900},{"Watched":false,"Name":"Exercise 15","Duration":"15m 55s","ChapterTopicVideoID":9590,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9590.jpeg","UploadDate":"2017-07-26T08:35:20.1570000","DurationForVideoObject":"PT15M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"This exercise, I put an asterisk next to it because"},{"Start":"00:04.410 ","End":"00:08.370","Text":"it\u0027s a little bit more difficult for example to have a proof by induction in it."},{"Start":"00:08.370 ","End":"00:14.100","Text":"Anyway, we\u0027re talking about a matrix which I call A_n by n,"},{"Start":"00:14.100 ","End":"00:17.010","Text":"which is an n by n matrix."},{"Start":"00:17.010 ","End":"00:19.290","Text":"The elements are a, i, j,"},{"Start":"00:19.290 ","End":"00:22.020","Text":"and it\u0027s given by this formula."},{"Start":"00:22.020 ","End":"00:25.470","Text":"But let\u0027s interpret this formula at the end."},{"Start":"00:25.470 ","End":"00:29.950","Text":"I\u0027ll just tell you what the matrix A looks like."},{"Start":"00:29.950 ","End":"00:39.000","Text":"Well, we already said that it\u0027s an n by n in size."},{"Start":"00:39.000 ","End":"00:48.320","Text":"What it is, it has 1s all along the secondary diagonal, minor diagonal."},{"Start":"00:48.320 ","End":"00:49.565","Text":"Think it has some other names,"},{"Start":"00:49.565 ","End":"00:53.015","Text":"not the principal, or the main diagonal,"},{"Start":"00:53.015 ","End":"00:55.220","Text":"and everything else is 0."},{"Start":"00:55.220 ","End":"00:57.080","Text":"Just put a big 0 here."},{"Start":"00:57.080 ","End":"00:59.690","Text":"It\u0027s 1 along the secondary diagonal."},{"Start":"00:59.690 ","End":"01:04.205","Text":"I\u0027ll return to this at the end, and show you why this is so,"},{"Start":"01:04.205 ","End":"01:05.990","Text":"how we deduce it from this."},{"Start":"01:05.990 ","End":"01:10.520","Text":"Meanwhile, let\u0027s continue, and see if we can figure out the determinant in"},{"Start":"01:10.520 ","End":"01:15.755","Text":"general for different values of n. Let\u0027s say n is bigger or equal to 2."},{"Start":"01:15.755 ","End":"01:19.670","Text":"For n equals 2, this is the matrix."},{"Start":"01:19.670 ","End":"01:22.520","Text":"We have ones along the secondary diagonal,"},{"Start":"01:22.520 ","End":"01:24.275","Text":"0, 1, 1, 0."},{"Start":"01:24.275 ","End":"01:31.945","Text":"Then straightforward computation, we can just multiply diagonals and get minus 1."},{"Start":"01:31.945 ","End":"01:36.195","Text":"Next we\u0027ll try n equals 3."},{"Start":"01:36.195 ","End":"01:38.100","Text":"Now how would we do this?"},{"Start":"01:38.100 ","End":"01:44.865","Text":"More than 1 way, but the easiest would be to switch row 1 with row 3."},{"Start":"01:44.865 ","End":"01:47.765","Text":"If we switch row 1 with row 3,"},{"Start":"01:47.765 ","End":"01:53.625","Text":"then we\u0027ll get ones along the main diagonal, and so the determinant is 1."},{"Start":"01:53.625 ","End":"01:55.800","Text":"But when you switch 2 rows,"},{"Start":"01:55.800 ","End":"01:57.675","Text":"you have to throw in a minus."},{"Start":"01:57.675 ","End":"02:00.465","Text":"This is minus 1."},{"Start":"02:00.465 ","End":"02:03.500","Text":"Meanwhile, we have minus 1, minus 1."},{"Start":"02:03.500 ","End":"02:05.600","Text":"I don\u0027t know if that\u0027s a pattern yet."},{"Start":"02:05.600 ","End":"02:08.615","Text":"If we suspect that it\u0027s always going to be minus 1,"},{"Start":"02:08.615 ","End":"02:10.910","Text":"let\u0027s check 1 more."},{"Start":"02:10.910 ","End":"02:19.080","Text":"When n equals 4, we have the ones along the secondary minor diagonal."},{"Start":"02:19.330 ","End":"02:23.165","Text":"This turns out to actually be plus 1."},{"Start":"02:23.165 ","End":"02:29.885","Text":"The reason is that if we do some row exchange operations,"},{"Start":"02:29.885 ","End":"02:34.670","Text":"exchange this 1 with this 1, and this 1 with this 1,"},{"Start":"02:34.670 ","End":"02:40.265","Text":"then we\u0027ll get 1s all along the main diagonal and the determinant of that is 1."},{"Start":"02:40.265 ","End":"02:46.035","Text":"Now, each switching of 2 rows introduces a factor of minus 1,"},{"Start":"02:46.035 ","End":"02:50.445","Text":"but we have 2 of them so minus 1 times minus 1 is plus 1."},{"Start":"02:50.445 ","End":"02:53.769","Text":"When we have an even number of row swaps,"},{"Start":"02:53.769 ","End":"02:55.635","Text":"it\u0027s going to be plus 1."},{"Start":"02:55.635 ","End":"02:59.845","Text":"This is what\u0027s going to happen in the next 1."},{"Start":"02:59.845 ","End":"03:03.460","Text":"Just clear some space for it."},{"Start":"03:03.460 ","End":"03:07.140","Text":"Here we are. The middle row could stay where it is."},{"Start":"03:07.140 ","End":"03:10.190","Text":"If we swap this with this, and this with this,"},{"Start":"03:10.190 ","End":"03:13.310","Text":"then we\u0027ll get 1s along the principle diagonal,"},{"Start":"03:13.310 ","End":"03:16.985","Text":"main diagonal, and that will be determinant 1."},{"Start":"03:16.985 ","End":"03:20.910","Text":"Because we had an even number of swaps we\u0027re okay."},{"Start":"03:21.190 ","End":"03:23.945","Text":"We now have 1 and 1."},{"Start":"03:23.945 ","End":"03:29.670","Text":"I\u0027ll just make a summary,"},{"Start":"03:29.670 ","End":"03:30.930","Text":"when n is 2,"},{"Start":"03:30.930 ","End":"03:33.465","Text":"we got minus 1."},{"Start":"03:33.465 ","End":"03:36.105","Text":"When n was 3,"},{"Start":"03:36.105 ","End":"03:38.640","Text":"we got minus 1."},{"Start":"03:38.640 ","End":"03:40.305","Text":"When n was 4,"},{"Start":"03:40.305 ","End":"03:43.550","Text":"we got plus 1, I\u0027ll emphasize it."},{"Start":"03:43.550 ","End":"03:47.405","Text":"Plus 1, 5 we just now got plus 1."},{"Start":"03:47.405 ","End":"03:52.220","Text":"Let\u0027s continue and see what else we can get."},{"Start":"03:52.220 ","End":"03:53.810","Text":"When n is 6,"},{"Start":"03:53.810 ","End":"04:01.430","Text":"we get again minus 1 because we have 3 row swaps."},{"Start":"04:01.430 ","End":"04:04.325","Text":"If I swap this with this, this with this,"},{"Start":"04:04.325 ","End":"04:05.705","Text":"and this with this,"},{"Start":"04:05.705 ","End":"04:09.970","Text":"then I\u0027ll get 1s along the main diagonal."},{"Start":"04:09.970 ","End":"04:12.930","Text":"3 swaps means minus 1, minus 1,"},{"Start":"04:12.930 ","End":"04:16.080","Text":"minus 1, which is minus 1."},{"Start":"04:16.080 ","End":"04:19.130","Text":"You can add another entry to this table."},{"Start":"04:19.130 ","End":"04:23.860","Text":"When n is 6, the determinant comes out to be minus 1."},{"Start":"04:23.860 ","End":"04:26.510","Text":"I think I\u0027ll go for more,"},{"Start":"04:26.510 ","End":"04:29.990","Text":"go for n equals 7 also."},{"Start":"04:29.990 ","End":"04:33.670","Text":"This comes out also as minus 1."},{"Start":"04:33.670 ","End":"04:36.380","Text":"Also because we have 3 swaps,"},{"Start":"04:36.380 ","End":"04:38.195","Text":"the middle row stays where it is."},{"Start":"04:38.195 ","End":"04:39.440","Text":"This swap with this,"},{"Start":"04:39.440 ","End":"04:40.895","Text":"this swap with this."},{"Start":"04:40.895 ","End":"04:43.055","Text":"This can swap with this."},{"Start":"04:43.055 ","End":"04:45.925","Text":"Then we\u0027ll get 1s along the main diagonal."},{"Start":"04:45.925 ","End":"04:50.625","Text":"3 swaps is minus 1 cubed so it\u0027s minus 1."},{"Start":"04:50.625 ","End":"04:53.940","Text":"It looks like the pattern is 2 minuses,"},{"Start":"04:53.940 ","End":"04:55.755","Text":"2 pluses, 2 minuses,"},{"Start":"04:55.755 ","End":"04:57.885","Text":"and then 2 pluses and so on."},{"Start":"04:57.885 ","End":"05:00.790","Text":"This is in fact correct."},{"Start":"05:00.790 ","End":"05:05.540","Text":"But we would like to have a more precise formula,"},{"Start":"05:05.540 ","End":"05:12.140","Text":"a formula involving n that will give us the plus or the minus 1."},{"Start":"05:12.140 ","End":"05:14.820","Text":"In fact there is 1."},{"Start":"05:15.170 ","End":"05:19.265","Text":"I\u0027ve pulled it like a rabbit out of a hat,"},{"Start":"05:19.265 ","End":"05:21.680","Text":"you wouldn\u0027t necessarily have gotten to this."},{"Start":"05:21.680 ","End":"05:23.195","Text":"I\u0027m going to prove it."},{"Start":"05:23.195 ","End":"05:24.980","Text":"Let\u0027s just see that it works so far."},{"Start":"05:24.980 ","End":"05:33.630","Text":"If n is 2, I\u0027ve got 2 minus 1 times 2 over 2 is 1 minus 1 to the 1, is minus 1."},{"Start":"05:33.630 ","End":"05:35.095","Text":"I put in 3,"},{"Start":"05:35.095 ","End":"05:37.655","Text":"2 times 3 over 2 is 3,"},{"Start":"05:37.655 ","End":"05:43.185","Text":"1 minus 1 cubed is minus 1. It\u0027s full."},{"Start":"05:43.185 ","End":"05:46.515","Text":"I\u0027ve got 3 times 4 over 2 is 6,"},{"Start":"05:46.515 ","End":"05:51.900","Text":"minus 1 to the 6 is 1."},{"Start":"05:51.900 ","End":"06:01.180","Text":"With 5, I\u0027ve got 4 times 5 over 2 is 10. Also a plus."},{"Start":"06:01.180 ","End":"06:03.265","Text":"Leave it to check for 6 and 7,"},{"Start":"06:03.265 ","End":"06:05.810","Text":"and it comes out correctly."},{"Start":"06:06.210 ","End":"06:11.720","Text":"I have to prove that this in fact works."},{"Start":"06:11.720 ","End":"06:15.750","Text":"I copy that formula here."},{"Start":"06:15.750 ","End":"06:18.835","Text":"We\u0027re going to prove it by induction."},{"Start":"06:18.835 ","End":"06:24.525","Text":"I\u0027m assuming that you learned it and maybe even remember it."},{"Start":"06:24.525 ","End":"06:30.005","Text":"This is going to hold for all natural numbers from 2 onwards."},{"Start":"06:30.005 ","End":"06:33.395","Text":"Here\u0027s the proof. Now by induction is 2 parts."},{"Start":"06:33.395 ","End":"06:37.580","Text":"There\u0027s the beginning part, starting the induction."},{"Start":"06:37.580 ","End":"06:43.645","Text":"The smallest possible value is 2 so we check that it\u0027s true for n equals 2."},{"Start":"06:43.645 ","End":"06:46.075","Text":"We already did this computation above."},{"Start":"06:46.075 ","End":"06:49.250","Text":"Never mind, it\u0027s minus 1 when n is 2."},{"Start":"06:49.250 ","End":"06:52.010","Text":"2 minus 1 times 2 over 2,"},{"Start":"06:52.010 ","End":"06:54.755","Text":"1 times 2 over 2 is 1 minus 1."},{"Start":"06:54.755 ","End":"06:56.465","Text":"The 1 is minus 1."},{"Start":"06:56.465 ","End":"06:58.295","Text":"For n equals 2,"},{"Start":"06:58.295 ","End":"07:03.700","Text":"it works because we\u0027ve already checked that we got minus 1."},{"Start":"07:03.700 ","End":"07:07.225","Text":"Now there\u0027s the induction stage,"},{"Start":"07:07.225 ","End":"07:10.460","Text":"which is confusing to many people."},{"Start":"07:10.460 ","End":"07:14.870","Text":"It\u0027s like we use what we\u0027re trying to prove, but not really."},{"Start":"07:14.870 ","End":"07:21.845","Text":"The induction step is to show that if it\u0027s true for a particular n,"},{"Start":"07:21.845 ","End":"07:27.760","Text":"if then it will also be true for the following n that is for n plus 1."},{"Start":"07:27.760 ","End":"07:35.160","Text":"We take for granted that for some n in our domain from 2 onwards,"},{"Start":"07:35.160 ","End":"07:40.540","Text":"that for some n that this formula holds"},{"Start":"07:40.650 ","End":"07:47.995","Text":"We have to show that it holds when we replace n by n plus 1."},{"Start":"07:47.995 ","End":"07:50.740","Text":"In other words, that the n plus 1 by n plus"},{"Start":"07:50.740 ","End":"07:55.960","Text":"1 matrix has determinant minus 1 to the power of note."},{"Start":"07:55.960 ","End":"07:59.770","Text":"If we put n plus 1 instead of n,"},{"Start":"07:59.770 ","End":"08:01.840","Text":"here we get, well,"},{"Start":"08:01.840 ","End":"08:11.050","Text":"it\u0027s n plus 1 minus 1 times n plus 1 over 2."},{"Start":"08:11.050 ","End":"08:13.930","Text":"The plus 1 and the minus 1 cancel,"},{"Start":"08:13.930 ","End":"08:15.580","Text":"it just gives us this."},{"Start":"08:15.580 ","End":"08:18.085","Text":"We have to show that if this is true,"},{"Start":"08:18.085 ","End":"08:20.630","Text":"then this is true."},{"Start":"08:21.960 ","End":"08:25.480","Text":"Let\u0027s get onto it."},{"Start":"08:25.480 ","End":"08:35.050","Text":"Now, here is our n plus 1 by n plus 1 matrix with 1s along the secondary diagonal."},{"Start":"08:35.050 ","End":"08:43.330","Text":"The way we\u0027ll do it is to expand along the first row."},{"Start":"08:43.330 ","End":"08:45.100","Text":"Oh, you could have done it by the last column,"},{"Start":"08:45.100 ","End":"08:51.735","Text":"but we\u0027ll do it by expansion along the first row."},{"Start":"08:51.735 ","End":"08:57.840","Text":"The only 1 that\u0027s not 0 that will contribute anything will be this element."},{"Start":"08:57.840 ","End":"09:03.040","Text":"We\u0027ll take the minor of this."},{"Start":"09:03.040 ","End":"09:06.175","Text":"We need 3 things we need a sign,"},{"Start":"09:06.175 ","End":"09:09.220","Text":"an element, and a minor."},{"Start":"09:09.220 ","End":"09:16.990","Text":"Then truly the 1 that\u0027s a bit tricky is the sign because we have plus,"},{"Start":"09:16.990 ","End":"09:23.365","Text":"minus, plus, minus, but then there\u0027s dot-dot-dot."},{"Start":"09:23.365 ","End":"09:28.240","Text":"Anyway at the moment, we don\u0027t know whether this is plus or minus,"},{"Start":"09:28.240 ","End":"09:33.100","Text":"but we\u0027ve actually seen this in the previous exercise in case you missed it,"},{"Start":"09:33.100 ","End":"09:34.330","Text":"I\u0027ll show you again."},{"Start":"09:34.330 ","End":"09:40.540","Text":"What we\u0027re looking for is some formula that when you put in 1,"},{"Start":"09:40.540 ","End":"09:42.895","Text":"2, 3, 4,"},{"Start":"09:42.895 ","End":"09:45.700","Text":"and so on, that we get,"},{"Start":"09:45.700 ","End":"09:47.500","Text":"I\u0027ll say plus 1,"},{"Start":"09:47.500 ","End":"09:53.110","Text":"minus 1, and so on."},{"Start":"09:53.110 ","End":"09:55.870","Text":"What will it be for a general value,"},{"Start":"09:55.870 ","End":"09:58.555","Text":"n, what is here?"},{"Start":"09:58.555 ","End":"10:04.510","Text":"We discovered earlier that it\u0027s got to be the minus 1 to the power of something."},{"Start":"10:04.510 ","End":"10:09.010","Text":"We usually start off with minus 1 to the n. If we\u0027re off by 1,"},{"Start":"10:09.010 ","End":"10:12.385","Text":"we adjust it because it could be maybe it\u0027s minus, plus, minus plus."},{"Start":"10:12.385 ","End":"10:16.510","Text":"Anyway. If we look at this expression minus 1 to the n,"},{"Start":"10:16.510 ","End":"10:20.185","Text":"then when n is even it\u0027s positive,"},{"Start":"10:20.185 ","End":"10:22.135","Text":"and odd is negative."},{"Start":"10:22.135 ","End":"10:27.970","Text":"Now we have exactly the opposite because when n is even it\u0027s negative and odd is"},{"Start":"10:27.970 ","End":"10:34.030","Text":"positive so we fix this by either adding or subtracting 1 doesn\u0027t matter."},{"Start":"10:34.030 ","End":"10:37.870","Text":"Minus 1 to the n minus 1 will give us what we wanted."},{"Start":"10:37.870 ","End":"10:39.340","Text":"Again, just for example,"},{"Start":"10:39.340 ","End":"10:45.700","Text":"if n is 3 minus 1 to the 3 minus 1 is minus 1 squared is plus 1."},{"Start":"10:45.700 ","End":"10:52.220","Text":"Now what we want actually is not n. For n,"},{"Start":"10:52.470 ","End":"10:55.525","Text":"what we just computed is this,"},{"Start":"10:55.525 ","End":"10:59.455","Text":"but this is of size n plus 1."},{"Start":"10:59.455 ","End":"11:01.570","Text":"For n plus 1,"},{"Start":"11:01.570 ","End":"11:07.630","Text":"what we will get is minus 1 to the power of n plus 1"},{"Start":"11:07.630 ","End":"11:13.825","Text":"minus 1 which is minus 1 to the n. This is the sign part."},{"Start":"11:13.825 ","End":"11:17.650","Text":"Now remember we said that we\u0027re going to have, when we expand here,"},{"Start":"11:17.650 ","End":"11:25.480","Text":"we need the sign times the element or the entry times the minor."},{"Start":"11:25.480 ","End":"11:32.020","Text":"The minor is the determinant that we get when we remove the row and column."},{"Start":"11:32.020 ","End":"11:37.255","Text":"The sign which we said was minus 1 to the n. Yeah,"},{"Start":"11:37.255 ","End":"11:40.630","Text":"the entry is 1. That\u0027s the 1."},{"Start":"11:40.630 ","End":"11:44.890","Text":"I guess we just didn\u0027t bother writing it and the minor,"},{"Start":"11:44.890 ","End":"11:48.655","Text":"which is this part here,"},{"Start":"11:48.655 ","End":"11:53.920","Text":"is the same thing just for n not n plus 1."},{"Start":"11:53.920 ","End":"11:58.180","Text":"We have n by n, so it\u0027s the n by n determinant."},{"Start":"11:58.180 ","End":"12:02.335","Text":"Now, by the induction hypothesis,"},{"Start":"12:02.335 ","End":"12:04.405","Text":"if you go back up,"},{"Start":"12:04.405 ","End":"12:08.890","Text":"we saw that this was the induction hypothesis that we assume"},{"Start":"12:08.890 ","End":"12:13.600","Text":"that it\u0027s true for n that we had minus 1 to the power of this,"},{"Start":"12:13.600 ","End":"12:20.185","Text":"and now we have times minus 1 to the n and so let\u0027s do a little bit of algebra here."},{"Start":"12:20.185 ","End":"12:21.640","Text":"By the law of exponents,"},{"Start":"12:21.640 ","End":"12:23.680","Text":"when we multiply with the same base,"},{"Start":"12:23.680 ","End":"12:25.510","Text":"we add the exponents."},{"Start":"12:25.510 ","End":"12:31.240","Text":"Here\u0027s the base minus 1 and here\u0027s the n plus this thing."},{"Start":"12:31.240 ","End":"12:36.790","Text":"Common denominator of 2 so this is 2n plus this."},{"Start":"12:36.790 ","End":"12:38.320","Text":"This I open the brackets,"},{"Start":"12:38.320 ","End":"12:42.400","Text":"it\u0027s n squared minus n and so we have n squared"},{"Start":"12:42.400 ","End":"12:47.440","Text":"minus n plus 2n is n squared plus n. This is what we got on."},{"Start":"12:47.440 ","End":"12:49.705","Text":"This is what we had to show."},{"Start":"12:49.705 ","End":"12:51.460","Text":"I\u0027m not going to scroll back,"},{"Start":"12:51.460 ","End":"12:58.900","Text":"but this is what we have to show for the induction hypothesis being true for n plus 1."},{"Start":"12:58.900 ","End":"13:01.735","Text":"We\u0027ve done with the induction part."},{"Start":"13:01.735 ","End":"13:06.485","Text":"The only thing that remains was what we had in the beginning."},{"Start":"13:06.485 ","End":"13:10.020","Text":"I have to show you why this formula,"},{"Start":"13:10.020 ","End":"13:15.660","Text":"a_ij is 1 or 0 according to this condition why this gives"},{"Start":"13:15.660 ","End":"13:22.360","Text":"me 1s along the secondary minor diagonal."},{"Start":"13:22.360 ","End":"13:25.360","Text":"I don\u0027t think I\u0027ll do a general proof,"},{"Start":"13:25.360 ","End":"13:27.250","Text":"let\u0027s just take as an example, you know what?"},{"Start":"13:27.250 ","End":"13:31.060","Text":"We\u0027ll go with n equals 4 and you\u0027ll see that the same logic"},{"Start":"13:31.060 ","End":"13:37.060","Text":"applies for all the other values of n. If n"},{"Start":"13:37.060 ","End":"13:43.039","Text":"is 4 what we\u0027re saying is if I copy this is that a_ij"},{"Start":"13:43.560 ","End":"13:52.675","Text":"is equal to 1 or 0 depending on if i plus j equals."},{"Start":"13:52.675 ","End":"13:54.160","Text":"Now if n is 4,"},{"Start":"13:54.160 ","End":"13:59.050","Text":"4 plus 1 is 5 and 0 otherwise,"},{"Start":"13:59.050 ","End":"14:02.470","Text":"sometimes we abbreviate OTW as otherwise."},{"Start":"14:02.470 ","End":"14:07.345","Text":"Let\u0027s see what it means that i plus j equals 5."},{"Start":"14:07.345 ","End":"14:10.060","Text":"Now i plus j equals 5."},{"Start":"14:10.060 ","End":"14:15.565","Text":"We can just make a table to see which values of i and j. I mean,"},{"Start":"14:15.565 ","End":"14:18.160","Text":"they have to be for at least 1."},{"Start":"14:18.160 ","End":"14:20.545","Text":"We could have I equals 1,"},{"Start":"14:20.545 ","End":"14:23.065","Text":"in which case j is 4."},{"Start":"14:23.065 ","End":"14:24.430","Text":"Or we could have 2,"},{"Start":"14:24.430 ","End":"14:26.470","Text":"3 or 3,"},{"Start":"14:26.470 ","End":"14:29.965","Text":"2 or 4, 1."},{"Start":"14:29.965 ","End":"14:33.520","Text":"Now these are the 1s which I want."},{"Start":"14:33.520 ","End":"14:41.575","Text":"I have that a_14 is 1,"},{"Start":"14:41.575 ","End":"14:44.170","Text":"a_23 is 1,"},{"Start":"14:44.170 ","End":"14:47.050","Text":"a_32 is 1,"},{"Start":"14:47.050 ","End":"14:55.285","Text":"and a_41 is 1 and the rest of the a_ij are 0 that are not from here."},{"Start":"14:55.285 ","End":"15:01.150","Text":"Look, a_14 is this because it\u0027s row 1 column 4,"},{"Start":"15:01.150 ","End":"15:03.595","Text":"then row 2 column 3,"},{"Start":"15:03.595 ","End":"15:06.295","Text":"then row 3 column 2,"},{"Start":"15:06.295 ","End":"15:08.965","Text":"row 4 column 1,"},{"Start":"15:08.965 ","End":"15:11.215","Text":"and through up to 0."},{"Start":"15:11.215 ","End":"15:19.150","Text":"You can see that this works in general because when I say that i plus j is fixed,"},{"Start":"15:19.150 ","End":"15:25.255","Text":"then I can increase i by 1 and reduce j by 1 to keep the sum the same."},{"Start":"15:25.255 ","End":"15:28.660","Text":"When you increase i by 1 and reduce j by 1,"},{"Start":"15:28.660 ","End":"15:30.610","Text":"you go downwards and left."},{"Start":"15:30.610 ","End":"15:33.790","Text":"Each time the 1 is going down and left,"},{"Start":"15:33.790 ","End":"15:36.860","Text":"down and left, down and left."},{"Start":"15:38.580 ","End":"15:40.995","Text":"Here we had 1, 4,"},{"Start":"15:40.995 ","End":"15:42.695","Text":"in general we\u0027d have 1,"},{"Start":"15:42.695 ","End":"15:45.880","Text":"n, first row nth column and so on."},{"Start":"15:45.880 ","End":"15:50.200","Text":"I think you get the idea so this shows this and I think this is"},{"Start":"15:50.200 ","End":"15:56.000","Text":"about completes the exercise. We\u0027re done."}],"ID":9901},{"Watched":false,"Name":"Exercise 16","Duration":"4m 41s","ChapterTopicVideoID":9591,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9591.jpeg","UploadDate":"2017-07-26T08:35:34.4470000","DurationForVideoObject":"PT4M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.390","Text":"This exercise, we have to figure the determinant of matrix A,"},{"Start":"00:06.390 ","End":"00:15.855","Text":"which is n by n and it\u0027s given in terms of its elements a i, j."},{"Start":"00:15.855 ","End":"00:23.190","Text":"Where a,i, j is equal to a if I equals j,"},{"Start":"00:23.190 ","End":"00:27.390","Text":"in other words, the row and the column are the same and b otherwise."},{"Start":"00:27.390 ","End":"00:28.680","Text":"What does this mean?"},{"Start":"00:28.680 ","End":"00:31.455","Text":"When is the row and the column number the same?"},{"Start":"00:31.455 ","End":"00:35.760","Text":"Think about it. That\u0027s the main diagonal, a i j."},{"Start":"00:35.760 ","End":"00:38.415","Text":"When I equals j, we get a_11,"},{"Start":"00:38.415 ","End":"00:40.845","Text":"we get a_22,"},{"Start":"00:40.845 ","End":"00:42.950","Text":"a_33, and so on."},{"Start":"00:42.950 ","End":"00:48.080","Text":"All the main diagonals are a and everything else is a b."},{"Start":"00:48.080 ","End":"00:52.130","Text":"Now I didn\u0027t want to write it with dot-dot."},{"Start":"00:52.130 ","End":"00:54.365","Text":"I did a 6 by 6"},{"Start":"00:54.365 ","End":"00:58.580","Text":"but really it\u0027s an n by n. This is"},{"Start":"00:58.580 ","End":"01:04.070","Text":"just so didn\u0027t mess it up with the ellipsis everywhere."},{"Start":"01:04.070 ","End":"01:08.780","Text":"Let\u0027s see how we can figure this 1 out."},{"Start":"01:08.780 ","End":"01:17.550","Text":"What I suggest doing is that we should subtract the top row from all the other rows."},{"Start":"01:17.550 ","End":"01:20.850","Text":"Mostly it\u0027s b\u0027s and they\u0027ll cancel each other out."},{"Start":"01:20.850 ","End":"01:26.315","Text":"We shouldn\u0027t have too many non-zero elements left after we do that."},{"Start":"01:26.315 ","End":"01:33.490","Text":"Precise notation for what I\u0027m going to do is this says that our R_2,"},{"Start":"01:33.490 ","End":"01:36.660","Text":"we subtract from into R_1."},{"Start":"01:36.660 ","End":"01:40.760","Text":"This says we take away row 1 from row"},{"Start":"01:40.760 ","End":"01:44.975","Text":"3 and put it in row 3 and so on up to the last row,"},{"Start":"01:44.975 ","End":"01:48.470","Text":"otherwise subtract row 1 from each of the other rows."},{"Start":"01:48.470 ","End":"01:50.150","Text":"This just says it precisely."},{"Start":"01:50.150 ","End":"01:53.900","Text":"If we do that, then this is what we have."},{"Start":"01:53.900 ","End":"01:59.720","Text":"Again, I want to remind you it\u0027s not a 6 by 6 matrix,"},{"Start":"01:59.720 ","End":"02:08.210","Text":"it\u0027s n by n. I just didn\u0027t want to put dot-dot-dot the ellipsis everywhere."},{"Start":"02:08.210 ","End":"02:11.795","Text":"Now, things are not looking too bad."},{"Start":"02:11.795 ","End":"02:16.820","Text":"Notice that we have b minus a everywhere"},{"Start":"02:16.820 ","End":"02:26.105","Text":"here and to the right of each of these b minus a\u0027s there\u0027s only 0\u0027s and then a minus b,"},{"Start":"02:26.105 ","End":"02:28.890","Text":"which is the negative of b minus a."},{"Start":"02:28.890 ","End":"02:32.750","Text":"If I add to each of these,"},{"Start":"02:32.750 ","End":"02:37.595","Text":"b minus a is all the rest of the entries,"},{"Start":"02:37.595 ","End":"02:39.545","Text":"then I\u0027ll get 0."},{"Start":"02:39.545 ","End":"02:44.794","Text":"What I\u0027m suggesting now is that we add to the first column,"},{"Start":"02:44.794 ","End":"02:47.075","Text":"all the rest of the columns,"},{"Start":"02:47.075 ","End":"02:49.730","Text":"which basically means taking the sum total of"},{"Start":"02:49.730 ","End":"02:54.035","Text":"these and squashing it onto the first column."},{"Start":"02:54.035 ","End":"02:58.570","Text":"Then we\u0027ll get zeros everywhere here."},{"Start":"02:58.570 ","End":"03:01.730","Text":"Here in precise forms what I\u0027m saying,"},{"Start":"03:01.730 ","End":"03:03.515","Text":"instead of column 1,"},{"Start":"03:03.515 ","End":"03:07.355","Text":"we\u0027re going to take column 1 and add to it all the rest of the columns from"},{"Start":"03:07.355 ","End":"03:13.250","Text":"2-n. Now adding other columns to a given column does not change the determinant,"},{"Start":"03:13.250 ","End":"03:14.825","Text":"so we can do that."},{"Start":"03:14.825 ","End":"03:17.770","Text":"After we do this addition,"},{"Start":"03:17.770 ","End":"03:20.475","Text":"we get, like I said,"},{"Start":"03:20.475 ","End":"03:27.515","Text":"0\u0027s in here in the first column except for the top term,"},{"Start":"03:27.515 ","End":"03:30.260","Text":"which is this, plus all of these."},{"Start":"03:30.260 ","End":"03:32.240","Text":"Now it looks like there\u0027s 5 b\u0027s,"},{"Start":"03:32.240 ","End":"03:33.470","Text":"but it\u0027s not 5,"},{"Start":"03:33.470 ","End":"03:34.955","Text":"it\u0027s 6 minus 1,"},{"Start":"03:34.955 ","End":"03:36.140","Text":"but it\u0027s not 6,"},{"Start":"03:36.140 ","End":"03:40.290","Text":"it\u0027s n. It\u0027s n minus 1 b\u0027s because"},{"Start":"03:40.290 ","End":"03:48.045","Text":"the number of there columns is n. That is just n minus one of them."},{"Start":"03:48.045 ","End":"03:51.105","Text":"Now if you look at it,"},{"Start":"03:51.105 ","End":"03:55.785","Text":"you can see it\u0027s an upper triangular."},{"Start":"03:55.785 ","End":"04:02.490","Text":"Everything below the main diagonal is 0."},{"Start":"04:02.490 ","End":"04:07.100","Text":"Since it\u0027s a triangular matrix,"},{"Start":"04:07.100 ","End":"04:10.550","Text":"the determinant is obtained by just multiplying"},{"Start":"04:10.550 ","End":"04:17.250","Text":"all the entries along the main diagonal, which is these."},{"Start":"04:17.250 ","End":"04:19.175","Text":"What do we have here?"},{"Start":"04:19.175 ","End":"04:22.830","Text":"Let\u0027s say it looks like 5a minus b\u0027s,"},{"Start":"04:22.830 ","End":"04:25.335","Text":"but it\u0027s really n minus 1 of them."},{"Start":"04:25.335 ","End":"04:31.895","Text":"We have to take this and multiply by this to the power of n minus 1."},{"Start":"04:31.895 ","End":"04:35.695","Text":"This is what it is first term."},{"Start":"04:35.695 ","End":"04:39.345","Text":"This to the power of n minus 1."},{"Start":"04:39.345 ","End":"04:42.040","Text":"That\u0027s the answer."}],"ID":9902},{"Watched":false,"Name":"Exercise 17","Duration":"3m 9s","ChapterTopicVideoID":9592,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9592.jpeg","UploadDate":"2017-07-26T08:35:44.8600000","DurationForVideoObject":"PT3M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.799","Text":"In this exercise, we have to figure out the determinant of this matrix."},{"Start":"00:04.799 ","End":"00:08.850","Text":"Notice that there\u0027s ones here, then twos here,"},{"Start":"00:08.850 ","End":"00:14.490","Text":"then threes here, and so on till we have a single n in the lower right corner."},{"Start":"00:14.490 ","End":"00:20.415","Text":"It\u0027s of course an n by n size matrix."},{"Start":"00:20.415 ","End":"00:24.075","Text":"What we\u0027re going to do is row operations."},{"Start":"00:24.075 ","End":"00:29.160","Text":"We\u0027re going actually bring it to upper diagonal form."},{"Start":"00:29.160 ","End":"00:34.940","Text":"The first thing we\u0027ll do is subtract the first row from all the other rows."},{"Start":"00:34.940 ","End":"00:38.735","Text":"Here is the row notation for it."},{"Start":"00:38.735 ","End":"00:42.530","Text":"This says that we subtract the first row from"},{"Start":"00:42.530 ","End":"00:46.595","Text":"the second row and it stays in the second row and then the third row,"},{"Start":"00:46.595 ","End":"00:48.605","Text":"and so on and so on up to the nth,"},{"Start":"00:48.605 ","End":"00:50.665","Text":"which is the last row."},{"Start":"00:50.665 ","End":"00:53.345","Text":"This is what we get."},{"Start":"00:53.345 ","End":"01:02.065","Text":"Notice that all these ones below the 1 here of all become zeros now."},{"Start":"01:02.065 ","End":"01:07.410","Text":"Also notice that this second column now has become all ones."},{"Start":"01:07.410 ","End":"01:10.910","Text":"If we repeat the trick of subtracting this time,"},{"Start":"01:10.910 ","End":"01:13.700","Text":"I\u0027ll subtract the second row from everything."},{"Start":"01:13.700 ","End":"01:17.140","Text":"We can get all these ones to be 0."},{"Start":"01:17.140 ","End":"01:24.830","Text":"I\u0027ll just write that down in row notation that we subtract row 2 from row 3,"},{"Start":"01:24.830 ","End":"01:30.445","Text":"from row 4, right the way down the line up to the nth row."},{"Start":"01:30.445 ","End":"01:34.700","Text":"You see that these ones below"},{"Start":"01:34.700 ","End":"01:41.045","Text":"the diagonal have become zeros and these twos have become ones."},{"Start":"01:41.045 ","End":"01:48.025","Text":"We can use the trick again and subtract the third row from all the rest"},{"Start":"01:48.025 ","End":"01:51.600","Text":"but not just the third row from all the rest,"},{"Start":"01:51.600 ","End":"01:53.165","Text":"we\u0027ll just keep on going."},{"Start":"01:53.165 ","End":"01:55.310","Text":"We don\u0027t know how large this thing is,"},{"Start":"01:55.310 ","End":"01:59.375","Text":"depends on n. But if we just keep on doing the same thing,"},{"Start":"01:59.375 ","End":"02:00.690","Text":"we get zeros,"},{"Start":"02:00.690 ","End":"02:03.200","Text":"zeros will get zeros here next,"},{"Start":"02:03.200 ","End":"02:06.380","Text":"until we get zeros all below the diagonal,"},{"Start":"02:06.380 ","End":"02:10.585","Text":"and what will be left will be ones along the main diagonal."},{"Start":"02:10.585 ","End":"02:12.090","Text":"In fact this is what will get."},{"Start":"02:12.090 ","End":"02:15.950","Text":"They\u0027ll be all ones and also above the diagonal."},{"Start":"02:15.950 ","End":"02:18.035","Text":"Anyway, we got a triangular matrix,"},{"Start":"02:18.035 ","End":"02:22.130","Text":"which is all ones,"},{"Start":"02:22.130 ","End":"02:24.110","Text":"but the main thing is the diagonal,"},{"Start":"02:24.110 ","End":"02:27.319","Text":"when we have a triangular matrix,"},{"Start":"02:27.319 ","End":"02:29.300","Text":"this 1 is an upper triangular matrix."},{"Start":"02:29.300 ","End":"02:32.390","Text":"We just multiply the elements of"},{"Start":"02:32.390 ","End":"02:36.170","Text":"the diagonal and I\u0027ve written the answer 1 times 1 times 1 over many times,"},{"Start":"02:36.170 ","End":"02:37.880","Text":"it comes out to be 1."},{"Start":"02:37.880 ","End":"02:39.410","Text":"If you\u0027re not sure about this,"},{"Start":"02:39.410 ","End":"02:44.465","Text":"you might want to try letting n equals some large numbers as 6,"},{"Start":"02:44.465 ","End":"02:50.360","Text":"and draw a 6-by-6 matrix and work through it without the dot"},{"Start":"02:50.360 ","End":"02:52.985","Text":"but I think this is straightforward enough."},{"Start":"02:52.985 ","End":"02:55.940","Text":"The main idea is that each time we subtract the row,"},{"Start":"02:55.940 ","End":"02:57.470","Text":"we get zeros below,"},{"Start":"02:57.470 ","End":"02:59.660","Text":"we get zeros below,"},{"Start":"02:59.660 ","End":"03:02.990","Text":"then we\u0027ll get zeros below here and so on."},{"Start":"03:02.990 ","End":"03:07.250","Text":"Until we get to all ones on the diagonal and all zeros below."},{"Start":"03:07.250 ","End":"03:10.720","Text":"Done."}],"ID":9903},{"Watched":false,"Name":"Exercise 18","Duration":"6m 12s","ChapterTopicVideoID":9593,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9593.jpeg","UploadDate":"2017-07-26T08:36:02.9500000","DurationForVideoObject":"PT6M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"In this exercise, we have to evaluate the following determinant."},{"Start":"00:05.280 ","End":"00:07.470","Text":"It\u0027s the determinant of this matrix,"},{"Start":"00:07.470 ","End":"00:11.760","Text":"which is an n by n matrix."},{"Start":"00:11.760 ","End":"00:16.560","Text":"One of our favorite tools or tricks is to try bringing"},{"Start":"00:16.560 ","End":"00:21.185","Text":"it to echelon or even diagonal form by using row operations."},{"Start":"00:21.185 ","End":"00:25.605","Text":"In this case, we\u0027re going to subtract the top row from"},{"Start":"00:25.605 ","End":"00:31.080","Text":"all the other rows and that will make everything here be a 0,"},{"Start":"00:31.080 ","End":"00:32.850","Text":"and then we might continue that way."},{"Start":"00:32.850 ","End":"00:37.500","Text":"Well, we\u0027ll see. Here\u0027s the formal notation."},{"Start":"00:37.500 ","End":"00:40.860","Text":"This one says that we subtract row 1 from row 2,"},{"Start":"00:40.860 ","End":"00:44.415","Text":"and it goes in row 2 of course, and so on."},{"Start":"00:44.415 ","End":"00:46.905","Text":"If we do all that,"},{"Start":"00:46.905 ","End":"00:54.470","Text":"then we will get this matrix where you see there are all 0 here."},{"Start":"00:54.470 ","End":"01:01.560","Text":"Now, if I subtract the 2nd row from all the rest from 3 onwards,"},{"Start":"01:01.560 ","End":"01:04.470","Text":"we\u0027re going to get more 0 here."},{"Start":"01:04.470 ","End":"01:07.885","Text":"First of all, I\u0027ll write down what I\u0027m going to do."},{"Start":"01:07.885 ","End":"01:09.470","Text":"I won\u0027t walk you through this,"},{"Start":"01:09.470 ","End":"01:13.085","Text":"you\u0027ve seen this enough times already so we\u0027re going to subtract inwards,"},{"Start":"01:13.085 ","End":"01:17.195","Text":"subtract the 2nd row from the 3rd row onwards."},{"Start":"01:17.195 ","End":"01:22.845","Text":"What we will get if we do that is this."},{"Start":"01:22.845 ","End":"01:30.950","Text":"You might think that we\u0027re going to keep on going that way and keep subtracting, but no."},{"Start":"01:30.950 ","End":"01:37.895","Text":"I\u0027m going to stop at this point and do something a bit different, a new trick."},{"Start":"01:37.895 ","End":"01:43.925","Text":"Let me put a line here and a line here to straighten that up a bit."},{"Start":"01:43.925 ","End":"01:46.130","Text":"Here we are with a bit more space."},{"Start":"01:46.130 ","End":"01:55.505","Text":"Now, what we have here is that this original big matrix is now broken up into blocks."},{"Start":"01:55.505 ","End":"01:57.905","Text":"This is 1 block,"},{"Start":"01:57.905 ","End":"02:00.385","Text":"the bit here,"},{"Start":"02:00.385 ","End":"02:05.805","Text":"and not only is it a block but it\u0027s a square matrix."},{"Start":"02:05.805 ","End":"02:09.710","Text":"Because I started out with square and I took off a square."},{"Start":"02:09.710 ","End":"02:12.305","Text":"Let me just also highlight this block."},{"Start":"02:12.305 ","End":"02:16.030","Text":"These are 2 square matrices."},{"Start":"02:16.030 ","End":"02:19.040","Text":"I also want you to note that this part,"},{"Start":"02:19.040 ","End":"02:22.280","Text":"this block here is all 0."},{"Start":"02:22.280 ","End":"02:28.415","Text":"There\u0027s a rule with determinants that if I have"},{"Start":"02:28.415 ","End":"02:36.079","Text":"a block matrix here and here,"},{"Start":"02:36.079 ","End":"02:38.195","Text":"and this part is A,"},{"Start":"02:38.195 ","End":"02:45.205","Text":"and this is a square matrix and this part B is also a square matrix as we have here,"},{"Start":"02:45.205 ","End":"02:50.040","Text":"and this 1 is usually not square."},{"Start":"02:50.040 ","End":"02:53.239","Text":"This, I could call it D,"},{"Start":"02:53.239 ","End":"02:57.475","Text":"but this happens to equal 0,"},{"Start":"02:57.475 ","End":"02:59.420","Text":"I will just label it 0."},{"Start":"02:59.420 ","End":"03:00.830","Text":"Then in this case,"},{"Start":"03:00.830 ","End":"03:05.390","Text":"when the matrix decompose into 4 blocks this way,"},{"Start":"03:05.390 ","End":"03:14.180","Text":"then the determinant of this is equal to the product of the determinants of these 2,"},{"Start":"03:14.180 ","End":"03:18.284","Text":"is the determinant of"},{"Start":"03:18.284 ","End":"03:26.300","Text":"A times the determinant of B."},{"Start":"03:26.300 ","End":"03:28.450","Text":"It also works, of course,"},{"Start":"03:28.450 ","End":"03:31.270","Text":"if C was 0 when D wasn\u0027t,"},{"Start":"03:31.270 ","End":"03:36.180","Text":"but 1 of these 2 non-square blocks has"},{"Start":"03:36.180 ","End":"03:41.485","Text":"to be 0 and then we can multiply the determinant of this by the determinant of this."},{"Start":"03:41.485 ","End":"03:44.200","Text":"Of course, it would make no sense if you tried"},{"Start":"03:44.200 ","End":"03:47.140","Text":"to figure some rule with the determinant of C"},{"Start":"03:47.140 ","End":"03:48.550","Text":"or D because they\u0027re not"},{"Start":"03:48.550 ","End":"03:52.600","Text":"square matrices and you can take determinants only of square matrices,"},{"Start":"03:52.600 ","End":"03:56.510","Text":"so don\u0027t do anything silly like that."},{"Start":"03:57.030 ","End":"04:02.490","Text":"Here, this breaks up like this."},{"Start":"04:02.490 ","End":"04:04.180","Text":"Instead of the word debt,"},{"Start":"04:04.180 ","End":"04:06.995","Text":"I just used the vertical bars."},{"Start":"04:06.995 ","End":"04:09.610","Text":"This is the green part."},{"Start":"04:09.610 ","End":"04:13.440","Text":"It doesn\u0027t hurt me to color it if that helps."},{"Start":"04:13.440 ","End":"04:19.660","Text":"This one is that one that was there, no trouble."},{"Start":"04:21.320 ","End":"04:23.520","Text":"As for the size,"},{"Start":"04:23.520 ","End":"04:25.665","Text":"this was n by n,"},{"Start":"04:25.665 ","End":"04:27.885","Text":"this of course is 2 by 2."},{"Start":"04:27.885 ","End":"04:33.260","Text":"This is going to be n minus 2 by n minus 2,"},{"Start":"04:33.260 ","End":"04:36.785","Text":"the size of the determinant of a matrix."},{"Start":"04:36.785 ","End":"04:40.585","Text":"Now, this is going to be easy enough,"},{"Start":"04:40.585 ","End":"04:43.380","Text":"but what are we going to do with this?"},{"Start":"04:43.380 ","End":"04:46.940","Text":"This is where the previous exercise comes in handy,"},{"Start":"04:46.940 ","End":"04:49.895","Text":"and I hope you\u0027ve done the previous exercise,"},{"Start":"04:49.895 ","End":"04:57.315","Text":"because we can reduce this to that if we just take 3s out of each row."},{"Start":"04:57.315 ","End":"05:01.140","Text":"I can take 3 out of this row and bring it to the front,"},{"Start":"05:01.140 ","End":"05:07.875","Text":"then take another 3 out of here and do that n minus 2 times and then we get this."},{"Start":"05:07.875 ","End":"05:11.420","Text":"Now, the 2 is the determinant of this because we"},{"Start":"05:11.420 ","End":"05:15.170","Text":"can just multiply this diagonal minus this diagonal;"},{"Start":"05:15.170 ","End":"05:19.205","Text":"2 minus 0 is 2, that\u0027s that bit."},{"Start":"05:19.205 ","End":"05:26.810","Text":"Now, this because I took 3 out of each row and there were n minus 2 rows,"},{"Start":"05:26.810 ","End":"05:32.260","Text":"this 3 comes out to the power of n minus 2."},{"Start":"05:32.260 ","End":"05:35.465","Text":"Then what we\u0027re left with is this."},{"Start":"05:35.465 ","End":"05:38.060","Text":"Now if you go back to the previous exercise,"},{"Start":"05:38.060 ","End":"05:39.320","Text":"this is what we had,"},{"Start":"05:39.320 ","End":"05:46.475","Text":"except there we went up to n and here it\u0027s only up to n minus 2,"},{"Start":"05:46.475 ","End":"05:49.220","Text":"but in all cases, for every n,"},{"Start":"05:49.220 ","End":"05:51.800","Text":"We found out that this was equal to"},{"Start":"05:51.800 ","End":"05:59.120","Text":"1 regardless of n. I guess we\u0027re assuming that n is bigger than 2,"},{"Start":"05:59.120 ","End":"06:01.100","Text":"otherwise this doesn\u0027t exist."},{"Start":"06:01.100 ","End":"06:03.770","Text":"Anyway, this times 1."},{"Start":"06:03.770 ","End":"06:08.930","Text":"I suppose that we don\u0027t even have to bother writing that."},{"Start":"06:08.930 ","End":"06:12.690","Text":"This is the answer, and we\u0027re done."}],"ID":9904},{"Watched":false,"Name":"Exercise 19","Duration":"14m 32s","ChapterTopicVideoID":9594,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9594.jpeg","UploadDate":"2017-07-26T08:36:43.6970000","DurationForVideoObject":"PT14M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"This exercise is a bit more challenging."},{"Start":"00:03.120 ","End":"00:07.530","Text":"I\u0027ll give it an asterisk because we\u0027re going to"},{"Start":"00:07.530 ","End":"00:12.855","Text":"use sequences and regressive or recursive equations."},{"Start":"00:12.855 ","End":"00:14.610","Text":"But even if you haven\u0027t,"},{"Start":"00:14.610 ","End":"00:16.935","Text":"you can still go along up to a point."},{"Start":"00:16.935 ","End":"00:22.485","Text":"What we have to do is to evaluate the determinant of the n by n matrix A,"},{"Start":"00:22.485 ","End":"00:26.280","Text":"where it\u0027s given element wise that"},{"Start":"00:26.280 ","End":"00:32.040","Text":"the a_ij element is given according to the following rule."},{"Start":"00:32.040 ","End":"00:34.230","Text":"When i equals j, it\u0027s a."},{"Start":"00:34.230 ","End":"00:36.285","Text":"When i is j plus 1, it\u0027s b."},{"Start":"00:36.285 ","End":"00:37.800","Text":"When j is i plus 1,"},{"Start":"00:37.800 ","End":"00:39.845","Text":"it\u0027s c, and 0 otherwise."},{"Start":"00:39.845 ","End":"00:42.590","Text":"Now let me just quickly interpret this."},{"Start":"00:42.590 ","End":"00:43.970","Text":"We\u0027ve done a few like this,"},{"Start":"00:43.970 ","End":"00:46.865","Text":"so it shouldn\u0027t be too bad."},{"Start":"00:46.865 ","End":"00:50.270","Text":"This tells me where I\u0027m going to put an a."},{"Start":"00:50.270 ","End":"00:51.920","Text":"Wherever i equals j,"},{"Start":"00:51.920 ","End":"00:54.355","Text":"which means the row equals the column,"},{"Start":"00:54.355 ","End":"00:57.300","Text":"these will be the elements a_11,"},{"Start":"00:57.300 ","End":"01:01.150","Text":"a_22, a_33, etc."},{"Start":"01:01.150 ","End":"01:03.410","Text":"These are all going to be a."},{"Start":"01:03.410 ","End":"01:07.160","Text":"Now, when i equals j plus 1,"},{"Start":"01:07.160 ","End":"01:11.720","Text":"it means that the row is 1 more"},{"Start":"01:11.720 ","End":"01:16.024","Text":"than the column and it\u0027s actually going to be just below the diagonal."},{"Start":"01:16.024 ","End":"01:17.585","Text":"These are on the diagonal."},{"Start":"01:17.585 ","End":"01:19.640","Text":"But let\u0027s say j is 1,"},{"Start":"01:19.640 ","End":"01:20.810","Text":"then i is 2."},{"Start":"01:20.810 ","End":"01:23.090","Text":"We have a_21,"},{"Start":"01:23.090 ","End":"01:28.755","Text":"a_32, a_43 and so on."},{"Start":"01:28.755 ","End":"01:30.185","Text":"Here\u0027s the other way."},{"Start":"01:30.185 ","End":"01:35.390","Text":"It\u0027s going to be just to the right or above the diagonal."},{"Start":"01:35.390 ","End":"01:43.580","Text":"We\u0027re going to have a_12, a_23, a_34."},{"Start":"01:43.580 ","End":"01:49.430","Text":"Let\u0027s see now how this comes out and everything else is going to be 0."},{"Start":"01:49.430 ","End":"01:51.940","Text":"That\u0027s not 1 of these cases."},{"Start":"01:51.940 ","End":"01:54.800","Text":"Let\u0027s just focus on the matrix part."},{"Start":"01:54.800 ","End":"01:58.640","Text":"Now, this where i equals j, a_11,"},{"Start":"01:58.640 ","End":"02:01.040","Text":"a_22, if you think about it,"},{"Start":"02:01.040 ","End":"02:02.525","Text":"is just the diagonal."},{"Start":"02:02.525 ","End":"02:04.745","Text":"Row and the column are the same."},{"Start":"02:04.745 ","End":"02:08.390","Text":"These are these elements."},{"Start":"02:08.390 ","End":"02:10.805","Text":"Now if I take a_21,"},{"Start":"02:10.805 ","End":"02:13.370","Text":"then row 2,"},{"Start":"02:13.370 ","End":"02:14.930","Text":"column 1, that\u0027s this."},{"Start":"02:14.930 ","End":"02:17.210","Text":"Row 3, column 2 is this."},{"Start":"02:17.210 ","End":"02:20.860","Text":"I guess I just should have highlighted also the a."},{"Start":"02:20.860 ","End":"02:26.650","Text":"As we were saying, these correspond to this,"},{"Start":"02:26.650 ","End":"02:31.360","Text":"where the row is 1 more than the column,"},{"Start":"02:31.360 ","End":"02:37.450","Text":"which means just below the diagonal or to the left,"},{"Start":"02:37.450 ","End":"02:39.655","Text":"whichever way you want to look at it."},{"Start":"02:39.655 ","End":"02:42.410","Text":"Those are all going to be b."},{"Start":"02:42.410 ","End":"02:46.320","Text":"These entries like row 1,"},{"Start":"02:46.320 ","End":"02:48.195","Text":"column 2 is this,"},{"Start":"02:48.195 ","End":"02:51.225","Text":"row 2, column 3 is this."},{"Start":"02:51.225 ","End":"02:53.460","Text":"Here we have cs,"},{"Start":"02:53.460 ","End":"02:55.470","Text":"just like it says here."},{"Start":"02:55.470 ","End":"02:58.300","Text":"Everything else is 0,"},{"Start":"02:58.300 ","End":"03:01.330","Text":"so I won\u0027t even bother highlighting them."},{"Start":"03:01.330 ","End":"03:06.450","Text":"Now, strictly speaking, I drew a 6 by 6,"},{"Start":"03:06.450 ","End":"03:08.790","Text":"but I don\u0027t mean this to be 6 by 6."},{"Start":"03:08.790 ","End":"03:10.050","Text":"You did it properly."},{"Start":"03:10.050 ","End":"03:15.355","Text":"There would be ellipses everywhere here and you would say ellipses and so on."},{"Start":"03:15.355 ","End":"03:19.440","Text":"I don\u0027t want to mess with that."},{"Start":"03:19.440 ","End":"03:21.090","Text":"I\u0027m drawing it 6 by 6,"},{"Start":"03:21.090 ","End":"03:23.400","Text":"but it\u0027s really n by n,"},{"Start":"03:23.400 ","End":"03:26.295","Text":"n rows and n columns."},{"Start":"03:26.295 ","End":"03:34.560","Text":"Let\u0027s label the result in sequence notation, D_n."},{"Start":"03:34.560 ","End":"03:42.420","Text":"The determinant of the n by n matrix that follows this rule will be D_n."},{"Start":"03:42.420 ","End":"03:46.470","Text":"Let\u0027s take n from 2 onwards and it could be 2, 3, 4."},{"Start":"03:46.470 ","End":"03:50.420","Text":"We usually don\u0027t mess with 1 by 1 matrices."},{"Start":"03:50.420 ","End":"03:51.920","Text":"It\u0027s just a single number."},{"Start":"03:51.920 ","End":"03:54.360","Text":"We start from 2."},{"Start":"03:54.460 ","End":"03:59.085","Text":"Let\u0027s take some special cases where n is"},{"Start":"03:59.085 ","End":"04:05.285","Text":"2 or 3 because the general case is going to have some difficulties."},{"Start":"04:05.285 ","End":"04:07.805","Text":"Let\u0027s start with easy, n is 2."},{"Start":"04:07.805 ","End":"04:09.365","Text":"We have the determinant,"},{"Start":"04:09.365 ","End":"04:14.850","Text":"a is along the 1 diagonal above c and b."},{"Start":"04:14.850 ","End":"04:16.695","Text":"This is what it comes out to be."},{"Start":"04:16.695 ","End":"04:20.210","Text":"The answer is just this times this minus this times this."},{"Start":"04:20.210 ","End":"04:22.745","Text":"We have what D_2 is."},{"Start":"04:22.745 ","End":"04:26.990","Text":"Now on to the 3 by 3,"},{"Start":"04:26.990 ","End":"04:32.885","Text":"the 3 by 3 we get a is along the diagonal,"},{"Start":"04:32.885 ","End":"04:35.135","Text":"c is above the diagonal,"},{"Start":"04:35.135 ","End":"04:36.440","Text":"and b is below,"},{"Start":"04:36.440 ","End":"04:38.255","Text":"and the rest 0."},{"Start":"04:38.255 ","End":"04:44.210","Text":"What I suggest is expanding by the first column,"},{"Start":"04:44.210 ","End":"04:46.370","Text":"could also do it by first row."},{"Start":"04:46.370 ","End":"04:48.440","Text":"The first column we have a 0 here,"},{"Start":"04:48.440 ","End":"04:51.650","Text":"so we\u0027re going to get an entry from the a and from the b."},{"Start":"04:51.650 ","End":"04:54.100","Text":"Let\u0027s start with the a."},{"Start":"04:54.100 ","End":"05:00.500","Text":"Remember, the contribution is first of all a sign like plus, minus, plus."},{"Start":"05:00.500 ","End":"05:01.925","Text":"It\u0027s a plus here."},{"Start":"05:01.925 ","End":"05:04.795","Text":"Then the entry which is an a."},{"Start":"05:04.795 ","End":"05:09.750","Text":"Then the minor, which is what we have not crossed out."},{"Start":"05:10.130 ","End":"05:14.130","Text":"In short, this is what it is."},{"Start":"05:14.130 ","End":"05:18.270","Text":"Next, we\u0027ll do the b."},{"Start":"05:18.270 ","End":"05:21.165","Text":"This one\u0027s going to be a minus."},{"Start":"05:21.165 ","End":"05:24.370","Text":"Minus, plus."},{"Start":"05:24.580 ","End":"05:27.724","Text":"We get the minus from the sign,"},{"Start":"05:27.724 ","End":"05:29.690","Text":"the entry is b,"},{"Start":"05:29.690 ","End":"05:30.920","Text":"and the minor is c,"},{"Start":"05:30.920 ","End":"05:33.130","Text":"0, b, a."},{"Start":"05:33.130 ","End":"05:35.255","Text":"If we compute it,"},{"Start":"05:35.255 ","End":"05:40.640","Text":"this is what we get and it simplifies to this expression."},{"Start":"05:40.640 ","End":"05:43.070","Text":"You know what? I just want to write them out again."},{"Start":"05:43.070 ","End":"05:49.590","Text":"I want to write what we found for D_2 and for D_3."},{"Start":"05:49.590 ","End":"05:54.450","Text":"D_2 we got was a squared minus bc."},{"Start":"05:54.450 ","End":"05:59.945","Text":"D_3 is a cubed minus twice abc."},{"Start":"05:59.945 ","End":"06:03.950","Text":"Now let\u0027s see if we can go over something more general."},{"Start":"06:03.950 ","End":"06:09.250","Text":"I\u0027m going to go over to the case of n in general,"},{"Start":"06:09.250 ","End":"06:15.240","Text":"assuming that n is bigger than 3."},{"Start":"06:15.240 ","End":"06:18.925","Text":"Here\u0027s our matrix, what it looked like."},{"Start":"06:18.925 ","End":"06:26.455","Text":"What I suggest doing is expanding along the first column with only 2 non-zero entries."},{"Start":"06:26.455 ","End":"06:29.720","Text":"Let\u0027s start with this 1."},{"Start":"06:30.000 ","End":"06:34.795","Text":"Now this contributes, remember that the checkerboard plus,"},{"Start":"06:34.795 ","End":"06:37.180","Text":"minus, plus, etc, plus,"},{"Start":"06:37.180 ","End":"06:39.040","Text":"so we don\u0027t write anything."},{"Start":"06:39.040 ","End":"06:41.110","Text":"Then there\u0027s the entry itself,"},{"Start":"06:41.110 ","End":"06:42.445","Text":"which is the a."},{"Start":"06:42.445 ","End":"06:46.850","Text":"Then the minor which is this determinant,"},{"Start":"06:46.850 ","End":"06:51.745","Text":"but it\u0027s size is n minus 1 by n minus 1."},{"Start":"06:51.745 ","End":"06:54.010","Text":"Still on the first column,"},{"Start":"06:54.010 ","End":"06:57.295","Text":"the other non-zero entry is this 1."},{"Start":"06:57.295 ","End":"07:00.770","Text":"This time it\u0027s going to be a minus."},{"Start":"07:00.770 ","End":"07:09.075","Text":"It contributes this minus with the b and then the minor which is this part,"},{"Start":"07:09.075 ","End":"07:13.930","Text":"c with 0s and then this part below it."},{"Start":"07:14.760 ","End":"07:24.819","Text":"Now, the first part is actually similar to the original matrix except 1 smaller."},{"Start":"07:24.819 ","End":"07:27.070","Text":"We have a\u0027s and then c\u0027s and then b\u0027s,"},{"Start":"07:27.070 ","End":"07:29.560","Text":"but this still needs a bit more work."},{"Start":"07:29.560 ","End":"07:33.790","Text":"What we\u0027re going to do is expand this 1 along"},{"Start":"07:33.790 ","End":"07:38.110","Text":"the first row where there\u0027s only a single non-zero elements."},{"Start":"07:38.110 ","End":"07:41.815","Text":"So I can already mark the column also,"},{"Start":"07:41.815 ","End":"07:44.380","Text":"and let\u0027s get some more space."},{"Start":"07:44.380 ","End":"07:48.040","Text":"So this is the a,"},{"Start":"07:48.040 ","End":"07:49.270","Text":"this, like I said,"},{"Start":"07:49.270 ","End":"07:51.820","Text":"is the same pattern as the original."},{"Start":"07:51.820 ","End":"07:53.950","Text":"It\u0027s just 1 size smaller."},{"Start":"07:53.950 ","End":"07:59.040","Text":"So it\u0027s D_n minus 1 minus b."},{"Start":"07:59.040 ","End":"08:01.185","Text":"Now we\u0027re going to do the expansion."},{"Start":"08:01.185 ","End":"08:07.645","Text":"So we have a plus for the c here."},{"Start":"08:07.645 ","End":"08:12.620","Text":"So we\u0027ll get c the element together with this."},{"Start":"08:12.720 ","End":"08:17.080","Text":"This part here is already 2 sizes smaller."},{"Start":"08:17.080 ","End":"08:19.885","Text":"It\u0027s n by 2 times n by 2,"},{"Start":"08:19.885 ","End":"08:22.720","Text":"and it\u0027s also the same pattern as"},{"Start":"08:22.720 ","End":"08:28.335","Text":"the original just 2 sizes smaller until we have just cleared some space."},{"Start":"08:28.335 ","End":"08:29.790","Text":"So like I was saying,"},{"Start":"08:29.790 ","End":"08:31.140","Text":"this is the same pattern,"},{"Start":"08:31.140 ","End":"08:35.505","Text":"2 sizes smaller so it\u0027s going to be D_n minus 2."},{"Start":"08:35.505 ","End":"08:37.440","Text":"Now, in a sense,"},{"Start":"08:37.440 ","End":"08:44.610","Text":"we\u0027ve solved the problem because we found D_2,"},{"Start":"08:44.610 ","End":"08:46.545","Text":"we found D_3,"},{"Start":"08:46.545 ","End":"08:51.655","Text":"and for n bigger than 3,"},{"Start":"08:51.655 ","End":"08:55.450","Text":"we have what is called a recursive or regressive formula."},{"Start":"08:55.450 ","End":"08:59.620","Text":"It gives us D_n in terms of smaller ones."},{"Start":"08:59.620 ","End":"09:03.565","Text":"For example, if n was 4,"},{"Start":"09:03.565 ","End":"09:08.710","Text":"then it would give us D_4 in terms of D_3 and D_2,"},{"Start":"09:08.710 ","End":"09:12.010","Text":"D_5 in terms of D_4 and D_3."},{"Start":"09:12.010 ","End":"09:18.010","Text":"We can keep on building them 1 by 1 as far as we want to go."},{"Start":"09:18.010 ","End":"09:20.740","Text":"Now all this is a bit too general."},{"Start":"09:20.740 ","End":"09:23.980","Text":"I mean, we have unknown constants a, b, and c,"},{"Start":"09:23.980 ","End":"09:28.030","Text":"and we have a varying n. Let\u0027s at least fix a, b,"},{"Start":"09:28.030 ","End":"09:32.260","Text":"and c, and hopefully this will become all clearer."},{"Start":"09:32.260 ","End":"09:34.585","Text":"So we\u0027ll take a to be 3,"},{"Start":"09:34.585 ","End":"09:38.545","Text":"b we\u0027ll take as 1 and c as 2."},{"Start":"09:38.545 ","End":"09:42.580","Text":"I want to just remember the terminology that this is"},{"Start":"09:42.580 ","End":"09:46.200","Text":"actually called a recurrence relation,"},{"Start":"09:46.200 ","End":"09:53.280","Text":"even though we also say it\u0027s a regressive or recursive formula,"},{"Start":"09:53.280 ","End":"09:56.440","Text":"regression, recurrence, recursion."},{"Start":"09:56.440 ","End":"09:58.390","Text":"It\u0027s all similar."},{"Start":"09:58.390 ","End":"10:01.420","Text":"So this is a recurrence relation."},{"Start":"10:01.420 ","End":"10:07.090","Text":"Yeah, if I take this formula and plug in a, b,"},{"Start":"10:07.090 ","End":"10:09.550","Text":"c as here,"},{"Start":"10:09.550 ","End":"10:14.050","Text":"and we can also compute D_2 and D_3."},{"Start":"10:14.050 ","End":"10:18.595","Text":"A squared minus bc with these values comes out 7,"},{"Start":"10:18.595 ","End":"10:21.070","Text":"and this 1 comes out 15."},{"Start":"10:21.070 ","End":"10:24.760","Text":"So here\u0027s basically everything that we need to know."},{"Start":"10:24.760 ","End":"10:31.090","Text":"We have that D_2 is 7, D_3 is 15."},{"Start":"10:31.090 ","End":"10:33.760","Text":"For anything higher than 3,"},{"Start":"10:33.760 ","End":"10:41.890","Text":"we can use the recurrence relation to get each n in terms of the previous ones."},{"Start":"10:41.890 ","End":"10:48.070","Text":"Now, for those of you who have studied recurrence,"},{"Start":"10:48.070 ","End":"10:56.140","Text":"we would prefer to get a closed formula to say straight away what D_100 is."},{"Start":"10:56.140 ","End":"11:02.665","Text":"I mean, this would just tell me how to get D_100 in terms of D_99, and D_98."},{"Start":"11:02.665 ","End":"11:05.695","Text":"That\u0027s to keep going back and back,"},{"Start":"11:05.695 ","End":"11:07.780","Text":"I\u0027d like to get there straight away."},{"Start":"11:07.780 ","End":"11:09.790","Text":"So for those of you who\u0027ve studied,"},{"Start":"11:09.790 ","End":"11:12.130","Text":"you can stay with me."},{"Start":"11:12.130 ","End":"11:17.155","Text":"For the rest, it\u0027s optional if you want to stay or not is what I\u0027m saying."},{"Start":"11:17.155 ","End":"11:24.475","Text":"I\u0027m now proceeding using the theory of linear recurrence relationships."},{"Start":"11:24.475 ","End":"11:26.350","Text":"It\u0027s called linear homogeneous."},{"Start":"11:26.350 ","End":"11:30.160","Text":"Anyway, there is a whole theory of how to do these,"},{"Start":"11:30.160 ","End":"11:32.860","Text":"and what we do is first of all,"},{"Start":"11:32.860 ","End":"11:35.890","Text":"find the characteristic polynomial."},{"Start":"11:35.890 ","End":"11:37.510","Text":"This we get from this."},{"Start":"11:37.510 ","End":"11:47.575","Text":"Well, might first of all write it as D_n minus 3D_n minus 1 plus 2D_n minus 2,"},{"Start":"11:47.575 ","End":"11:51.400","Text":"and because we go down from n to n minus 2,"},{"Start":"11:51.400 ","End":"11:58.660","Text":"we need a quadratic and we just copy the coefficients like 1 minus 3 and 2."},{"Start":"11:58.660 ","End":"12:04.405","Text":"We also set this to 0, this characteristic polynomial."},{"Start":"12:04.405 ","End":"12:07.420","Text":"What we want to find are its roots."},{"Start":"12:07.420 ","End":"12:09.130","Text":"I\u0027ll tell you what they are."},{"Start":"12:09.130 ","End":"12:11.530","Text":"We\u0027re not going to waste time with quadratic equations."},{"Start":"12:11.530 ","End":"12:13.915","Text":"The answers are 1 and 2."},{"Start":"12:13.915 ","End":"12:16.300","Text":"When we have the roots,"},{"Start":"12:16.300 ","End":"12:19.420","Text":"then the theory says that the general term is of"},{"Start":"12:19.420 ","End":"12:25.335","Text":"the form some constant A times 1 root to the n,"},{"Start":"12:25.335 ","End":"12:28.170","Text":"plus another constant B times the other root to"},{"Start":"12:28.170 ","End":"12:33.390","Text":"the n. So we still have to find 2 constants,"},{"Start":"12:33.390 ","End":"12:35.615","Text":"A and B,"},{"Start":"12:35.615 ","End":"12:46.135","Text":"and the way we do that is by plugging n equals 2 and 3 into here."},{"Start":"12:46.135 ","End":"12:48.910","Text":"Then we have 2 equations and 2 unknowns,"},{"Start":"12:48.910 ","End":"12:51.759","Text":"A and B as follows."},{"Start":"12:51.759 ","End":"12:55.015","Text":"D_2, according to this formula,"},{"Start":"12:55.015 ","End":"12:56.950","Text":"is A times 1 squared,"},{"Start":"12:56.950 ","End":"12:59.770","Text":"I\u0027m putting n equals 2 plus B times 2 squared."},{"Start":"12:59.770 ","End":"13:03.070","Text":"On the other hand, D_2 from here is 7."},{"Start":"13:03.070 ","End":"13:06.835","Text":"So we got that this is 7."},{"Start":"13:06.835 ","End":"13:10.690","Text":"Similarly with the 3, if we put n equals 3,"},{"Start":"13:10.690 ","End":"13:12.310","Text":"we have a cubed here,"},{"Start":"13:12.310 ","End":"13:15.730","Text":"cubed here, and we have the 15 from here."},{"Start":"13:15.730 ","End":"13:18.430","Text":"To save time, I\u0027ve given you the answers."},{"Start":"13:18.430 ","End":"13:20.870","Text":"I mean, we don\u0027t need this part."},{"Start":"13:23.250 ","End":"13:25.870","Text":"Well, I\u0027ll give you a bit of how we would do it."},{"Start":"13:25.870 ","End":"13:30.430","Text":"We would say that A plus 4B is 7 because 2 squared,"},{"Start":"13:30.430 ","End":"13:36.415","Text":"and then we would say that A plus 2 cubed is 8, B is 15."},{"Start":"13:36.415 ","End":"13:42.940","Text":"Then you might subtract the second from the first and get 4B equals 8, so B is 2."},{"Start":"13:42.940 ","End":"13:44.830","Text":"Plug in B equals 2."},{"Start":"13:44.830 ","End":"13:47.260","Text":"Here, A plus 8 is 7,"},{"Start":"13:47.260 ","End":"13:48.655","Text":"so A is minus 1,"},{"Start":"13:48.655 ","End":"13:50.440","Text":"and here we are."},{"Start":"13:50.440 ","End":"13:52.765","Text":"Now that we have A and B,"},{"Start":"13:52.765 ","End":"13:56.395","Text":"we have to plug them into here and here,"},{"Start":"13:56.395 ","End":"14:06.460","Text":"and then we get the closed form that D_n is minus 1 plus B is 2."},{"Start":"14:06.460 ","End":"14:11.110","Text":"So I can just jack up the index by 1,"},{"Start":"14:11.110 ","End":"14:14.380","Text":"so it\u0027s minus 1 plus 2 to the n plus 1."},{"Start":"14:14.380 ","End":"14:20.920","Text":"As an example, suppose I wanted to know what D_20 is,"},{"Start":"14:20.920 ","End":"14:24.460","Text":"and I would just plug in 20 to the formula,"},{"Start":"14:24.460 ","End":"14:28.570","Text":"and it\u0027s minus 1 plus 2 to the power of 21,"},{"Start":"14:28.570 ","End":"14:32.480","Text":"and we\u0027re finally done with this exercise."}],"ID":9905},{"Watched":false,"Name":"Exercise 20","Duration":"3m 19s","ChapterTopicVideoID":9595,"CourseChapterTopicPlaylistID":7286,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9595.jpeg","UploadDate":"2017-07-26T08:36:52.2400000","DurationForVideoObject":"PT3M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"In this exercise, we have to evaluate the following."},{"Start":"00:04.065 ","End":"00:14.115","Text":"It\u0027s a determinant of a 5 by 5 matrix plus another determinant of a 5 by 5 matrix."},{"Start":"00:14.115 ","End":"00:19.545","Text":"Now, the temptation is just to add the 2 matrices."},{"Start":"00:19.545 ","End":"00:23.700","Text":"That would be a mistake because although there is a rule with"},{"Start":"00:23.700 ","End":"00:27.300","Text":"multiplication that the determinant of"},{"Start":"00:27.300 ","End":"00:36.615","Text":"A times B is equal to the determinant of A times the determinant of B."},{"Start":"00:36.615 ","End":"00:42.600","Text":"Here, assuming that the 2 matrices are of the same size."},{"Start":"00:42.600 ","End":"00:44.880","Text":"This is true,"},{"Start":"00:44.880 ","End":"00:49.460","Text":"but to say that the determinant of A plus"},{"Start":"00:49.460 ","End":"00:56.550","Text":"B is equal to the determinant of A plus the determinant of B,"},{"Start":"00:57.350 ","End":"01:00.710","Text":"that is definitely wrong."},{"Start":"01:00.710 ","End":"01:05.780","Text":"I mean, it might be in some cases where it would work,"},{"Start":"01:05.780 ","End":"01:07.895","Text":"I could think of examples."},{"Start":"01:07.895 ","End":"01:12.185","Text":"If A and B were both 0 matrices, it would work,"},{"Start":"01:12.185 ","End":"01:14.525","Text":"but in general, not."},{"Start":"01:14.525 ","End":"01:18.920","Text":"However, there is something that we can do,"},{"Start":"01:18.920 ","End":"01:21.040","Text":"at least in this case."},{"Start":"01:21.040 ","End":"01:28.175","Text":"If the 2 matrices involved that we want to add are almost the same,"},{"Start":"01:28.175 ","End":"01:29.990","Text":"and by which I mean they\u0027re the same,"},{"Start":"01:29.990 ","End":"01:33.215","Text":"except for a single row or a single column,"},{"Start":"01:33.215 ","End":"01:34.768","Text":"then there\u0027s something we can do."},{"Start":"01:34.768 ","End":"01:41.014","Text":"In this case, it\u0027s the last row that is different,"},{"Start":"01:41.014 ","End":"01:43.100","Text":"but all the rest is the same."},{"Start":"01:43.100 ","End":"01:46.325","Text":"You can see a, b, c, d, e, and so on."},{"Start":"01:46.325 ","End":"01:50.495","Text":"All identical except for that single last row."},{"Start":"01:50.495 ","End":"01:55.549","Text":"In this case, the rule is that you copy"},{"Start":"01:55.549 ","End":"02:01.820","Text":"the common part and add the part that differs."},{"Start":"02:01.820 ","End":"02:05.615","Text":"I\u0027ve hidden the fifth row here."},{"Start":"02:05.615 ","End":"02:07.460","Text":"I just want to show you the common part,"},{"Start":"02:07.460 ","End":"02:10.280","Text":"which is this, is just copied."},{"Start":"02:10.280 ","End":"02:16.310","Text":"Now, I\u0027ll reveal the fifth row."},{"Start":"02:16.310 ","End":"02:22.085","Text":"We get this just by adding corresponding entries in these 2."},{"Start":"02:22.085 ","End":"02:33.095","Text":"Look, 2a plus 1 plus minus a minus 1 gives us a, and so on."},{"Start":"02:33.095 ","End":"02:40.682","Text":"I\u0027ll do 1 more. Y plus e minus y gives us e."},{"Start":"02:40.682 ","End":"02:46.476","Text":"All the other 3, I mean, we get a, b, c, d, e."},{"Start":"02:46.476 ","End":"02:53.040","Text":"Now, we can proceed because you should observe,"},{"Start":"02:53.350 ","End":"02:56.240","Text":"and you might not, but hopefully you do,"},{"Start":"02:56.240 ","End":"03:02.450","Text":"that the top and the bottom row, they are identical."},{"Start":"03:02.450 ","End":"03:05.225","Text":"This and this are just the same."},{"Start":"03:05.225 ","End":"03:09.680","Text":"When you have 2 identical rows in a determinant,"},{"Start":"03:09.680 ","End":"03:15.385","Text":"then the result is, that\u0027s right, 0."},{"Start":"03:15.385 ","End":"03:19.620","Text":"That\u0027s the answer and we\u0027re done."}],"ID":9906}],"Thumbnail":null,"ID":7286},{"Name":"More Rules of Determinants","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"More Rules of Determinants","Duration":"5m 1s","ChapterTopicVideoID":9565,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9565.jpeg","UploadDate":"2017-07-26T08:33:57.0300000","DurationForVideoObject":"PT5M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"Here are some more rules for determinants."},{"Start":"00:03.930 ","End":"00:08.580","Text":"Although we might have possibly used some of them before I gave them,"},{"Start":"00:08.580 ","End":"00:10.810","Text":"I know that the first 1 we have,"},{"Start":"00:10.810 ","End":"00:13.530","Text":"that the determinant of"},{"Start":"00:13.530 ","End":"00:20.340","Text":"a transpose matrix is the same as the determinant of the matrix itself."},{"Start":"00:20.340 ","End":"00:23.070","Text":"T here is transpose."},{"Start":"00:23.070 ","End":"00:26.790","Text":"The inverse of a matrix,"},{"Start":"00:26.790 ","End":"00:28.530","Text":"if we take its determinant,"},{"Start":"00:28.530 ","End":"00:33.395","Text":"it\u0027s just the reciprocal of the determinant of the original matrix."},{"Start":"00:33.395 ","End":"00:36.265","Text":"This relates to inverse."},{"Start":"00:36.265 ","End":"00:40.174","Text":"Now you might say, what if this is 0?"},{"Start":"00:40.174 ","End":"00:45.710","Text":"Well, it turns out that if a determinant of a matrix is 0,"},{"Start":"00:45.710 ","End":"00:47.135","Text":"it doesn\u0027t have an inverse,"},{"Start":"00:47.135 ","End":"00:49.295","Text":"so this won\u0027t happen."},{"Start":"00:49.295 ","End":"00:53.390","Text":"It has an inverse, it can\u0027t have determinants 0, so we okay."},{"Start":"00:53.390 ","End":"00:55.615","Text":"Rule number 3,"},{"Start":"00:55.615 ","End":"00:57.765","Text":"I used A, B, C,"},{"Start":"00:57.765 ","End":"01:03.020","Text":"meaning I used 3 factors in a product,"},{"Start":"01:03.020 ","End":"01:04.910","Text":"but it worked for just A,"},{"Start":"01:04.910 ","End":"01:06.170","Text":"B or A, B,"},{"Start":"01:06.170 ","End":"01:07.925","Text":"C, D, or however many."},{"Start":"01:07.925 ","End":"01:10.925","Text":"If I have a product of matrices,"},{"Start":"01:10.925 ","End":"01:13.550","Text":"all square and of the same size,"},{"Start":"01:13.550 ","End":"01:16.580","Text":"then the determinant of the product is"},{"Start":"01:16.580 ","End":"01:21.275","Text":"just the product of the determinants of each separate matrix."},{"Start":"01:21.275 ","End":"01:22.865","Text":"I gave 3,"},{"Start":"01:22.865 ","End":"01:24.680","Text":"A, B, C as an example,"},{"Start":"01:24.680 ","End":"01:29.330","Text":"as I said it works for 2 or for any other number."},{"Start":"01:29.330 ","End":"01:35.300","Text":"Now, rule number 4 is actually a special case of rule 3,"},{"Start":"01:35.300 ","End":"01:39.770","Text":"where I take B to equal A and C equals A,"},{"Start":"01:39.770 ","End":"01:42.410","Text":"and I just take n times A,"},{"Start":"01:42.410 ","End":"01:46.370","Text":"then I\u0027ll get that the determinant of A^n is"},{"Start":"01:46.370 ","End":"01:51.040","Text":"determinant of A times determinant of A n times which is this."},{"Start":"01:51.040 ","End":"01:54.350","Text":"If I combine rule 4 with rule 2,"},{"Start":"01:54.350 ","End":"01:57.080","Text":"then we also get this rule that the determinant of"},{"Start":"01:57.080 ","End":"02:02.760","Text":"A^minus n is 1 over the determinant of A^n."},{"Start":"02:02.870 ","End":"02:09.500","Text":"Now, rule 5 says that if I take a matrix A, a square matrix,"},{"Start":"02:09.500 ","End":"02:13.330","Text":"but specifically n-by-n,"},{"Start":"02:13.540 ","End":"02:18.815","Text":"and I multiply that square matrix by a constant k,"},{"Start":"02:18.815 ","End":"02:21.815","Text":"the determinant is not what you might think."},{"Start":"02:21.815 ","End":"02:24.745","Text":"It isn\u0027t just k times the determinant,"},{"Start":"02:24.745 ","End":"02:27.995","Text":"k has to be raised to the power of n,"},{"Start":"02:27.995 ","End":"02:29.899","Text":"where n is the dimension,"},{"Start":"02:29.899 ","End":"02:35.150","Text":"the number of rows or columns of the matrix."},{"Start":"02:35.150 ","End":"02:38.540","Text":"This rule, labeled 6,"},{"Start":"02:38.540 ","End":"02:43.150","Text":"says that the determinant of the identity matrix is 1."},{"Start":"02:43.150 ","End":"02:47.230","Text":"Although I should really generalize or explain more,"},{"Start":"02:47.230 ","End":"02:50.035","Text":"identity matrix could be of any size."},{"Start":"02:50.035 ","End":"02:52.910","Text":"Sometimes we say I with a n,"},{"Start":"02:52.910 ","End":"02:57.264","Text":"to mean the n-by-n identity matrix,"},{"Start":"02:57.264 ","End":"03:01.850","Text":"which means that it has 1, 1, 1,"},{"Start":"03:01.850 ","End":"03:05.230","Text":"1, and so on and everything else is 0,"},{"Start":"03:05.230 ","End":"03:13.480","Text":"but the size is n-by-n. That\u0027s the identity matrix of size n. In any event,"},{"Start":"03:13.480 ","End":"03:19.710","Text":"all of them are equal to 1 because for 1 thing they\u0027re triangular matrices,"},{"Start":"03:19.710 ","End":"03:25.130","Text":"all these identities and we can multiply the diagonal and productive 1s is just 1."},{"Start":"03:25.130 ","End":"03:33.970","Text":"That\u0027s that, and there\u0027s something called the adjoint."},{"Start":"03:36.470 ","End":"03:40.545","Text":"Actually there\u0027s more than 1 adjoint."},{"Start":"03:40.545 ","End":"03:45.285","Text":"This is called the classical adjoint,"},{"Start":"03:45.285 ","End":"03:48.960","Text":"and it actually has a newish name,"},{"Start":"03:48.960 ","End":"03:56.000","Text":"it\u0027s called the adjugate matrix."},{"Start":"03:56.000 ","End":"03:58.910","Text":"If I take the determinant of the,"},{"Start":"03:58.910 ","End":"04:01.370","Text":"I\u0027ll call it the adjugate,"},{"Start":"04:01.370 ","End":"04:08.930","Text":"then we get it from the determinant of A by raising it to the power of n minus 1."},{"Start":"04:08.930 ","End":"04:15.270","Text":"The end is assuming that A is an n-by-n square matrix,"},{"Start":"04:15.270 ","End":"04:17.370","Text":"so we have to raise it to this exponent."},{"Start":"04:17.370 ","End":"04:21.280","Text":"Note, you might not expect that."},{"Start":"04:21.280 ","End":"04:24.650","Text":"The last rule, I think of it,"},{"Start":"04:24.650 ","End":"04:27.660","Text":"it\u0027s a bit of a joke, it\u0027s like duh."},{"Start":"04:27.790 ","End":"04:30.875","Text":"I mean, if 2 things are equal,"},{"Start":"04:30.875 ","End":"04:35.090","Text":"then any property of equal things would be the same."},{"Start":"04:35.090 ","End":"04:40.605","Text":"I mean, the determinant of A if A is equal to B and just substitute B for A,"},{"Start":"04:40.605 ","End":"04:44.630","Text":"it goes without saying because as I said,"},{"Start":"04:44.630 ","End":"04:47.510","Text":"any mathematical property of A would be the"},{"Start":"04:47.510 ","End":"04:50.660","Text":"same as the mathematical property of B because they\u0027re the same."},{"Start":"04:50.660 ","End":"04:52.625","Text":"In particular they\u0027re determinants,"},{"Start":"04:52.625 ","End":"04:53.930","Text":"but even though it\u0027s obvious,"},{"Start":"04:53.930 ","End":"04:57.900","Text":"I would like to have it written so yeah."},{"Start":"04:57.900 ","End":"04:59.055","Text":"That\u0027s the last 1,"},{"Start":"04:59.055 ","End":"05:01.630","Text":"so we\u0027re done here."}],"ID":9907},{"Watched":false,"Name":"Exercise 1 Parts 1-2","Duration":"3m 15s","ChapterTopicVideoID":9570,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9570.jpeg","UploadDate":"2017-07-26T08:34:41.3370000","DurationForVideoObject":"PT3M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.670","Text":"In this exercise, we have 3 by 3 matrices, A and B."},{"Start":"00:07.970 ","End":"00:10.740","Text":"The determinant of A is 4,"},{"Start":"00:10.740 ","End":"00:13.005","Text":"the determinant of B is 2."},{"Start":"00:13.005 ","End":"00:19.200","Text":"Then we have to evaluate 4 different determinants based on A and B."},{"Start":"00:19.200 ","End":"00:22.410","Text":"The reason that 3 and 4 are grayed out is I\u0027m going"},{"Start":"00:22.410 ","End":"00:26.250","Text":"to do those in the next clip and I\u0027ll do 1 and 2 here."},{"Start":"00:26.250 ","End":"00:29.400","Text":"Just to summarize what it says here,"},{"Start":"00:29.400 ","End":"00:31.500","Text":"that\u0027s what we have here."},{"Start":"00:31.500 ","End":"00:34.230","Text":"This is the set theory notation,"},{"Start":"00:34.230 ","End":"00:37.290","Text":"and this is a set of 3 by 3 matrices."},{"Start":"00:37.290 ","End":"00:40.065","Text":"We\u0027re starting with Number 1,"},{"Start":"00:40.065 ","End":"00:42.700","Text":"and let\u0027s get some space."},{"Start":"00:43.700 ","End":"00:48.390","Text":"We have here a product of 4 different things."},{"Start":"00:48.390 ","End":"00:50.550","Text":"We have A, B,"},{"Start":"00:50.550 ","End":"00:53.080","Text":"A inverse, and B transpose."},{"Start":"00:53.080 ","End":"00:55.720","Text":"We can use the rule for products,"},{"Start":"00:55.720 ","End":"01:01.465","Text":"like so take the determined to be each of the 4 factors separately."},{"Start":"01:01.465 ","End":"01:06.775","Text":"Now, there is a rule for the determinant of an inverse,"},{"Start":"01:06.775 ","End":"01:11.230","Text":"and that is 1 over the determinant of the matrix itself,"},{"Start":"01:11.230 ","End":"01:15.245","Text":"and for transpose it doesn\u0027t change the determinant."},{"Start":"01:15.245 ","End":"01:17.525","Text":"Here we get the reciprocal,"},{"Start":"01:17.525 ","End":"01:20.350","Text":"and here we can just get rid of the transpose."},{"Start":"01:20.350 ","End":"01:23.540","Text":"Now stuff cancels out."},{"Start":"01:24.290 ","End":"01:29.475","Text":"Here, this determinant of A cancels with this in the denominator."},{"Start":"01:29.475 ","End":"01:32.855","Text":"Then we have just this twice,"},{"Start":"01:32.855 ","End":"01:35.735","Text":"which makes it the determinant of B squared,"},{"Start":"01:35.735 ","End":"01:38.150","Text":"but we have the determinant of B here,"},{"Start":"01:38.150 ","End":"01:41.925","Text":"it\u0027s 2, which gives us our answers,"},{"Start":"01:41.925 ","End":"01:44.040","Text":"2 squared or 4."},{"Start":"01:44.040 ","End":"01:51.770","Text":"Now let\u0027s get on to the second part of the question which is this."},{"Start":"01:51.770 ","End":"01:55.790","Text":"Now here, not only do we have products,"},{"Start":"01:55.790 ","End":"01:58.370","Text":"but we also have a constant."},{"Start":"01:58.370 ","End":"02:05.750","Text":"Here it\u0027s important to remember that our matrices are 3 by 3 because the rule for"},{"Start":"02:05.750 ","End":"02:09.410","Text":"a constant times the matrix is"},{"Start":"02:09.410 ","End":"02:13.880","Text":"that it comes out of the determinant raised to the power of n,"},{"Start":"02:13.880 ","End":"02:16.490","Text":"where here n is 3."},{"Start":"02:16.490 ","End":"02:20.075","Text":"Here is this 4, it\u0027s 4 cubed."},{"Start":"02:20.075 ","End":"02:23.240","Text":"Now we need the product after we take"},{"Start":"02:23.240 ","End":"02:27.125","Text":"determinant of A squared and the determinant of B cubed"},{"Start":"02:27.125 ","End":"02:29.960","Text":"but exponents are okay with determinants,"},{"Start":"02:29.960 ","End":"02:32.185","Text":"they just come right out."},{"Start":"02:32.185 ","End":"02:35.055","Text":"This 2 I push outside here,"},{"Start":"02:35.055 ","End":"02:36.510","Text":"and 3 out here,"},{"Start":"02:36.510 ","End":"02:41.245","Text":"and now we remember that we know the determinant of A and B."},{"Start":"02:41.245 ","End":"02:47.100","Text":"This 1 was 4, and this 1 was 2 so we now have this expression."},{"Start":"02:47.100 ","End":"02:49.985","Text":"Then a bit of algebra gives us this."},{"Start":"02:49.985 ","End":"02:51.845","Text":"If you want me to spell it out for you,"},{"Start":"02:51.845 ","End":"02:55.970","Text":"4 cubed is 2 squared cubed, so it\u0027s 2^6."},{"Start":"02:55.970 ","End":"02:58.520","Text":"This is 2 squared squared,"},{"Start":"02:58.520 ","End":"03:01.840","Text":"so it\u0027s 2^4 and then 2 cubed."},{"Start":"03:01.840 ","End":"03:07.350","Text":"Then all I have to do is 6 plus 4 plus 3 is 13."},{"Start":"03:07.350 ","End":"03:11.360","Text":"I think you can see why the answer is 2^13."},{"Start":"03:11.360 ","End":"03:15.030","Text":"We\u0027re going to continue in the next clip."}],"ID":9908},{"Watched":false,"Name":"Exercise 1 Parts 3-4","Duration":"3m 28s","ChapterTopicVideoID":9571,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9571.jpeg","UploadDate":"2017-07-26T08:34:50.4500000","DurationForVideoObject":"PT3M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"This is a continuation of the previous clip where did parts 1 and 2,"},{"Start":"00:04.860 ","End":"00:07.785","Text":"and now we\u0027re going to do parts 3 and 4."},{"Start":"00:07.785 ","End":"00:15.015","Text":"Part 3, we have to compute the determinant of this expression."},{"Start":"00:15.015 ","End":"00:17.415","Text":"Now, this minus,"},{"Start":"00:17.415 ","End":"00:25.755","Text":"it\u0027s really like a constant because it\u0027s like we had minus 1 times these 3."},{"Start":"00:25.755 ","End":"00:29.865","Text":"Because we\u0027re with 3-by-3 matrices,"},{"Start":"00:29.865 ","End":"00:32.400","Text":"that minus 1 comes out cubed,"},{"Start":"00:32.400 ","End":"00:34.710","Text":"here it is, and the rest of it,"},{"Start":"00:34.710 ","End":"00:37.415","Text":"we just use a product of determinants,"},{"Start":"00:37.415 ","End":"00:40.370","Text":"so I put each of these 3 pieces in bars,"},{"Start":"00:40.370 ","End":"00:43.955","Text":"and now we\u0027re going to use rules of exponents for this and this,"},{"Start":"00:43.955 ","End":"00:47.110","Text":"and rule for transpose here."},{"Start":"00:47.110 ","End":"00:53.195","Text":"The determinant of a transpose is just the same as for the original matrix."},{"Start":"00:53.195 ","End":"00:55.460","Text":"When we have an exponent,"},{"Start":"00:55.460 ","End":"00:57.560","Text":"it just comes right out,"},{"Start":"00:57.560 ","End":"00:59.600","Text":"and then if it\u0027s a negative exponent,"},{"Start":"00:59.600 ","End":"01:01.865","Text":"then we put this in the denominator."},{"Start":"01:01.865 ","End":"01:06.635","Text":"Let\u0027s go to the page of rules for determinants."},{"Start":"01:06.635 ","End":"01:09.830","Text":"You\u0027ll find all these rules."},{"Start":"01:09.830 ","End":"01:17.625","Text":"Now, stuff cancels out because this determinant of A squared,"},{"Start":"01:17.625 ","End":"01:18.870","Text":"and here it\u0027s cubed,"},{"Start":"01:18.870 ","End":"01:21.880","Text":"so I can just cross out the 3."},{"Start":"01:22.760 ","End":"01:28.685","Text":"Now I just have to remember that we know what the determinant of A and B are,"},{"Start":"01:28.685 ","End":"01:32.440","Text":"and also that minus 1 cubed is minus,"},{"Start":"01:32.440 ","End":"01:35.840","Text":"the determinant of B from here is 2,"},{"Start":"01:35.840 ","End":"01:39.840","Text":"the determinant of A from here is 4,"},{"Start":"01:40.250 ","End":"01:44.180","Text":"this is minus 1 times 2 times 4,"},{"Start":"01:44.180 ","End":"01:46.460","Text":"so minus 8,"},{"Start":"01:46.460 ","End":"01:49.340","Text":"and now we\u0027re on to part 4,"},{"Start":"01:49.340 ","End":"01:51.320","Text":"which is the last."},{"Start":"01:51.320 ","End":"01:55.580","Text":"This is how it was, I just copied it."},{"Start":"01:55.580 ","End":"01:59.720","Text":"You don\u0027t even have to remember what this adjoint means,"},{"Start":"01:59.720 ","End":"02:02.300","Text":"you just have to remember the rules for it."},{"Start":"02:02.300 ","End":"02:04.355","Text":"We\u0027ll get to in a moment."},{"Start":"02:04.355 ","End":"02:08.920","Text":"First of all, there\u0027s a constant minus 2."},{"Start":"02:08.920 ","End":"02:14.895","Text":"Remember we are working on 3-by-3 matrices,"},{"Start":"02:14.895 ","End":"02:18.720","Text":"so this minus 2 comes out cubed,"},{"Start":"02:18.720 ","End":"02:21.185","Text":"then I take the determinant of this bit,"},{"Start":"02:21.185 ","End":"02:23.390","Text":"then this bit, then this bit."},{"Start":"02:23.390 ","End":"02:25.370","Text":"Now we have an exponent,"},{"Start":"02:25.370 ","End":"02:27.635","Text":"a transpose, and an adjoint."},{"Start":"02:27.635 ","End":"02:30.830","Text":"Each of them has a different rule."},{"Start":"02:30.830 ","End":"02:33.500","Text":"For the power,"},{"Start":"02:33.500 ","End":"02:35.135","Text":"it just comes right out,"},{"Start":"02:35.135 ","End":"02:37.855","Text":"the transpose we can ignore,"},{"Start":"02:37.855 ","End":"02:42.350","Text":"that\u0027s the rule if the determinant of the transpose is the same as for the original,"},{"Start":"02:42.350 ","End":"02:43.970","Text":"and for the adjoint,"},{"Start":"02:43.970 ","End":"02:49.670","Text":"we have to raise the determinant of the original to the power of n minus 1,"},{"Start":"02:49.670 ","End":"02:51.380","Text":"where n is 3 here,"},{"Start":"02:51.380 ","End":"02:55.195","Text":"so it\u0027s 3 minus 1, which is 2."},{"Start":"02:55.195 ","End":"02:58.170","Text":"The determinant of A is 4,"},{"Start":"02:58.170 ","End":"02:59.820","Text":"if you look back,"},{"Start":"02:59.820 ","End":"03:03.244","Text":"squared and to the power of 1 is cubed,"},{"Start":"03:03.244 ","End":"03:05.450","Text":"the determinant of B was 2,"},{"Start":"03:05.450 ","End":"03:06.950","Text":"and 3 minus 1 is 2,"},{"Start":"03:06.950 ","End":"03:09.305","Text":"and minus 2 cubed is minus 8,"},{"Start":"03:09.305 ","End":"03:12.380","Text":"and if you multiply all this together,"},{"Start":"03:12.380 ","End":"03:15.830","Text":"this is what we get after a little bit of algebra."},{"Start":"03:15.830 ","End":"03:17.300","Text":"In case you\u0027re not sure,"},{"Start":"03:17.300 ","End":"03:18.830","Text":"just write 8 is 2 cubed,"},{"Start":"03:18.830 ","End":"03:20.270","Text":"and write 4 as 2 squared,"},{"Start":"03:20.270 ","End":"03:21.560","Text":"and use your exponents,"},{"Start":"03:21.560 ","End":"03:24.390","Text":"and you\u0027ll get this."},{"Start":"03:24.920 ","End":"03:28.930","Text":"That concludes this exercise."}],"ID":9909},{"Watched":false,"Name":"Exercise 2","Duration":"2m 45s","ChapterTopicVideoID":9572,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9572.jpeg","UploadDate":"2017-07-26T08:34:57.6270000","DurationForVideoObject":"PT2M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:12.010","Text":"In this exercise, we\u0027re given that a certain condition holds PQ^-1 APQ equals B."},{"Start":"00:12.770 ","End":"00:16.680","Text":"It goes without saying we\u0027re in this chapter on matrices"},{"Start":"00:16.680 ","End":"00:19.970","Text":"and determinants that all these A, B,"},{"Start":"00:19.970 ","End":"00:26.525","Text":"P, and Q are square matrices that the bars done for determinant."},{"Start":"00:26.525 ","End":"00:31.470","Text":"We have to prove that the determinant of A equals the determinant of B."},{"Start":"00:31.470 ","End":"00:35.870","Text":"Let me first copy what we know, which is this."},{"Start":"00:35.870 ","End":"00:41.895","Text":"Now I\u0027d like you to remember that when we take the inverse of a product,"},{"Start":"00:41.895 ","End":"00:44.730","Text":"this is the inverse of a product,"},{"Start":"00:44.730 ","End":"00:49.075","Text":"we have to take the inverse of each but also reverse the order."},{"Start":"00:49.075 ","End":"00:50.600","Text":"Be careful with that."},{"Start":"00:50.600 ","End":"00:52.370","Text":"Not only do we take the inverse of each,"},{"Start":"00:52.370 ","End":"00:54.185","Text":"but also reversed the order."},{"Start":"00:54.185 ","End":"00:58.040","Text":"It\u0027s easy to show that if PQ has an inverse,"},{"Start":"00:58.040 ","End":"01:02.030","Text":"then each of Q and P is also invertible."},{"Start":"01:02.030 ","End":"01:03.995","Text":"Well, I\u0027ll throw this as an extra."},{"Start":"01:03.995 ","End":"01:06.170","Text":"If PQ is invertible,"},{"Start":"01:06.170 ","End":"01:10.760","Text":"then the determinant of PQ is not equal to 0,"},{"Start":"01:10.760 ","End":"01:17.930","Text":"which means that the determinant of P times the determinant of Q is not equal to 0,"},{"Start":"01:17.930 ","End":"01:25.520","Text":"which implies that the determinant of P is not 0 and the determinant of Q is not 0."},{"Start":"01:25.520 ","End":"01:27.740","Text":"Because if a product is non-zero,"},{"Start":"01:27.740 ","End":"01:30.125","Text":"then each of the factors is non-zero,"},{"Start":"01:30.125 ","End":"01:32.860","Text":"so each of these now is invertible."},{"Start":"01:32.860 ","End":"01:37.010","Text":"We can put bars around both side."},{"Start":"01:37.010 ","End":"01:38.750","Text":"In other words, things are equal,"},{"Start":"01:38.750 ","End":"01:41.405","Text":"then their determinants are also equal."},{"Start":"01:41.405 ","End":"01:45.940","Text":"Now we can use the rule for the determinant of a product."},{"Start":"01:45.940 ","End":"01:49.700","Text":"It works for 5 factors and any number of factors,"},{"Start":"01:49.700 ","End":"01:51.995","Text":"just take the determinant of each piece."},{"Start":"01:51.995 ","End":"01:56.360","Text":"Now we have to use a rule that"},{"Start":"01:56.360 ","End":"02:02.145","Text":"the determinant of an inverse is the reciprocal of the determinant."},{"Start":"02:02.145 ","End":"02:05.930","Text":"Yeah, if I have an inverse of Q and I want the determinant,"},{"Start":"02:05.930 ","End":"02:09.860","Text":"I just take 1 over the reciprocal of the determinant of Q."},{"Start":"02:09.860 ","End":"02:11.440","Text":"The same for P,"},{"Start":"02:11.440 ","End":"02:14.000","Text":"I have an inverse and I take 1 over."},{"Start":"02:14.000 ","End":"02:15.730","Text":"The rest of it is the same."},{"Start":"02:15.730 ","End":"02:18.105","Text":"Now a lot of stuff cancels."},{"Start":"02:18.105 ","End":"02:20.795","Text":"This cancels with this,"},{"Start":"02:20.795 ","End":"02:28.310","Text":"and this P cancels with this P. We\u0027ve already talked about the fact that these are not 0."},{"Start":"02:28.310 ","End":"02:29.840","Text":"What are we left with?"},{"Start":"02:29.840 ","End":"02:36.490","Text":"That the determinant of A equals the determinant of B, just as required."},{"Start":"02:36.490 ","End":"02:41.090","Text":"Sometimes when we prove what we have to prove, you write QED."},{"Start":"02:41.090 ","End":"02:44.220","Text":"It\u0027s Latin to something."}],"ID":9910},{"Watched":false,"Name":"Exercise 3","Duration":"2m 38s","ChapterTopicVideoID":9573,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9573.jpeg","UploadDate":"2017-07-26T08:35:04.7100000","DurationForVideoObject":"PT2M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this exercise, we have 2 matrices, A and B,"},{"Start":"00:03.750 ","End":"00:06.480","Text":"and they\u0027re both invertible and of order 4,"},{"Start":"00:06.480 ","End":"00:09.495","Text":"meaning they\u0027re square 4 by 4 matrices."},{"Start":"00:09.495 ","End":"00:15.765","Text":"We\u0027re given 2 things that this equation holds."},{"Start":"00:15.765 ","End":"00:18.300","Text":"Twice A times B plus 3 times"},{"Start":"00:18.300 ","End":"00:23.130","Text":"the identity matrix is the 0 matrix,"},{"Start":"00:23.130 ","End":"00:27.380","Text":"and the determinant of A is 2."},{"Start":"00:27.380 ","End":"00:31.040","Text":"I guess I should have somehow marked the fact that this is the 0 matrix,"},{"Start":"00:31.040 ","End":"00:32.795","Text":"but it\u0027s clear, anyway."},{"Start":"00:32.795 ","End":"00:37.175","Text":"We have to compute the determinant of B."},{"Start":"00:37.175 ","End":"00:45.390","Text":"Here, I just rephrased what was given that A and B are 4 by 4 matrices,"},{"Start":"00:45.390 ","End":"00:47.990","Text":"they belong to the set of 4 by 4 matrices and"},{"Start":"00:47.990 ","End":"00:52.550","Text":"determinant of A is 2 and that this equation holds."},{"Start":"00:52.550 ","End":"00:55.595","Text":"We got all we need here,"},{"Start":"00:55.595 ","End":"01:00.170","Text":"and we have to compute the determinant of B."},{"Start":"01:00.170 ","End":"01:04.010","Text":"Now, we want to do something with this and there\u0027s no point taking"},{"Start":"01:04.010 ","End":"01:08.374","Text":"the determinant of both sides because the determinant of a sum,"},{"Start":"01:08.374 ","End":"01:10.145","Text":"there\u0027s no formula for that."},{"Start":"01:10.145 ","End":"01:18.590","Text":"But if we bring this to the other side and write it like this, that\u0027s more helpful."},{"Start":"01:18.590 ","End":"01:20.810","Text":"I is the identity matrix, well,"},{"Start":"01:20.810 ","End":"01:23.735","Text":"specifically it\u0027s the 4 by 4 identity matrix."},{"Start":"01:23.735 ","End":"01:29.945","Text":"Now, we have a matrix equal to the matrix and we can take the determinant of both sides."},{"Start":"01:29.945 ","End":"01:39.200","Text":"Now, on each side, we have the determinant of a constant times the matrix."},{"Start":"01:39.200 ","End":"01:47.190","Text":"The constant comes out with the power of the order, which is 4."},{"Start":"01:47.190 ","End":"01:48.990","Text":"When we pull the 2 out,"},{"Start":"01:48.990 ","End":"01:52.860","Text":"it\u0027s 2 to the fourth and the minus 3 is minus 3 to the fourth."},{"Start":"01:52.860 ","End":"01:54.765","Text":"I also did something else."},{"Start":"01:54.765 ","End":"01:56.490","Text":"The determinant of a product,"},{"Start":"01:56.490 ","End":"01:59.350","Text":"I broke it up as the product of the determinants,"},{"Start":"01:59.350 ","End":"02:02.345","Text":"so this is what we\u0027ve got so far."},{"Start":"02:02.345 ","End":"02:04.270","Text":"Now, there\u0027s things that we know."},{"Start":"02:04.270 ","End":"02:11.140","Text":"We know the determinant of A is 2 and the determinant of I is always 1,"},{"Start":"02:11.140 ","End":"02:13.360","Text":"so plug those values in."},{"Start":"02:13.360 ","End":"02:15.570","Text":"Here\u0027s the 1, here\u0027s the 2."},{"Start":"02:15.570 ","End":"02:18.975","Text":"Now, 2 to the fourth times 2 is 2 to the fifth,"},{"Start":"02:18.975 ","End":"02:24.145","Text":"and we just bring that to the other side,"},{"Start":"02:24.145 ","End":"02:26.335","Text":"and we have the determinant of B."},{"Start":"02:26.335 ","End":"02:28.240","Text":"There\u0027s the 2 to the fifth is 32,"},{"Start":"02:28.240 ","End":"02:29.710","Text":"it\u0027s on the denominator,"},{"Start":"02:29.710 ","End":"02:31.700","Text":"and negative 3 to the fourth,"},{"Start":"02:31.700 ","End":"02:36.650","Text":"it\u0027s even power, so it\u0027s a plus 81/32,"},{"Start":"02:36.650 ","End":"02:38.970","Text":"and we are done."}],"ID":9911},{"Watched":false,"Name":"Exercise 4","Duration":"4m 43s","ChapterTopicVideoID":9574,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9574.jpeg","UploadDate":"2017-07-26T08:35:18.4270000","DurationForVideoObject":"PT4M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this exercise,"},{"Start":"00:02.040 ","End":"00:03.870","Text":"we have 2 matrices, A and B."},{"Start":"00:03.870 ","End":"00:09.585","Text":"They\u0027re both invertible and they\u0027re both 3 by 3 matrices,"},{"Start":"00:09.585 ","End":"00:17.715","Text":"and we have a 2 equations relations that they must hold."},{"Start":"00:17.715 ","End":"00:20.100","Text":"A plus 3B is 0,"},{"Start":"00:20.100 ","End":"00:22.740","Text":"that\u0027s the 0 matrix."},{"Start":"00:22.740 ","End":"00:28.620","Text":"Also b squared minus twice the inverse of a is also 0."},{"Start":"00:28.620 ","End":"00:31.785","Text":"Again that\u0027s the 0 matrix not the number 0."},{"Start":"00:31.785 ","End":"00:36.805","Text":"We have to compute the determinant of each of them of A and of B."},{"Start":"00:36.805 ","End":"00:43.365","Text":"Here I just summarize what we were given that A and B are 3 by 3 matrices."},{"Start":"00:43.365 ","End":"00:45.990","Text":"Well, and invertible,"},{"Start":"00:45.990 ","End":"00:53.510","Text":"and we have 2 equations that hold this here and this one here."},{"Start":"00:53.510 ","End":"00:59.370","Text":"Now, what we can do is well,"},{"Start":"00:59.370 ","End":"01:00.640","Text":"split this up into 2."},{"Start":"01:00.640 ","End":"01:02.210","Text":"Bring this to the other side,"},{"Start":"01:02.210 ","End":"01:04.220","Text":"because we don\u0027t want to take a determinant of"},{"Start":"01:04.220 ","End":"01:06.530","Text":"a sum or a difference there\u0027s no rule for that."},{"Start":"01:06.530 ","End":"01:08.930","Text":"We can bring things to the other side."},{"Start":"01:08.930 ","End":"01:11.670","Text":"Then from here we got A is minus 3B,"},{"Start":"01:11.670 ","End":"01:17.820","Text":"and from here B squared is twice A to the minus 1."},{"Start":"01:17.820 ","End":"01:21.820","Text":"Now, we can apply determinants to both sides to these,"},{"Start":"01:21.820 ","End":"01:23.430","Text":"and we get this,"},{"Start":"01:23.430 ","End":"01:26.900","Text":"and now we can use some of the rules of determinants."},{"Start":"01:26.900 ","End":"01:31.040","Text":"For example, when we have a constant,"},{"Start":"01:31.040 ","End":"01:35.885","Text":"then it comes out to an exponent."},{"Start":"01:35.885 ","End":"01:38.870","Text":"This minus 3 is like a constant times B,"},{"Start":"01:38.870 ","End":"01:43.230","Text":"and this 2 is a constant times inverse of A"},{"Start":"01:43.230 ","End":"01:45.510","Text":"but because these are 3 by 3,"},{"Start":"01:45.510 ","End":"01:47.010","Text":"let me go back and show you."},{"Start":"01:47.010 ","End":"01:49.850","Text":"See they were 3 by 3 matrices,"},{"Start":"01:49.850 ","End":"01:52.870","Text":"then it comes out to the power of 3."},{"Start":"01:52.870 ","End":"01:59.600","Text":"Here\u0027s the 2 cubed and here\u0027s the minus 3 cube to the power of 3."},{"Start":"01:59.600 ","End":"02:05.665","Text":"Now, we can also apply the rule for an exponent and for an inverse."},{"Start":"02:05.665 ","End":"02:08.495","Text":"The first equation I left as is"},{"Start":"02:08.495 ","End":"02:09.740","Text":"but the second 1,"},{"Start":"02:09.740 ","End":"02:14.655","Text":"I put the power of 2 outside."},{"Start":"02:14.655 ","End":"02:17.285","Text":"I got some more space here,"},{"Start":"02:17.285 ","End":"02:23.210","Text":"and the inverse. Now let me go back."},{"Start":"02:23.210 ","End":"02:27.150","Text":"The inverse becomes 1 over."},{"Start":"02:27.380 ","End":"02:32.705","Text":"Let\u0027s continue. We have to find the determinant of A and the determinant of B."},{"Start":"02:32.705 ","End":"02:35.570","Text":"Now, these are just numbers."},{"Start":"02:35.570 ","End":"02:38.210","Text":"The fact that there\u0027s something in bars there,"},{"Start":"02:38.210 ","End":"02:42.245","Text":"it might be easier to follow if we just give them a single letter."},{"Start":"02:42.245 ","End":"02:46.550","Text":"That determinant of A is X and determinant of B is Y,"},{"Start":"02:46.550 ","End":"02:48.710","Text":"and then when we rewrite this in this form,"},{"Start":"02:48.710 ","End":"02:49.880","Text":"it looks a bit simpler."},{"Start":"02:49.880 ","End":"02:53.190","Text":"They don\u0027t have all these determinant bars in."},{"Start":"02:53.190 ","End":"02:55.895","Text":"Now we have 2 equations and 2 unknowns."},{"Start":"02:55.895 ","End":"03:04.360","Text":"What I suggest is take the X from here and substitute it in here,"},{"Start":"03:04.360 ","End":"03:09.450","Text":"and that gives us that Y squared is 8 over."},{"Start":"03:09.450 ","End":"03:16.280","Text":"We have the X and the denominator replaced by what\u0027s equal to it."},{"Start":"03:16.280 ","End":"03:23.935","Text":"Now I\u0027m going to multiply both sides by minus 27Y."},{"Start":"03:23.935 ","End":"03:27.600","Text":"Change of mind, just multiply by Y,"},{"Start":"03:27.600 ","End":"03:31.610","Text":"and so we have Y cubed equals this number."},{"Start":"03:31.610 ","End":"03:33.080","Text":"It\u0027s a negative number."},{"Start":"03:33.080 ","End":"03:35.360","Text":"We can take the cube root of both sides."},{"Start":"03:35.360 ","End":"03:38.210","Text":"Cube root of a negative is okay, it\u0027s negative,"},{"Start":"03:38.210 ","End":"03:39.530","Text":"and when we have a fraction,"},{"Start":"03:39.530 ","End":"03:43.375","Text":"we can take the cube root of numerator and denominator separately,"},{"Start":"03:43.375 ","End":"03:48.270","Text":"so Y is minus 2/3 and we also want X and"},{"Start":"03:48.270 ","End":"03:55.175","Text":"besides should remember that Y is just the determinant of B."},{"Start":"03:55.175 ","End":"03:59.035","Text":"Anyway, we can plug Y into,"},{"Start":"03:59.035 ","End":"04:08.565","Text":"say here and would get that X is minus 27 times minus 2/3,"},{"Start":"04:08.565 ","End":"04:12.545","Text":"and let\u0027s see, minus with minus is plus,"},{"Start":"04:12.545 ","End":"04:15.350","Text":"and 3 into 27 goes 9 times,"},{"Start":"04:15.350 ","End":"04:18.065","Text":"9 times 2 is 18,"},{"Start":"04:18.065 ","End":"04:22.660","Text":"and don\u0027t forget that X is the determinant of A."},{"Start":"04:22.660 ","End":"04:26.840","Text":"We found everything, the determinant of B and the determinant of"},{"Start":"04:26.840 ","End":"04:31.355","Text":"A. I don\u0027t know why I just felt like putting it in the box anyway,"},{"Start":"04:31.355 ","End":"04:33.425","Text":"so to highlight the answer,"},{"Start":"04:33.425 ","End":"04:43.080","Text":"determinant of B is minus 2/3 and the determinant of A is 18. We\u0027re done."}],"ID":9912},{"Watched":false,"Name":"Exercise 5","Duration":"4m 3s","ChapterTopicVideoID":9566,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9566.jpeg","UploadDate":"2017-07-26T08:34:09.7270000","DurationForVideoObject":"PT4M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.775","Text":"This exercise is a bit different."},{"Start":"00:02.775 ","End":"00:06.510","Text":"We\u0027re given actually 2 of the rules"},{"Start":"00:06.510 ","End":"00:11.670","Text":"for properties of determinants and we have to prove them."},{"Start":"00:11.670 ","End":"00:15.000","Text":"Obviously we can\u0027t use them to prove themselves,"},{"Start":"00:15.000 ","End":"00:19.860","Text":"but we can prove them using the other rules of determinants."},{"Start":"00:19.860 ","End":"00:22.470","Text":"We\u0027ll start with the first 1,"},{"Start":"00:22.470 ","End":"00:24.240","Text":"which is about the inverse."},{"Start":"00:24.240 ","End":"00:27.550","Text":"The determinant is the reciprocal."},{"Start":"00:27.920 ","End":"00:35.520","Text":"I\u0027ll start from this identity that when you take any matrix and multiply by its inverse,"},{"Start":"00:35.520 ","End":"00:38.520","Text":"then you get the identity matrix."},{"Start":"00:38.520 ","End":"00:42.380","Text":"Now let\u0027s take the determinant of both sides,"},{"Start":"00:42.380 ","End":"00:49.960","Text":"like so now we\u0027ll use the property of the product and we will also use the property that"},{"Start":"00:49.960 ","End":"00:58.130","Text":"the determinant of the identity matrix, I, its 1."},{"Start":"00:58.130 ","End":"01:01.420","Text":"We get that the product of the determinant"},{"Start":"01:01.420 ","End":"01:04.555","Text":"of a times the determinant of the inverse is 1."},{"Start":"01:04.555 ","End":"01:08.080","Text":"All you have to do now is divide by the determinant of A."},{"Start":"01:08.080 ","End":"01:13.640","Text":"Obviously it\u0027s not 0 because 0 times something can\u0027t be 1."},{"Start":"01:14.060 ","End":"01:18.130","Text":"This is what we get to choose what we had to prove."},{"Start":"01:18.130 ","End":"01:23.365","Text":"Now let\u0027s move on to part B and hopefully,"},{"Start":"01:23.365 ","End":"01:27.070","Text":"you know what adj means."},{"Start":"01:27.070 ","End":"01:30.250","Text":"It\u0027s the adjoint, but actually more than 1 adjoint,"},{"Start":"01:30.250 ","End":"01:35.165","Text":"this is called the classical adjoint."},{"Start":"01:35.165 ","End":"01:37.075","Text":"There are other words for it."},{"Start":"01:37.075 ","End":"01:45.885","Text":"It\u0027s sometimes called the adjugate like conjugate but adjugate."},{"Start":"01:45.885 ","End":"01:48.880","Text":"See if I spelled that right."},{"Start":"01:49.610 ","End":"01:51.960","Text":"Note that in part b,"},{"Start":"01:51.960 ","End":"01:58.420","Text":"it depends on the size of the order of the matrix that it\u0027s n by n,"},{"Start":"01:58.420 ","End":"02:01.870","Text":"the order is n, and here we\u0027ll have an n minus 1."},{"Start":"02:01.870 ","End":"02:08.450","Text":"Anyway, so let\u0027s move on to part B."},{"Start":"02:08.450 ","End":"02:15.005","Text":"Here I\u0027ve written the main property of this classical adjoint."},{"Start":"02:15.005 ","End":"02:21.290","Text":"If you multiply a matrix by its adjoint on either side,"},{"Start":"02:21.290 ","End":"02:23.285","Text":"on the left or the right it doesn\u0027t matter,"},{"Start":"02:23.285 ","End":"02:30.860","Text":"you get the determinant of a times the identity matrix I."},{"Start":"02:30.860 ","End":"02:37.715","Text":"This would be the n by n identity matrix."},{"Start":"02:37.715 ","End":"02:43.220","Text":"Now, just think of this determinant of a,"},{"Start":"02:43.220 ","End":"02:47.060","Text":"like a constant because we\u0027re going to take the determinant of"},{"Start":"02:47.060 ","End":"02:52.200","Text":"both sides and might be confusing to have a determinant within a determinant."},{"Start":"02:52.370 ","End":"03:01.939","Text":"Let me just highlight this part and think of this as being like a constant k,"},{"Start":"03:01.939 ","End":"03:06.320","Text":"like we had in the rule that the determinant of"},{"Start":"03:06.320 ","End":"03:11.765","Text":"k times a matrix is k to the n times the determinant."},{"Start":"03:11.765 ","End":"03:13.670","Text":"That explains the right-hand side."},{"Start":"03:13.670 ","End":"03:15.620","Text":"When I take this outside the brackets,"},{"Start":"03:15.620 ","End":"03:18.395","Text":"forget the fact that the determinant is just a number."},{"Start":"03:18.395 ","End":"03:21.110","Text":"It\u0027s that number to the power of n,"},{"Start":"03:21.110 ","End":"03:24.065","Text":"k to the n times the determinant of this."},{"Start":"03:24.065 ","End":"03:28.205","Text":"Here I just split it up as a product."},{"Start":"03:28.205 ","End":"03:30.950","Text":"Now I can do a couple of things."},{"Start":"03:30.950 ","End":"03:34.205","Text":"The determinant of I is just 1."},{"Start":"03:34.205 ","End":"03:37.110","Text":"I can just throw it out."},{"Start":"03:37.110 ","End":"03:38.870","Text":"Here I have a to the n,"},{"Start":"03:38.870 ","End":"03:41.480","Text":"and here I have a to the 1 rather than"},{"Start":"03:41.480 ","End":"03:45.920","Text":"determinant to the n. If I bring this to the other side,"},{"Start":"03:45.920 ","End":"03:49.330","Text":"it will become the power n minus 1"},{"Start":"03:49.330 ","End":"03:54.455","Text":"and in short we\u0027re left with this and this is what we had to prove."},{"Start":"03:54.455 ","End":"03:59.960","Text":"We\u0027re done with this part as we were with part A,"},{"Start":"03:59.960 ","End":"04:03.180","Text":"and so we\u0027re finished with this exercise."}],"ID":9913},{"Watched":false,"Name":"Exercise 6","Duration":"2m 22s","ChapterTopicVideoID":9567,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9567.jpeg","UploadDate":"2017-07-26T08:34:15.7100000","DurationForVideoObject":"PT2M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this exercise, we\u0027re given a matrix A,"},{"Start":"00:04.110 ","End":"00:09.030","Text":"a square matrix with odd order."},{"Start":"00:09.030 ","End":"00:12.840","Text":"Meaning it\u0027s like 3 by 3, 5 by 5,"},{"Start":"00:12.840 ","End":"00:14.550","Text":"or 7 by 7,"},{"Start":"00:14.550 ","End":"00:17.085","Text":"some odd number order."},{"Start":"00:17.085 ","End":"00:20.530","Text":"We have to show that its determinant is"},{"Start":"00:20.530 ","End":"00:26.955","Text":"0 and I\u0027ll start by reminding you what anti-symmetric means."},{"Start":"00:26.955 ","End":"00:33.385","Text":"The word anti-symmetric is actually anti-symmetric."},{"Start":"00:33.385 ","End":"00:40.235","Text":"If it\u0027s symmetric, then A is equal to its transpose,"},{"Start":"00:40.235 ","End":"00:45.470","Text":"but when there\u0027s a minus also, then it\u0027s anti-symmetric."},{"Start":"00:45.470 ","End":"00:49.365","Text":"The minus here is what makes it the anti."},{"Start":"00:49.365 ","End":"00:50.750","Text":"If it didn\u0027t have the minus,"},{"Start":"00:50.750 ","End":"00:52.355","Text":"it would just be symmetric."},{"Start":"00:52.355 ","End":"00:56.880","Text":"Here we\u0027re reminded that M is an n by n matrix,"},{"Start":"00:56.880 ","End":"01:02.355","Text":"meaning it\u0027s of order n. Which is odd, n is odd."},{"Start":"01:02.355 ","End":"01:06.725","Text":"Then we start by taking the determinant of both sides and of course,"},{"Start":"01:06.725 ","End":"01:13.355","Text":"minus a matrix is the same as minus 1 times that matrix."},{"Start":"01:13.355 ","End":"01:15.020","Text":"You can tell where I\u0027m heading."},{"Start":"01:15.020 ","End":"01:18.065","Text":"I want this constant to come out so I do the determinant,"},{"Start":"01:18.065 ","End":"01:23.210","Text":"but I have to raise it to the power of n. We have that"},{"Start":"01:23.210 ","End":"01:29.165","Text":"the determinant of A is minus 1 to the n times the determinant of A transpose,"},{"Start":"01:29.165 ","End":"01:34.070","Text":"but remember there\u0027s a rule that the determinant of A transpose is the same as"},{"Start":"01:34.070 ","End":"01:40.400","Text":"the determinant of A and so I can replace this by what\u0027s equal to it."},{"Start":"01:40.400 ","End":"01:47.390","Text":"Now we are going to figure that n is odd and when n is odd,"},{"Start":"01:47.390 ","End":"01:50.065","Text":"we know what minus 1 to the n is."},{"Start":"01:50.065 ","End":"01:53.060","Text":"Minus 1 to the odd power is minus 1,"},{"Start":"01:53.060 ","End":"01:54.905","Text":"which I can write just as a minus."},{"Start":"01:54.905 ","End":"01:58.550","Text":"It was crucial that we took n is odd because if n was even,"},{"Start":"01:58.550 ","End":"02:01.925","Text":"you wouldn\u0027t get the minus and wouldn\u0027t be able to proceed."},{"Start":"02:01.925 ","End":"02:07.610","Text":"Now, the easiest thing to do is just to or 1 way to go is to bring this to"},{"Start":"02:07.610 ","End":"02:12.290","Text":"the other side and so we get this determinant plus itself"},{"Start":"02:12.290 ","End":"02:17.795","Text":"and it was twice the determinant of A is 0 and if twice something is 0,"},{"Start":"02:17.795 ","End":"02:23.140","Text":"then it itself is 0 and that\u0027s what we had to show, so we\u0027re done.1"}],"ID":9914},{"Watched":false,"Name":"Exercise 7","Duration":"2m 57s","ChapterTopicVideoID":9568,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9568.jpeg","UploadDate":"2017-07-26T08:34:24.7970000","DurationForVideoObject":"PT2M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.399","Text":"In this exercise, we\u0027re given several pieces of information about two matrices,"},{"Start":"00:05.399 ","End":"00:08.970","Text":"A and B, that they are of order n, or if you like,"},{"Start":"00:08.970 ","End":"00:12.345","Text":"they are square matrices of order n by n,"},{"Start":"00:12.345 ","End":"00:16.770","Text":"size n by n. We\u0027re given that B is invertible,"},{"Start":"00:16.770 ","End":"00:22.380","Text":"we\u0027re given that the determinant of A is 128,"},{"Start":"00:22.380 ","End":"00:31.255","Text":"and we\u0027re given the matrix equation that twice A times B is B transpose times A squared."},{"Start":"00:31.255 ","End":"00:37.880","Text":"From all this, we have to deduce or to find what n is."},{"Start":"00:37.880 ","End":"00:40.760","Text":"Here\u0027s how we go about it."},{"Start":"00:40.760 ","End":"00:47.015","Text":"I just wrote again the important bits that the determinant of A is 128."},{"Start":"00:47.015 ","End":"00:52.940","Text":"It\u0027s of order n, meaning it\u0027s a square matrix n by n. Now,"},{"Start":"00:52.940 ","End":"00:58.855","Text":"let\u0027s take the determinant of both sides of this."},{"Start":"00:58.855 ","End":"01:03.875","Text":"Well, I copied it first and now apply the bars to each side."},{"Start":"01:03.875 ","End":"01:07.320","Text":"Now let\u0027s use the properties of determinants."},{"Start":"01:08.030 ","End":"01:11.240","Text":"Well, there\u0027s two things we can use."},{"Start":"01:11.240 ","End":"01:14.015","Text":"We can use the property that the product,"},{"Start":"01:14.015 ","End":"01:17.389","Text":"when you take a determinant it\u0027s just the product of the determinants."},{"Start":"01:17.389 ","End":"01:18.950","Text":"When you have a constant,"},{"Start":"01:18.950 ","End":"01:22.460","Text":"it comes out to the power of n,"},{"Start":"01:22.460 ","End":"01:24.280","Text":"which is the order."},{"Start":"01:24.280 ","End":"01:26.370","Text":"That explains the left side,"},{"Start":"01:26.370 ","End":"01:27.540","Text":"and then the right-hand side,"},{"Start":"01:27.540 ","End":"01:31.680","Text":"just the product of the determinants."},{"Start":"01:31.680 ","End":"01:33.965","Text":"Now we\u0027re going to use some other rules."},{"Start":"01:33.965 ","End":"01:37.180","Text":"That when we have"},{"Start":"01:37.180 ","End":"01:44.270","Text":"the determinant of a transpose is the same as the thing itself."},{"Start":"01:44.270 ","End":"01:46.040","Text":"Also when we have an exponent,"},{"Start":"01:46.040 ","End":"01:48.275","Text":"we can take the exponent out."},{"Start":"01:48.275 ","End":"01:53.960","Text":"Here\u0027s where I\u0027ve dropped the T and here\u0027s where I\u0027ve taken the two outside the bars."},{"Start":"01:53.960 ","End":"01:59.885","Text":"Now you might recall that we said in the beginning that B is invertible."},{"Start":"01:59.885 ","End":"02:02.690","Text":"Why do you think they told us that?"},{"Start":"02:02.690 ","End":"02:05.540","Text":"Now we can find the reason if B is invertible,"},{"Start":"02:05.540 ","End":"02:07.220","Text":"its determinant is not 0."},{"Start":"02:07.220 ","End":"02:10.595","Text":"Then we can cancel both sides."},{"Start":"02:10.595 ","End":"02:16.610","Text":"Not only that, but we can also bring the determinant of A to the other side,"},{"Start":"02:16.610 ","End":"02:18.680","Text":"so it\u0027s not squared."},{"Start":"02:18.680 ","End":"02:25.160","Text":"Just get 2 to the n equals the determinant of A. I think we can still see it here."},{"Start":"02:25.160 ","End":"02:28.330","Text":"Here it is that\u0027s 128."},{"Start":"02:28.330 ","End":"02:34.390","Text":"We\u0027re looking for a number n such that 2 to the power of it is 128."},{"Start":"02:34.390 ","End":"02:38.465","Text":"Either use experience or trial and error,"},{"Start":"02:38.465 ","End":"02:40.595","Text":"2 squared, 2 cubed, 2 to the 4th."},{"Start":"02:40.595 ","End":"02:43.205","Text":"When you hit 2 to the 7th, bingo,"},{"Start":"02:43.205 ","End":"02:46.760","Text":"it\u0027s 128 and so n equals 7."},{"Start":"02:46.760 ","End":"02:49.310","Text":"Of course, we could have also done it with logarithms and"},{"Start":"02:49.310 ","End":"02:52.445","Text":"said that n log to the base 2 of 128."},{"Start":"02:52.445 ","End":"02:56.910","Text":"I don\u0027t know if that\u0027s any easier. That\u0027s it."}],"ID":9915},{"Watched":false,"Name":"Exercise 8","Duration":"1m 51s","ChapterTopicVideoID":9569,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9569.jpeg","UploadDate":"2017-07-26T08:34:29.0330000","DurationForVideoObject":"PT1M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.760","Text":"This exercise is just a straight forward exercise in the properties of determinants."},{"Start":"00:05.760 ","End":"00:10.155","Text":"We were given 2 matrices, A and B."},{"Start":"00:10.155 ","End":"00:20.825","Text":"They\u0027re both of size n by n. The determinant of A is 2 and the determinant of B is 1/3."},{"Start":"00:20.825 ","End":"00:24.330","Text":"We have to compute the determinant of this expression,"},{"Start":"00:24.330 ","End":"00:25.890","Text":"which has a constant,"},{"Start":"00:25.890 ","End":"00:28.830","Text":"it has exponents, positive and negative."},{"Start":"00:28.830 ","End":"00:32.065","Text":"We\u0027ll just use the rules to figure this out."},{"Start":"00:32.065 ","End":"00:36.860","Text":"Here I just copied the data so that I can scroll down."},{"Start":"00:36.860 ","End":"00:42.095","Text":"I left too much space here, probably never mind."},{"Start":"00:42.095 ","End":"00:45.425","Text":"Now for this determinant, I\u0027m going to break it up."},{"Start":"00:45.425 ","End":"00:52.520","Text":"The constant comes out to the power of n. Then here I have a determinant of the product,"},{"Start":"00:52.520 ","End":"00:56.985","Text":"so I break it up into separate determinants."},{"Start":"00:56.985 ","End":"01:00.550","Text":"Then I can use exponent rules."},{"Start":"01:00.550 ","End":"01:02.580","Text":"For positive exponent,"},{"Start":"01:02.580 ","End":"01:04.770","Text":"we just put it outside the bars."},{"Start":"01:04.770 ","End":"01:05.980","Text":"When we have a negative,"},{"Start":"01:05.980 ","End":"01:09.210","Text":"we also put it in the denominator."},{"Start":"01:10.250 ","End":"01:15.850","Text":"We\u0027ve just about still see on screen that the determinant of A is 2,"},{"Start":"01:15.850 ","End":"01:17.155","Text":"so that will go there."},{"Start":"01:17.155 ","End":"01:20.345","Text":"B is 1/3 will go there."},{"Start":"01:20.345 ","End":"01:22.985","Text":"We get this expression,"},{"Start":"01:22.985 ","End":"01:26.990","Text":"but we can do a bit of counseling because look"},{"Start":"01:26.990 ","End":"01:34.000","Text":"the 1/3 to the n in the numerator cancels with the one in the denominator."},{"Start":"01:34.000 ","End":"01:38.170","Text":"We could have left the answer as 2^2n,"},{"Start":"01:38.170 ","End":"01:42.015","Text":"but 2^2 is 4."},{"Start":"01:42.015 ","End":"01:45.345","Text":"We could also write it as 4^n,"},{"Start":"01:45.345 ","End":"01:51.820","Text":"whichever of these 2 forms you think is simpler. We\u0027re done."}],"ID":9916},{"Watched":false,"Name":"Exercise 9","Duration":"11m 50s","ChapterTopicVideoID":28074,"CourseChapterTopicPlaylistID":7287,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/28074.jpeg","UploadDate":"2021-12-20T08:25:11.7900000","DurationForVideoObject":"PT11M50S","Description":null,"VideoComments":[],"Subtitles":[],"ID":29273}],"Thumbnail":null,"ID":7287},{"Name":"Cramer\u0027s Rule","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Cramer\u0027s Rule","Duration":"10m 42s","ChapterTopicVideoID":9575,"CourseChapterTopicPlaylistID":7288,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9575.jpeg","UploadDate":"2019-02-18T06:06:25.8770000","DurationForVideoObject":"PT10M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"In this clip, I\u0027ll be talking about an application of"},{"Start":"00:03.180 ","End":"00:07.500","Text":"determinants to solving linear systems of"},{"Start":"00:07.500 ","End":"00:11.730","Text":"several equations in several unknowns"},{"Start":"00:11.730 ","End":"00:17.295","Text":"like 2 equations and 2 unknowns and then later 3 equations and 3 unknowns and so on."},{"Start":"00:17.295 ","End":"00:23.550","Text":"But this will only work when the system has a unique solution."},{"Start":"00:23.550 ","End":"00:28.020","Text":"Let me get started and then we\u0027ll see what this all means."},{"Start":"00:28.020 ","End":"00:32.175","Text":"By the way, I thought it wouldn\u0027t hurt this to put a picture of him."},{"Start":"00:32.175 ","End":"00:34.290","Text":"It\u0027s Gabriel Cramer."},{"Start":"00:34.290 ","End":"00:39.375","Text":"Anyway. Let\u0027s take the 2-by-2 case."},{"Start":"00:39.375 ","End":"00:43.835","Text":"I mean 2 equations in 2 unknowns and then we\u0027ll work our way up."},{"Start":"00:43.835 ","End":"00:46.430","Text":"This is what the general equation looks like,"},{"Start":"00:46.430 ","End":"00:51.965","Text":"where x and y are the unknowns and the other letters are numbers."},{"Start":"00:51.965 ","End":"00:53.990","Text":"The a\u0027s and the b\u0027s are coefficients,"},{"Start":"00:53.990 ","End":"00:55.700","Text":"and these are just constants,"},{"Start":"00:55.700 ","End":"00:59.015","Text":"and you\u0027ll see why I put them in another color."},{"Start":"00:59.015 ","End":"01:02.690","Text":"Now I want to say again that this whole theory is"},{"Start":"01:02.690 ","End":"01:05.990","Text":"going to work when the system has a unique solution."},{"Start":"01:05.990 ","End":"01:09.715","Text":"Because in general, we could have no solution,"},{"Start":"01:09.715 ","End":"01:12.705","Text":"1 solution or infinite solutions."},{"Start":"01:12.705 ","End":"01:17.085","Text":"This only works when it has just 1 solution."},{"Start":"01:17.085 ","End":"01:23.570","Text":"There is a simple condition for this to hold is that the coefficient matrix,"},{"Start":"01:23.570 ","End":"01:30.290","Text":"I mean the matrix with just the a\u0027s and the b\u0027s is non-0."},{"Start":"01:30.290 ","End":"01:35.420","Text":"This is also called the restricted matrix."},{"Start":"01:35.420 ","End":"01:40.070","Text":"The reason it\u0027s called the restricted matrix is if I throw in k_1 and k_2,"},{"Start":"01:40.070 ","End":"01:42.380","Text":"that\u0027s the augmented matrix."},{"Start":"01:42.380 ","End":"01:43.940","Text":"Augmented meaning increased."},{"Start":"01:43.940 ","End":"01:47.880","Text":"Anyway. We\u0027re going to assume this condition."},{"Start":"01:48.080 ","End":"01:54.050","Text":"What Cramer discovered were exact expressions,"},{"Start":"01:54.050 ","End":"02:00.440","Text":"closed form expressions for the answers x and y. I\u0027ll give them to you."},{"Start":"02:00.440 ","End":"02:05.255","Text":"Here they are. Each of x and y is just a computation,"},{"Start":"02:05.255 ","End":"02:08.000","Text":"a determinant divided by a determinant."},{"Start":"02:08.000 ","End":"02:12.320","Text":"There\u0027s no procedure to follow, no cookbook algorithm,"},{"Start":"02:12.320 ","End":"02:16.250","Text":"just an exact formula for the answer for x and"},{"Start":"02:16.250 ","End":"02:22.414","Text":"y. I\u0027m saying for a third time on the condition that the system has a unique solution."},{"Start":"02:22.414 ","End":"02:25.820","Text":"Now let\u0027s look at this more closely and see what it is."},{"Start":"02:25.820 ","End":"02:27.980","Text":"If you look at these 2 denominators,"},{"Start":"02:27.980 ","End":"02:33.755","Text":"they\u0027re the same and they are this coefficient matrix or restricted matrix,"},{"Start":"02:33.755 ","End":"02:36.020","Text":"which we already know is non-0."},{"Start":"02:36.020 ","End":"02:38.645","Text":"We can put it on the denominator."},{"Start":"02:38.645 ","End":"02:47.120","Text":"Now, the numerator, what we do is it starts off like the denominator,"},{"Start":"02:47.120 ","End":"02:54.375","Text":"except that we replace the first column here by these numbers k_1,"},{"Start":"02:54.375 ","End":"02:57.155","Text":"k_2, and that will give us x."},{"Start":"02:57.155 ","End":"02:59.720","Text":"But if we replace the second column,"},{"Start":"02:59.720 ","End":"03:02.990","Text":"the a\u0027s from here and k\u0027s from here,"},{"Start":"03:02.990 ","End":"03:04.820","Text":"then it gives us y."},{"Start":"03:04.820 ","End":"03:09.915","Text":"In each case, all but 1 column is from here,"},{"Start":"03:09.915 ","End":"03:13.985","Text":"several but 1 because in the 3-by-3 in higher case,"},{"Start":"03:13.985 ","End":"03:17.090","Text":"all the columns will be the same as below,"},{"Start":"03:17.090 ","End":"03:21.900","Text":"except for a single column which will be in blue, so to speak."},{"Start":"03:23.120 ","End":"03:29.640","Text":"Let\u0027s see an example and here\u0027s our example."},{"Start":"03:29.640 ","End":"03:35.500","Text":"Let me just keep the formula so it doesn\u0027t disappear."},{"Start":"03:37.220 ","End":"03:40.140","Text":"This is the system,"},{"Start":"03:40.140 ","End":"03:49.560","Text":"and this is a_1 and a_2 and b_1 and b_2 and k_1 and k_2."},{"Start":"03:49.560 ","End":"03:51.720","Text":"This is what I do."},{"Start":"03:51.720 ","End":"03:54.195","Text":"I take fraction,"},{"Start":"03:54.195 ","End":"03:55.410","Text":"a determinant on the top,"},{"Start":"03:55.410 ","End":"03:57.035","Text":"a determinant on the bottom."},{"Start":"03:57.035 ","End":"04:02.060","Text":"On the bottom I put 5 minus 2 to 3 in each of them."},{"Start":"04:02.060 ","End":"04:05.315","Text":"That\u0027s only 1 computation, it appears twice."},{"Start":"04:05.315 ","End":"04:12.260","Text":"Now, for x, I copy the column from the denominator,"},{"Start":"04:12.260 ","End":"04:14.375","Text":"but on the numerator,"},{"Start":"04:14.375 ","End":"04:18.680","Text":"I put the minus 1, 3 from here."},{"Start":"04:18.680 ","End":"04:22.180","Text":"Similarly, for the y,"},{"Start":"04:22.180 ","End":"04:25.230","Text":"the denominator is, yeah,"},{"Start":"04:25.230 ","End":"04:28.355","Text":"I mean the determinant of this."},{"Start":"04:28.355 ","End":"04:29.990","Text":"Once again, the same as this."},{"Start":"04:29.990 ","End":"04:36.030","Text":"But here it\u0027s the second column that is replaced by the minus 1,"},{"Start":"04:36.030 ","End":"04:37.985","Text":"3 and this is copied."},{"Start":"04:37.985 ","End":"04:42.440","Text":"Now we have 3 determinants to compute. This 1."},{"Start":"04:42.440 ","End":"04:49.110","Text":"You can do it mentally as 15 takeaway minus 4 gives us 19."},{"Start":"04:49.360 ","End":"04:52.475","Text":"Here we have 19."},{"Start":"04:52.475 ","End":"04:55.640","Text":"Now I just have 2 more to compute."},{"Start":"04:55.640 ","End":"04:59.450","Text":"I\u0027ll just tell you what they are because I"},{"Start":"04:59.450 ","End":"05:03.575","Text":"don\u0027t want to waste time with the computations."},{"Start":"05:03.575 ","End":"05:08.945","Text":"This is how we get the answer for a system of 2 by 2,"},{"Start":"05:08.945 ","End":"05:12.350","Text":"2 equations and 2 unknowns linear."},{"Start":"05:12.350 ","End":"05:17.315","Text":"Let\u0027s move on to 3-by-3."},{"Start":"05:17.315 ","End":"05:22.505","Text":"You see this is fairly straightforward and easy to compute."},{"Start":"05:22.505 ","End":"05:26.045","Text":"It gets a bit more difficult as we go with"},{"Start":"05:26.045 ","End":"05:28.640","Text":"the higher dimensions because the determinant become more"},{"Start":"05:28.640 ","End":"05:32.940","Text":"difficult anyway let\u0027s jump to the 3-by-3."},{"Start":"05:33.530 ","End":"05:36.180","Text":"In the 3-by-3 case,"},{"Start":"05:36.180 ","End":"05:37.755","Text":"the 3 equations, 3 unknowns,"},{"Start":"05:37.755 ","End":"05:40.610","Text":"I\u0027m going to repeat the warning that this whole system"},{"Start":"05:40.610 ","End":"05:43.790","Text":"only works when there\u0027s a unique solution."},{"Start":"05:43.790 ","End":"05:49.105","Text":"This is how a system looks like with the variables x, y, and z."},{"Start":"05:49.105 ","End":"05:54.530","Text":"We have a similar condition that we know when this thing has a unique solution,"},{"Start":"05:54.530 ","End":"06:00.260","Text":"it\u0027s if and only if the determinant of this matrix is non-0."},{"Start":"06:00.260 ","End":"06:05.030","Text":"The determinant of the 3-by-3 matrix that we get"},{"Start":"06:05.030 ","End":"06:09.305","Text":"from here is called the co-efficient matrix or the restricted matrix,"},{"Start":"06:09.305 ","End":"06:12.780","Text":"and that has to be non-0."},{"Start":"06:12.780 ","End":"06:21.680","Text":"If this holds, then Cramer\u0027s rule for the 3-by-3 situation looks like this."},{"Start":"06:21.680 ","End":"06:24.860","Text":"I\u0027ll go into some more details."},{"Start":"06:24.860 ","End":"06:28.390","Text":"Let me just, yeah."},{"Start":"06:28.390 ","End":"06:33.740","Text":"Now once again, you notice that the 3 denominator you just saw the same and"},{"Start":"06:33.740 ","End":"06:40.505","Text":"this denominator is exactly this coefficient matrix,"},{"Start":"06:40.505 ","End":"06:44.920","Text":"the determinant of it, which is non-0."},{"Start":"06:44.920 ","End":"06:47.750","Text":"There\u0027s no problem with dividing by 0."},{"Start":"06:47.750 ","End":"06:50.510","Text":"Now the pattern I told you about before also"},{"Start":"06:50.510 ","End":"06:55.340","Text":"works because if we want the x actually like the first variable,"},{"Start":"06:55.340 ","End":"07:01.020","Text":"then we replaced the first column here by the k_1, k_2, k_3."},{"Start":"07:01.020 ","End":"07:05.805","Text":"Maybe I\u0027ll just show you again the k1 and fit,"},{"Start":"07:05.805 ","End":"07:08.555","Text":"just about fit everything on here."},{"Start":"07:08.555 ","End":"07:11.495","Text":"That\u0027s the k_1, k_2, k_3 from here."},{"Start":"07:11.495 ","End":"07:16.490","Text":"Similarly, the y first and last column of the same is here,"},{"Start":"07:16.490 ","End":"07:18.980","Text":"but the second column is this."},{"Start":"07:18.980 ","End":"07:20.615","Text":"In the case of z,"},{"Start":"07:20.615 ","End":"07:24.395","Text":"the third column, because this is the third variable."},{"Start":"07:24.395 ","End":"07:27.740","Text":"We just have a determinant over determinant,"},{"Start":"07:27.740 ","End":"07:31.500","Text":"these for computation\u0027s because this denominator is the same for all 3."},{"Start":"07:31.500 ","End":"07:35.930","Text":"Then we have 1, 2, 3 more computations of 3-by-3 determinants."},{"Start":"07:35.930 ","End":"07:37.520","Text":"Not so much work,"},{"Start":"07:37.520 ","End":"07:42.180","Text":"but it starts to get to bit more difficult."},{"Start":"07:42.180 ","End":"07:44.960","Text":"When we get into higher dimensions,"},{"Start":"07:44.960 ","End":"07:47.300","Text":"the same principle applies to 4,"},{"Start":"07:47.300 ","End":"07:49.430","Text":"5 and so on,"},{"Start":"07:49.430 ","End":"07:51.800","Text":"except that the determinant start getting difficult to"},{"Start":"07:51.800 ","End":"07:55.680","Text":"compute and it might not be the best way to do."},{"Start":"07:57.290 ","End":"08:02.445","Text":"Let\u0027s do an example to just keep the formula."},{"Start":"08:02.445 ","End":"08:08.640","Text":"Here\u0027s a 3-by-3 linear system of equations."},{"Start":"08:08.640 ","End":"08:11.164","Text":"Because there\u0027s something missing,"},{"Start":"08:11.164 ","End":"08:14.210","Text":"you might want to put 0\u0027s in, for example,"},{"Start":"08:14.210 ","End":"08:22.655","Text":"say plus 0y and maybe I also want to put 1x here and minus 1x here,"},{"Start":"08:22.655 ","End":"08:27.355","Text":"just so we can see the numbers when we take the matrix."},{"Start":"08:27.355 ","End":"08:30.890","Text":"This matrix and the denominator is 1,"},{"Start":"08:30.890 ","End":"08:34.850","Text":"0, 2 and so on. Let\u0027s see."},{"Start":"08:34.850 ","End":"08:42.140","Text":"As long as I can see this, yeah."},{"Start":"08:42.140 ","End":"08:46.970","Text":"Cramer\u0027s rule says that on the denominator for x, y, and z,"},{"Start":"08:46.970 ","End":"08:50.750","Text":"we have this determinant, 1, 0,"},{"Start":"08:50.750 ","End":"08:52.070","Text":"2 minus 3, 4,"},{"Start":"08:52.070 ","End":"08:55.805","Text":"6 minus 1, minus 2, 3."},{"Start":"08:55.805 ","End":"08:57.200","Text":"That\u0027s the same everywhere."},{"Start":"08:57.200 ","End":"09:02.585","Text":"I\u0027m not going to spend a lot of time in the computation here."},{"Start":"09:02.585 ","End":"09:06.470","Text":"We\u0027ll do more of that in the exercises."},{"Start":"09:06.470 ","End":"09:11.705","Text":"I\u0027ll give you the answer that this comes out to be 11,"},{"Start":"09:11.705 ","End":"09:16.055","Text":"and it would also be 11 here and 11 here."},{"Start":"09:16.055 ","End":"09:21.105","Text":"Now the numerators, as you see,"},{"Start":"09:21.105 ","End":"09:24.670","Text":"the 2 columns are always the same,"},{"Start":"09:24.670 ","End":"09:27.755","Text":"all but 1 column are the same as here."},{"Start":"09:27.755 ","End":"09:31.370","Text":"In this case it\u0027s the first column has replaced and it\u0027s by"},{"Start":"09:31.370 ","End":"09:36.755","Text":"these numbers here and you see why I used the coloring. It helps."},{"Start":"09:36.755 ","End":"09:41.960","Text":"Here, y would be the second column and z is the third column."},{"Start":"09:41.960 ","End":"09:46.520","Text":"Again, I\u0027m not going to spend time computing the determinant."},{"Start":"09:46.520 ","End":"09:51.365","Text":"I\u0027ll just give you the answer that here it comes out to be minus 10,"},{"Start":"09:51.365 ","End":"09:54.550","Text":"here it comes out to be 18,"},{"Start":"09:54.550 ","End":"09:58.275","Text":"and here it comes out to be 38."},{"Start":"09:58.275 ","End":"10:02.495","Text":"That gives us the answer for x, y, and z."},{"Start":"10:02.495 ","End":"10:08.095","Text":"Finally, we can reduce the fractions by dividing top and bottom by 4."},{"Start":"10:08.095 ","End":"10:14.955","Text":"This is Cramer\u0027s rule and you can see how to extend it to higher than 3."},{"Start":"10:14.955 ","End":"10:18.390","Text":"But I\u0027ll just say that it\u0027s not always the best."},{"Start":"10:18.390 ","End":"10:20.160","Text":"It\u0027s the most straightforward,"},{"Start":"10:20.160 ","End":"10:23.630","Text":"but sometimes the determinants are a nuisance to"},{"Start":"10:23.630 ","End":"10:27.410","Text":"compute especially when you get larger than 3 by 3."},{"Start":"10:27.410 ","End":"10:31.729","Text":"It\u0027s a technique, it\u0027s a tool you can use it or use other systems."},{"Start":"10:31.729 ","End":"10:36.500","Text":"In any event, I\u0027m done with this introduction which is 1 of"},{"Start":"10:36.500 ","End":"10:42.870","Text":"the main applications of determinants and some examples will follow."}],"ID":9917},{"Watched":false,"Name":"Exercise 1","Duration":"1m 33s","ChapterTopicVideoID":9577,"CourseChapterTopicPlaylistID":7288,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9577.jpeg","UploadDate":"2017-07-26T08:35:10.0270000","DurationForVideoObject":"PT1M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.275","Text":"In this exercise we\u0027re given a system of 2 equations and 2 unknowns, X and Y."},{"Start":"00:07.275 ","End":"00:11.400","Text":"We have to solve this system using Cramer\u0027s rule."},{"Start":"00:11.400 ","End":"00:16.920","Text":"In Cramer\u0027s rule, we have a formula explicitly for X and for Y."},{"Start":"00:16.920 ","End":"00:20.430","Text":"Each of them is something over the determinant of"},{"Start":"00:20.430 ","End":"00:23.970","Text":"the coefficient matrix, or restricted matrix."},{"Start":"00:23.970 ","End":"00:25.710","Text":"In any event, well,"},{"Start":"00:25.710 ","End":"00:29.110","Text":"you don\u0027t see the 1 here, but there\u0027s 1x,1,2,3,4."},{"Start":"00:31.340 ","End":"00:37.370","Text":"On the numerator, we have a similar determinant except that for X,"},{"Start":"00:37.370 ","End":"00:44.850","Text":"I replaced the first column by the coefficients on the right, 5, 11."},{"Start":"00:44.850 ","End":"00:46.860","Text":"We get this."},{"Start":"00:46.860 ","End":"00:52.790","Text":"On the Y we replace the second column,"},{"Start":"00:52.790 ","End":"00:55.160","Text":"the 2, 4 by 5,"},{"Start":"00:55.160 ","End":"00:57.770","Text":"11, and we get this."},{"Start":"00:57.770 ","End":"01:00.785","Text":"Both the denominators are the same."},{"Start":"01:00.785 ","End":"01:05.615","Text":"It\u0027s minus 2 because it\u0027s 4 minus 6,"},{"Start":"01:05.615 ","End":"01:08.150","Text":"the difference of the product of the diagonals,"},{"Start":"01:08.150 ","End":"01:10.670","Text":"and here also minus 2."},{"Start":"01:10.670 ","End":"01:15.420","Text":"Here we have 20 minus 22,"},{"Start":"01:15.420 ","End":"01:16.960","Text":"is minus 2,"},{"Start":"01:16.960 ","End":"01:22.890","Text":"and here we have 11 minus 15 is minus 4."},{"Start":"01:23.780 ","End":"01:26.460","Text":"I also wrote the answers,"},{"Start":"01:26.460 ","End":"01:29.655","Text":"obviously this is 1 and this is 2."},{"Start":"01:29.655 ","End":"01:32.920","Text":"That\u0027s it, we\u0027re done."}],"ID":9918},{"Watched":false,"Name":"Exercise 2","Duration":"4m 33s","ChapterTopicVideoID":9578,"CourseChapterTopicPlaylistID":7288,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9578.jpeg","UploadDate":"2017-07-26T08:35:24.0600000","DurationForVideoObject":"PT4M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.860","Text":"In this exercise, we have to use Cramer\u0027s rule to solve the system of linear equations,"},{"Start":"00:09.860 ","End":"00:12.080","Text":"3 equations and 3 unknowns,"},{"Start":"00:12.080 ","End":"00:14.610","Text":"x, y, and z."},{"Start":"00:15.650 ","End":"00:20.790","Text":"Just want you to note that we do this almost without saying,"},{"Start":"00:20.790 ","End":"00:24.960","Text":"but you have to at least mentally imagine there\u0027s a 1 here,"},{"Start":"00:24.960 ","End":"00:26.250","Text":"and a 1 here,"},{"Start":"00:26.250 ","End":"00:31.695","Text":"and also that there\u0027s a 0y, and what else?"},{"Start":"00:31.695 ","End":"00:34.485","Text":"Also here we have a 0y?"},{"Start":"00:34.485 ","End":"00:35.730","Text":"You didn\u0027t mention it,"},{"Start":"00:35.730 ","End":"00:36.825","Text":"but once in a while,"},{"Start":"00:36.825 ","End":"00:40.835","Text":"the 1s and the 0s you have to watch out for."},{"Start":"00:40.835 ","End":"00:43.910","Text":"Cramer\u0027s rule says that each of x, y,"},{"Start":"00:43.910 ","End":"00:46.730","Text":"and z is a fraction where the denominator is"},{"Start":"00:46.730 ","End":"00:52.240","Text":"the determinant of the coefficient matrix or restricted matrix."},{"Start":"00:52.240 ","End":"00:57.099","Text":"Here it is. We just take the coefficients from here, 1, 0, 1,"},{"Start":"00:57.099 ","End":"01:07.125","Text":"4, 1, 8, and then at 2,0,3."},{"Start":"01:07.125 ","End":"01:11.380","Text":"The same thing here and here just copied it."},{"Start":"01:11.380 ","End":"01:13.850","Text":"Now on the numerator,"},{"Start":"01:13.850 ","End":"01:17.060","Text":"we have in each case something similar to the denominator,"},{"Start":"01:17.060 ","End":"01:24.110","Text":"except that each time we replace a different column with these numbers 3, 21, 8."},{"Start":"01:24.110 ","End":"01:27.865","Text":"Here we replace the first column,"},{"Start":"01:27.865 ","End":"01:30.360","Text":"and we get this."},{"Start":"01:30.360 ","End":"01:35.930","Text":"I used the coloring in blue just so you can see more clearly what\u0027s happening,"},{"Start":"01:35.930 ","End":"01:37.775","Text":"and everything else is the same."},{"Start":"01:37.775 ","End":"01:44.375","Text":"Similarly for y, the same except for the middle column which are these."},{"Start":"01:44.375 ","End":"01:46.805","Text":"The numerator for z,"},{"Start":"01:46.805 ","End":"01:49.175","Text":"also the same pattern."},{"Start":"01:49.175 ","End":"01:53.495","Text":"Now, this denominator we only have to compute once."},{"Start":"01:53.495 ","End":"01:56.270","Text":"The easiest is to expand along"},{"Start":"01:56.270 ","End":"02:02.090","Text":"the middle column because everything is 0 except for this entry."},{"Start":"02:02.090 ","End":"02:04.130","Text":"This is the only entry that counts."},{"Start":"02:04.130 ","End":"02:09.590","Text":"I take the minor by removing this row also."},{"Start":"02:09.590 ","End":"02:11.450","Text":"I\u0027ve got 1, 1,"},{"Start":"02:11.450 ","End":"02:13.580","Text":"2, 3 is the minor."},{"Start":"02:13.580 ","End":"02:18.455","Text":"Its determinant is 3 minus 2 is 1."},{"Start":"02:18.455 ","End":"02:20.840","Text":"Then I have to multiply by this 1,"},{"Start":"02:20.840 ","End":"02:22.325","Text":"so it\u0027s still 1."},{"Start":"02:22.325 ","End":"02:24.979","Text":"Then there\u0027s the matter of the sign, the checkerboard,"},{"Start":"02:24.979 ","End":"02:27.670","Text":"remember, plus, minus, plus."},{"Start":"02:27.670 ","End":"02:32.380","Text":"All together this comes out to 1."},{"Start":"02:32.380 ","End":"02:37.890","Text":"Similarly here and here it\u0027s the same determinant."},{"Start":"02:37.890 ","End":"02:44.880","Text":"Now here we can also expand along the middle column."},{"Start":"02:44.880 ","End":"02:50.045","Text":"In this case, we get also a plus times 1,"},{"Start":"02:50.045 ","End":"02:54.845","Text":"and the minor is 3 times 3 minus 8 times 1."},{"Start":"02:54.845 ","End":"02:57.225","Text":"Let\u0027s leave this 1 in a minute."},{"Start":"02:57.225 ","End":"03:03.150","Text":"This 1 also has the middle column is easiest."},{"Start":"03:03.150 ","End":"03:05.810","Text":"Again we get plus times 1,"},{"Start":"03:05.810 ","End":"03:09.200","Text":"and this time 1 times 8 minus 2 times 3,"},{"Start":"03:09.200 ","End":"03:12.090","Text":"8 minus 6 is 2."},{"Start":"03:12.590 ","End":"03:16.465","Text":"This determinant is a little bit more difficult."},{"Start":"03:16.465 ","End":"03:19.765","Text":"Do that at the side here or at the bottom."},{"Start":"03:19.765 ","End":"03:22.660","Text":"What I\u0027ll do is combination."},{"Start":"03:22.660 ","End":"03:29.350","Text":"First I\u0027ll do some row operations as if trying to bring it to echelon form."},{"Start":"03:29.350 ","End":"03:32.845","Text":"I\u0027ll subtract 4 times this row,"},{"Start":"03:32.845 ","End":"03:34.805","Text":"from this row,"},{"Start":"03:34.805 ","End":"03:37.410","Text":"4 minus 4 will be 0,"},{"Start":"03:37.410 ","End":"03:39.360","Text":"21 minus 12 is 9,"},{"Start":"03:39.360 ","End":"03:43.110","Text":"8 minus 4 is 4."},{"Start":"03:43.110 ","End":"03:47.400","Text":"Similarly, this minus twice this,"},{"Start":"03:47.400 ","End":"03:51.765","Text":"and that brings us to this determinant."},{"Start":"03:51.765 ","End":"03:57.154","Text":"Now the obvious thing to do is to expand along the first column."},{"Start":"03:57.154 ","End":"04:00.625","Text":"There\u0027s only the 1 which is significant."},{"Start":"04:00.625 ","End":"04:09.655","Text":"It\u0027s a plus, so it\u0027s plus times 1 times the determinant of 9 minus 8 is 1,"},{"Start":"04:09.655 ","End":"04:11.445","Text":"so that\u0027s 1,"},{"Start":"04:11.445 ","End":"04:14.760","Text":"and that\u0027s what belongs over here."},{"Start":"04:14.760 ","End":"04:17.295","Text":"Basically, we\u0027re done,"},{"Start":"04:17.295 ","End":"04:20.060","Text":"except you might want to write it nicer,"},{"Start":"04:20.060 ","End":"04:23.075","Text":"and say that x equals 1,"},{"Start":"04:23.075 ","End":"04:26.599","Text":"y equals 1 over 1 is 1,"},{"Start":"04:26.599 ","End":"04:30.710","Text":"and z equals 2 over 1 is 2,"},{"Start":"04:30.710 ","End":"04:34.080","Text":"and that\u0027s our answer. We\u0027re done."}],"ID":9919},{"Watched":false,"Name":"Exercise 3","Duration":"10m 13s","ChapterTopicVideoID":9579,"CourseChapterTopicPlaylistID":7288,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9579.jpeg","UploadDate":"2017-07-26T08:35:53.1800000","DurationForVideoObject":"PT10M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"In this exercise, we\u0027re going to use Cramer\u0027s rule to solve the system"},{"Start":"00:05.370 ","End":"00:10.770","Text":"of 4 linear equations in 4 unknowns, x,"},{"Start":"00:10.770 ","End":"00:18.875","Text":"y, z, and t. Don\u0027t forget that where there\u0027s missing terms,"},{"Start":"00:18.875 ","End":"00:22.985","Text":"you just think of it as 0s like there\u0027s no y here,"},{"Start":"00:22.985 ","End":"00:26.695","Text":"and here there\u0027s no z and t, and so on."},{"Start":"00:26.695 ","End":"00:31.145","Text":"Now, Cramer\u0027s rule gives us explicit form"},{"Start":"00:31.145 ","End":"00:38.300","Text":"of each of the 4 variables and they all have the same denominator,"},{"Start":"00:38.300 ","End":"00:41.780","Text":"which is the coefficient matrix,"},{"Start":"00:41.780 ","End":"00:43.730","Text":"also called the restricted matrix."},{"Start":"00:43.730 ","End":"00:48.485","Text":"Let me just go back up and see that that\u0027s correct."},{"Start":"00:48.485 ","End":"00:52.715","Text":"Notice that here I have like 1,"},{"Start":"00:52.715 ","End":"00:54.605","Text":"0, there\u0027s no y,"},{"Start":"00:54.605 ","End":"00:58.085","Text":"and 2 and 5, and so on."},{"Start":"00:58.085 ","End":"01:03.650","Text":"Also, let me just write down these numbers."},{"Start":"01:03.650 ","End":"01:06.650","Text":"There was an 8, an 8,"},{"Start":"01:06.650 ","End":"01:10.880","Text":"2 minus 8, a 5 and a 51."},{"Start":"01:10.880 ","End":"01:14.250","Text":"When I scroll, I still have them."},{"Start":"01:15.470 ","End":"01:24.199","Text":"We get the numerators by copying from the denominator except for a single column."},{"Start":"01:24.199 ","End":"01:30.960","Text":"For x, it\u0027s the first column which is replaced by these numbers."},{"Start":"01:30.960 ","End":"01:34.845","Text":"Then this column keeps sliding to the right."},{"Start":"01:34.845 ","End":"01:36.960","Text":"Here it is. For y,"},{"Start":"01:36.960 ","End":"01:38.820","Text":"it\u0027s in the second column."},{"Start":"01:38.820 ","End":"01:41.310","Text":"For z, it\u0027s in the third column,"},{"Start":"01:41.310 ","End":"01:44.985","Text":"and for t, it\u0027s in the fourth and last column."},{"Start":"01:44.985 ","End":"01:48.980","Text":"Now, essentially, this is already the answer,"},{"Start":"01:48.980 ","End":"01:53.930","Text":"except that we have to do the tedious work of computing determinants."},{"Start":"01:53.930 ","End":"01:55.910","Text":"Now there\u0027s not 8 of them."},{"Start":"01:55.910 ","End":"01:57.815","Text":"There\u0027s actually 5 to compute,"},{"Start":"01:57.815 ","End":"02:00.530","Text":"because all 4 denominators are the same."},{"Start":"02:00.530 ","End":"02:02.150","Text":"That\u0027s one computation."},{"Start":"02:02.150 ","End":"02:04.440","Text":"Then 2, 3, 4,"},{"Start":"02:04.440 ","End":"02:10.910","Text":"5 computations, each of them are 4 by 4 determinant, which is a lot of work."},{"Start":"02:10.910 ","End":"02:16.810","Text":"That\u0027s why often we don\u0027t use Cramer\u0027s rule for large systems."},{"Start":"02:16.810 ","End":"02:21.785","Text":"Anyway, I\u0027m going to do some of the tedious work."},{"Start":"02:21.785 ","End":"02:23.195","Text":"Maybe I won\u0027t do all of them."},{"Start":"02:23.195 ","End":"02:25.340","Text":"Well, first of all, I\u0027ll give you the answers."},{"Start":"02:25.340 ","End":"02:28.910","Text":"It turns out that all of the determinants come out to be minus"},{"Start":"02:28.910 ","End":"02:35.060","Text":"4,310 which makes all the variables come out to be one."},{"Start":"02:35.060 ","End":"02:38.780","Text":"Let\u0027s do some of the calculations."},{"Start":"02:38.780 ","End":"02:45.565","Text":"For example, this one I\u0027ll do first because it appears in all 4 them."},{"Start":"02:45.565 ","End":"02:48.675","Text":"Here it is in a new page."},{"Start":"02:48.675 ","End":"02:51.110","Text":"There\u0027s more than one way of doing this."},{"Start":"02:51.110 ","End":"02:54.500","Text":"What I\u0027ll do is I\u0027ll do some row operations."},{"Start":"02:54.500 ","End":"02:58.130","Text":"First of all, I\u0027ll add twice the top row to the second row."},{"Start":"02:58.130 ","End":"03:00.890","Text":"Then I\u0027ll subtract 5 times this from"},{"Start":"03:00.890 ","End":"03:05.885","Text":"this row and subtract twice the first row from the last row."},{"Start":"03:05.885 ","End":"03:09.905","Text":"If I do those 3 row operations,"},{"Start":"03:09.905 ","End":"03:15.865","Text":"we get this determinant because this row operation doesn\u0027t change the determinant."},{"Start":"03:15.865 ","End":"03:17.150","Text":"Let\u0027s give one example."},{"Start":"03:17.150 ","End":"03:21.695","Text":"Let\u0027s say in this place we have minus 7,"},{"Start":"03:21.695 ","End":"03:24.950","Text":"minus 5, times 2 is minus 7,"},{"Start":"03:24.950 ","End":"03:27.455","Text":"minus 10 is minus 17."},{"Start":"03:27.455 ","End":"03:29.785","Text":"Similarly all the rest."},{"Start":"03:29.785 ","End":"03:33.405","Text":"You could expand along the first column,"},{"Start":"03:33.405 ","End":"03:40.940","Text":"but I noticed that all of this row divides by 2 and the last row I can take 5 out."},{"Start":"03:40.940 ","End":"03:44.930","Text":"It\u0027s often comes out easier if you take common factors out."},{"Start":"03:44.930 ","End":"03:47.360","Text":"We get, this is 2 times 5,"},{"Start":"03:47.360 ","End":"03:50.005","Text":"not 2.5, it\u0027s 10."},{"Start":"03:50.005 ","End":"03:52.955","Text":"This is divided by 2 to give this."},{"Start":"03:52.955 ","End":"03:56.790","Text":"This is divided by 5 to give this."},{"Start":"03:56.870 ","End":"03:59.960","Text":"Why don\u0027t I continue with row operations?"},{"Start":"03:59.960 ","End":"04:04.490","Text":"If I add the second row to the third row, let me write that."},{"Start":"04:04.490 ","End":"04:09.495","Text":"I\u0027ll do row 2 plus"},{"Start":"04:09.495 ","End":"04:15.405","Text":"row 3 and put that into row 3,"},{"Start":"04:15.405 ","End":"04:17.790","Text":"that will make this one a 0."},{"Start":"04:17.790 ","End":"04:26.070","Text":"I can also get a 0 here if I take 3 times this row plus this row."},{"Start":"04:26.070 ","End":"04:36.680","Text":"In other words, I take 3 times row 4 plus row 2 into row 4."},{"Start":"04:36.680 ","End":"04:40.099","Text":"But if I do that,"},{"Start":"04:40.099 ","End":"04:44.105","Text":"the determinant changes because of this 3."},{"Start":"04:44.105 ","End":"04:49.345","Text":"I have to compensate by multiplying by a 1/3."},{"Start":"04:49.345 ","End":"04:51.600","Text":"Okay, I\u0027ll explain. This is what we get."},{"Start":"04:51.600 ","End":"04:56.290","Text":"The 3 here becomes 1/3 to compensate."},{"Start":"04:56.290 ","End":"05:02.150","Text":"Now, this row here is fairly clear."},{"Start":"05:02.150 ","End":"05:04.170","Text":"Just adding these 2."},{"Start":"05:04.170 ","End":"05:05.660","Text":"0 plus 0 is 0."},{"Start":"05:05.660 ","End":"05:09.725","Text":"This plus this is 0, this plus this is minus 15."},{"Start":"05:09.725 ","End":"05:12.455","Text":"This and this gives me minus 16."},{"Start":"05:12.455 ","End":"05:18.630","Text":"The last row is when we take 3 times this plus this."},{"Start":"05:18.630 ","End":"05:22.520","Text":"3 times 1 plus minus 3 is 0."},{"Start":"05:22.520 ","End":"05:24.230","Text":"3 times 8 is 24,"},{"Start":"05:24.230 ","End":"05:26.645","Text":"plus 2 is 26."},{"Start":"05:26.645 ","End":"05:32.175","Text":"Here we have minus 6 plus 5 is minus one."},{"Start":"05:32.175 ","End":"05:37.179","Text":"Now, notice that I have a block matrix."},{"Start":"05:37.179 ","End":"05:41.575","Text":"If I divide it up this way into blocks,"},{"Start":"05:41.575 ","End":"05:43.885","Text":"this is a 0 block."},{"Start":"05:43.885 ","End":"05:47.140","Text":"All I have to do is take the determinant of this 2 by"},{"Start":"05:47.140 ","End":"05:51.370","Text":"2 and multiply it by this determinant of a 2 by 2."},{"Start":"05:51.370 ","End":"05:59.545","Text":"Now, this is what minus 3 take away 0 is minus 3."},{"Start":"05:59.545 ","End":"06:10.015","Text":"This block is plus 15 and comes out to be negative 416."},{"Start":"06:10.015 ","End":"06:15.120","Text":"I\u0027ve got 15 plus 416 is"},{"Start":"06:15.120 ","End":"06:24.975","Text":"431 times minus 3 is minus 1293."},{"Start":"06:24.975 ","End":"06:33.420","Text":"Divide by 3, you get 431 and then multiply by 10 is 4,310."},{"Start":"06:33.420 ","End":"06:39.380","Text":"Anyway, that\u0027s the answer to the first determinant,"},{"Start":"06:39.380 ","End":"06:42.065","Text":"which was the denominator on all 4."},{"Start":"06:42.065 ","End":"06:45.830","Text":"Now we\u0027ve got 4 more determinants to do."},{"Start":"06:45.830 ","End":"06:47.975","Text":"Well, I\u0027ll do a couple more."},{"Start":"06:47.975 ","End":"06:55.145","Text":"I\u0027ll do this one which was the numerator for x. I see these 2 0s here."},{"Start":"06:55.145 ","End":"06:57.980","Text":"I think I\u0027ll do an expansion along"},{"Start":"06:57.980 ","End":"07:05.110","Text":"the second row that will give us 2 entries that are non-zero."},{"Start":"07:05.110 ","End":"07:10.470","Text":"First of all, I\u0027ll do this for the minus 8."},{"Start":"07:11.140 ","End":"07:15.670","Text":"Notice that it\u0027s a minus on the checkerboard."},{"Start":"07:15.670 ","End":"07:18.380","Text":"I\u0027ve got minus, minus 8,"},{"Start":"07:18.380 ","End":"07:22.735","Text":"which is 8 times a 3 by 3 determinant."},{"Start":"07:22.735 ","End":"07:25.125","Text":"That\u0027s this one."},{"Start":"07:25.125 ","End":"07:29.060","Text":"For this element, I need to cross out this column."},{"Start":"07:29.060 ","End":"07:30.590","Text":"That\u0027s a minus 6,"},{"Start":"07:30.590 ","End":"07:34.880","Text":"but with a plus times this determinant."},{"Start":"07:34.880 ","End":"07:37.130","Text":"This is what we call the minor."},{"Start":"07:37.130 ","End":"07:40.315","Text":"Yeah, it\u0027s the ones that we haven\u0027t crossed out."},{"Start":"07:40.315 ","End":"07:44.210","Text":"I\u0027ll continue with the expansion along rows."},{"Start":"07:44.210 ","End":"07:48.095","Text":"Let\u0027s say this one I\u0027ll do with the first row."},{"Start":"07:48.095 ","End":"07:52.345","Text":"This one I\u0027ll expand along the last row."},{"Start":"07:52.345 ","End":"07:54.840","Text":"We\u0027ll basically get 4 pieces."},{"Start":"07:54.840 ","End":"07:58.624","Text":"I mean from this column,"},{"Start":"07:58.624 ","End":"08:02.675","Text":"we\u0027ll get minus 2 times this minor."},{"Start":"08:02.675 ","End":"08:09.455","Text":"Then we\u0027ll get the plus 5 times this minor and so on."},{"Start":"08:09.455 ","End":"08:12.050","Text":"For this one, we\u0027ll get something for you here,"},{"Start":"08:12.050 ","End":"08:14.210","Text":"and then something from here,"},{"Start":"08:14.210 ","End":"08:17.570","Text":"and it gets quite tedious."},{"Start":"08:17.570 ","End":"08:25.845","Text":"I\u0027ll leave you to check the computation of the 2 minors and then the product."},{"Start":"08:25.845 ","End":"08:30.040","Text":"We still have the 8 and the minus 6."},{"Start":"08:30.290 ","End":"08:35.315","Text":"After we open up these determinants, expand them,"},{"Start":"08:35.315 ","End":"08:38.245","Text":"compute them, and we get this."},{"Start":"08:38.245 ","End":"08:40.965","Text":"It\u0027s just arithmetic."},{"Start":"08:40.965 ","End":"08:48.475","Text":"Like this times this is 835 plus 40 is 875 times 8 is 7,000."},{"Start":"08:48.475 ","End":"08:51.600","Text":"I\u0027ll leave you to check this."},{"Start":"08:51.600 ","End":"08:57.645","Text":"Anyway in the end we get to minus 4,310 like I said."},{"Start":"08:57.645 ","End":"09:02.120","Text":"All the determinants in this exercise come out to be minus 4,310."},{"Start":"09:02.120 ","End":"09:05.300","Text":"Let\u0027s do another one."},{"Start":"09:05.300 ","End":"09:11.990","Text":"This is the one from the numerator of y. I\u0027ll go more quickly."},{"Start":"09:11.990 ","End":"09:17.310","Text":"We\u0027ll do an expansion along the second row."},{"Start":"09:17.310 ","End":"09:20.539","Text":"Once with this entry here which is a minus,"},{"Start":"09:20.539 ","End":"09:25.350","Text":"and once with this entry here which is a plus."},{"Start":"09:25.520 ","End":"09:28.550","Text":"Each of these 3 by 3 determinants,"},{"Start":"09:28.550 ","End":"09:32.180","Text":"I\u0027ll expand along the last row,"},{"Start":"09:32.180 ","End":"09:35.635","Text":"where at least I have a 0."},{"Start":"09:35.635 ","End":"09:41.570","Text":"This one gives me these 2 pieces and so on here."},{"Start":"09:41.570 ","End":"09:43.790","Text":"Then we expand the 2 by 2s."},{"Start":"09:43.790 ","End":"09:50.360","Text":"For example, 8 minus minus 35 is 43, and so on."},{"Start":"09:50.360 ","End":"09:53.330","Text":"Then we compute the curly brace here,"},{"Start":"09:53.330 ","End":"09:55.790","Text":"comes out to be this curly brace here,"},{"Start":"09:55.790 ","End":"09:58.700","Text":"that multiplied by 8 by 2."},{"Start":"09:58.700 ","End":"10:05.565","Text":"The final thing is once again, minus 4,310."},{"Start":"10:05.565 ","End":"10:07.700","Text":"Here there\u0027s two more to go,"},{"Start":"10:07.700 ","End":"10:10.925","Text":"but we just let it go at that."},{"Start":"10:10.925 ","End":"10:13.950","Text":"I think that\u0027s enough."}],"ID":9920},{"Watched":false,"Name":"Exercise 4 Part a","Duration":"6m 50s","ChapterTopicVideoID":9580,"CourseChapterTopicPlaylistID":7288,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9580.jpeg","UploadDate":"2017-07-26T08:36:12.7730000","DurationForVideoObject":"PT6M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"This exercise is important."},{"Start":"00:03.300 ","End":"00:05.894","Text":"It has several parts."},{"Start":"00:05.894 ","End":"00:09.540","Text":"It covers a lot of material that mainly in Part A,"},{"Start":"00:09.540 ","End":"00:13.290","Text":"we\u0027re going to talk about a theorem which is very"},{"Start":"00:13.290 ","End":"00:19.060","Text":"basic about when a system has unique solutions."},{"Start":"00:19.160 ","End":"00:28.140","Text":"The exercise involves a system of 5 equations, 5 unknowns linear,"},{"Start":"00:28.140 ","End":"00:30.420","Text":"and the unknown are x,"},{"Start":"00:30.420 ","End":"00:31.770","Text":"y, z, t,"},{"Start":"00:31.770 ","End":"00:38.085","Text":"and r. We\u0027ll start with part a,"},{"Start":"00:38.085 ","End":"00:42.310","Text":"and for this, I need a theorem."},{"Start":"00:42.310 ","End":"00:49.340","Text":"Actually, it\u0027s a theorem I\u0027ve hinted at or even mentioned before on the tutorial,"},{"Start":"00:49.340 ","End":"00:52.005","Text":"but didn\u0027t actually state it formally."},{"Start":"00:52.005 ","End":"00:56.840","Text":"It relates to a general system of linear equations,"},{"Start":"00:56.840 ","End":"01:02.720","Text":"n equations in n unknowns for an arbitrary n. We"},{"Start":"01:02.720 ","End":"01:09.315","Text":"need a set of double subscripts indices for the coefficients,"},{"Start":"01:09.315 ","End":"01:11.750","Text":"and for the right-hand side,"},{"Start":"01:11.750 ","End":"01:19.665","Text":"we just need 1 through n. Here goes the theorem."},{"Start":"01:19.665 ","End":"01:22.215","Text":"It gives a precise condition,"},{"Start":"01:22.215 ","End":"01:26.330","Text":"an if and only if condition for this system to have"},{"Start":"01:26.330 ","End":"01:34.910","Text":"a unique solution and that condition is that the determinant of all these coefficients."},{"Start":"01:34.910 ","End":"01:39.470","Text":"This is called the restricted matrix,"},{"Start":"01:39.470 ","End":"01:44.985","Text":"or sometimes the coefficient matrix."},{"Start":"01:44.985 ","End":"01:47.500","Text":"Its determinant should be non-zero."},{"Start":"01:47.500 ","End":"01:50.765","Text":"If I take all the numbers here with the x\u0027s,"},{"Start":"01:50.765 ","End":"01:53.530","Text":"not the numbers on the right."},{"Start":"01:53.530 ","End":"01:56.360","Text":"If this determinant is non-zero,"},{"Start":"01:56.360 ","End":"01:59.405","Text":"then it has unique solution otherwise not."},{"Start":"01:59.405 ","End":"02:01.940","Text":"That\u0027s a very important theorem."},{"Start":"02:01.940 ","End":"02:07.505","Text":"Now let\u0027s use this theorem in our exercise."},{"Start":"02:07.505 ","End":"02:13.175","Text":"If you go back and look at the system of equations,"},{"Start":"02:13.175 ","End":"02:15.870","Text":"we had 5 equations and 5 unknowns,"},{"Start":"02:15.870 ","End":"02:22.905","Text":"and the coefficients were k\u0027s along the diagonal and 1\u0027s everywhere else."},{"Start":"02:22.905 ","End":"02:31.040","Text":"All we have to do now is find conditions on k such that this determinant is non-zero."},{"Start":"02:31.040 ","End":"02:33.305","Text":"I\u0027ll move on to the next page,"},{"Start":"02:33.305 ","End":"02:36.260","Text":"where I will try and compute this,"},{"Start":"02:36.260 ","End":"02:43.465","Text":"at least in terms of k. We start off with 4-row operations."},{"Start":"02:43.465 ","End":"02:46.820","Text":"The first row stays as is,"},{"Start":"02:46.820 ","End":"02:51.410","Text":"and then we subtract it from each of the other 4 rows that\u0027s indicated here."},{"Start":"02:51.410 ","End":"02:53.645","Text":"For example, row 2,"},{"Start":"02:53.645 ","End":"02:58.100","Text":"subtract row 1 in place of what was row 2,"},{"Start":"02:58.100 ","End":"03:00.110","Text":"and so on up to the fifth 1,"},{"Start":"03:00.110 ","End":"03:01.910","Text":"we subtract row 1,"},{"Start":"03:01.910 ","End":"03:05.180","Text":"and if we do that, well,"},{"Start":"03:05.180 ","End":"03:10.280","Text":"these operations don\u0027t change the determinant and we get this,"},{"Start":"03:10.280 ","End":"03:14.060","Text":"which has quite a few 0\u0027s in it."},{"Start":"03:14.060 ","End":"03:16.250","Text":"Well, it has certain patterns,"},{"Start":"03:16.250 ","End":"03:19.310","Text":"as you can see as 1\u0027s here 1 minus k\u0027s here."},{"Start":"03:19.310 ","End":"03:24.140","Text":"You might have encountered this in 1 of the previous exercises,"},{"Start":"03:24.140 ","End":"03:27.990","Text":"but I\u0027m not going to assume that."},{"Start":"03:28.400 ","End":"03:30.950","Text":"There\u0027s a trick to this."},{"Start":"03:30.950 ","End":"03:36.740","Text":"Notice that 1 minus k and k minus 1 are negatives of each other."},{"Start":"03:36.740 ","End":"03:38.390","Text":"If you add this to this, you get 0."},{"Start":"03:38.390 ","End":"03:41.780","Text":"Similarly this with this and this with this, this with this."},{"Start":"03:41.780 ","End":"03:47.060","Text":"Now, if we add up all the entries on a given row,"},{"Start":"03:47.060 ","End":"03:51.440","Text":"we\u0027ll get 0 except for the first row."},{"Start":"03:51.440 ","End":"03:56.015","Text":"What I\u0027m suggesting is we add up all the columns,"},{"Start":"03:56.015 ","End":"04:01.895","Text":"and what we\u0027ll do with this sum is just put it on the first column."},{"Start":"04:01.895 ","End":"04:10.975","Text":"Another way of saying that is to add these 4 columns to the first column."},{"Start":"04:10.975 ","End":"04:15.890","Text":"The sum of all the columns will be in column 1."},{"Start":"04:15.890 ","End":"04:22.080","Text":"Then we\u0027ll get these full columns are unchanged of course."},{"Start":"04:22.080 ","End":"04:29.465","Text":"Here we talked about these 0\u0027s because 1 minus k plus k minus 1 is 0, 4 times."},{"Start":"04:29.465 ","End":"04:32.540","Text":"The only thing is the entry here,"},{"Start":"04:32.540 ","End":"04:37.220","Text":"which is gotten by k plus 1 plus 1 plus 1 plus 1,"},{"Start":"04:37.220 ","End":"04:39.750","Text":"which is k plus 4."},{"Start":"04:40.160 ","End":"04:42.830","Text":"Now we\u0027re in good shape because look,"},{"Start":"04:42.830 ","End":"04:48.980","Text":"this is an upper triangular matrix inside the determinant."},{"Start":"04:48.980 ","End":"04:51.950","Text":"I mean, if I put a line here,"},{"Start":"04:51.950 ","End":"04:59.900","Text":"you can see that everything below the main diagonal is 0\u0027s."},{"Start":"04:59.980 ","End":"05:04.225","Text":"I\u0027ll just complete the triangle here."},{"Start":"05:04.225 ","End":"05:07.150","Text":"This is the triangle."},{"Start":"05:07.730 ","End":"05:12.575","Text":"When we have a triangular matrix,"},{"Start":"05:12.575 ","End":"05:19.860","Text":"then the determinant is just the product of the entries along the diagonal."},{"Start":"05:19.860 ","End":"05:24.350","Text":"If we multiply these together and we have k minus 1, k minus 1,"},{"Start":"05:24.350 ","End":"05:26.585","Text":"k minus 1, k minus 1,"},{"Start":"05:26.585 ","End":"05:29.225","Text":"and then k plus 4."},{"Start":"05:29.225 ","End":"05:35.925","Text":"That makes it k minus 1 to the 4th,"},{"Start":"05:35.925 ","End":"05:38.730","Text":"and there\u0027s also a k plus 4."},{"Start":"05:38.730 ","End":"05:40.380","Text":"We have this product."},{"Start":"05:40.380 ","End":"05:42.875","Text":"Now our question is,"},{"Start":"05:42.875 ","End":"05:48.020","Text":"is that when is this 0 and when is it non-zero?"},{"Start":"05:48.020 ","End":"05:53.435","Text":"Well, it\u0027s pretty clear that this is 0."},{"Start":"05:53.435 ","End":"05:54.710","Text":"There\u0027s only 2 cases."},{"Start":"05:54.710 ","End":"05:58.115","Text":"It can be 0 if our product is 0."},{"Start":"05:58.115 ","End":"06:00.919","Text":"Then either the k plus 4 is 0,"},{"Start":"06:00.919 ","End":"06:05.835","Text":"which is k equaling minus 4,"},{"Start":"06:05.835 ","End":"06:08.040","Text":"or this is 0,"},{"Start":"06:08.040 ","End":"06:10.760","Text":"to the power of 4 doesn\u0027t make any difference."},{"Start":"06:10.760 ","End":"06:12.890","Text":"If something to the power of 4 is 0,"},{"Start":"06:12.890 ","End":"06:16.010","Text":"then it has to be 0 and vice versa,"},{"Start":"06:16.010 ","End":"06:21.280","Text":"which means that we have k equals 1."},{"Start":"06:22.460 ","End":"06:27.440","Text":"What it means is that if k is not minus 4,"},{"Start":"06:27.440 ","End":"06:33.230","Text":"or 1, then the original system has a unique solution."},{"Start":"06:33.230 ","End":"06:37.520","Text":"I just wrote this out, that\u0027s our conclusion."},{"Start":"06:37.520 ","End":"06:42.965","Text":"Unique solution if and only if k is not equal to 1 and k is not equal to minus 4."},{"Start":"06:42.965 ","End":"06:47.125","Text":"That completes part a."},{"Start":"06:47.125 ","End":"06:51.190","Text":"We\u0027ll continue on the next clip."}],"ID":9921},{"Watched":false,"Name":"Exercise 4 Part b","Duration":"4m 53s","ChapterTopicVideoID":9581,"CourseChapterTopicPlaylistID":7288,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9581.jpeg","UploadDate":"2017-07-26T08:36:25.1400000","DurationForVideoObject":"PT4M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"This exercise is the continuation of the previous clip."},{"Start":"00:04.965 ","End":"00:07.110","Text":"It was a full part exercise."},{"Start":"00:07.110 ","End":"00:10.860","Text":"We did part a and now we\u0027re on to part b,"},{"Start":"00:10.860 ","End":"00:15.600","Text":"where in this system of 5 equations with 5 unknowns,"},{"Start":"00:15.600 ","End":"00:19.320","Text":"we\u0027re asked for which value of k does"},{"Start":"00:19.320 ","End":"00:23.925","Text":"the system have a unique solution with x equals 0.5?"},{"Start":"00:23.925 ","End":"00:29.005","Text":"Now there are 5 unknowns and we\u0027re just asked about x."},{"Start":"00:29.005 ","End":"00:34.295","Text":"This is a good indicator that you should use Cramer\u0027s rule."},{"Start":"00:34.295 ","End":"00:40.125","Text":"Normally with large systems 4 and 5 and higher,"},{"Start":"00:40.125 ","End":"00:43.925","Text":"Cramer\u0027s rule is a lot of work a little determinants to compute."},{"Start":"00:43.925 ","End":"00:48.180","Text":"But when you only have to compute 1 of the variables,"},{"Start":"00:48.180 ","End":"00:49.965","Text":"it\u0027s not so bad."},{"Start":"00:49.965 ","End":"00:53.620","Text":"The regular system is a lot of work too."},{"Start":"00:53.620 ","End":"01:00.805","Text":"I mean to find x with all sorts of row operations and everything is also a lot of work."},{"Start":"01:00.805 ","End":"01:09.110","Text":"Let\u0027s go for Cramer\u0027s rule where we have an explicit expression for x. I\u0027m"},{"Start":"01:09.110 ","End":"01:17.810","Text":"using the result from the previous part a for the determinant in the denominator."},{"Start":"01:17.810 ","End":"01:26.240","Text":"We have the determinant of all k\u0027s here corresponding to this and that\u0027s here."},{"Start":"01:26.240 ","End":"01:30.470","Text":"What we\u0027ve done, according to Cramer\u0027s, for x,"},{"Start":"01:30.470 ","End":"01:35.630","Text":"x is represented by the first column and we replace it by all 1s."},{"Start":"01:35.630 ","End":"01:45.370","Text":"Really if x is just changing what was a k here to a 1 and we want this to equal 0.5."},{"Start":"01:45.370 ","End":"01:48.410","Text":"Now to compute this determinant,"},{"Start":"01:48.410 ","End":"01:54.275","Text":"I suggest subtracting the top row from each of the other rows."},{"Start":"01:54.275 ","End":"01:55.955","Text":"Because if we do that,"},{"Start":"01:55.955 ","End":"01:58.530","Text":"we\u0027ll get a lot of 0s."},{"Start":"01:58.530 ","End":"02:04.145","Text":"What I mean is I\u0027ll just write it\u0027s like R_2"},{"Start":"02:04.145 ","End":"02:10.010","Text":"will be replaced by R_2 minus 1."},{"Start":"02:10.010 ","End":"02:15.845","Text":"In other words, subtract and put that into the second place and so on."},{"Start":"02:15.845 ","End":"02:22.900","Text":"Row 5 minus row 1 will go into the old row 5 and if we"},{"Start":"02:22.900 ","End":"02:30.320","Text":"do all these 4 row operations which don\u0027t change the value of the determinant,"},{"Start":"02:30.320 ","End":"02:34.760","Text":"we get, and I\u0027m just talking about the numerator part here."},{"Start":"02:36.300 ","End":"02:42.530","Text":"All these are 0s and taking away 1 from everything,"},{"Start":"02:42.530 ","End":"02:45.470","Text":"all the 1s become 0 and wherever I had k,"},{"Start":"02:45.470 ","End":"02:47.810","Text":"I have k minus 1 now."},{"Start":"02:47.810 ","End":"02:54.385","Text":"Observe that this is now an upper diagonal matrix."},{"Start":"02:54.385 ","End":"03:00.035","Text":"Below the main diagonal, everything is 0."},{"Start":"03:00.035 ","End":"03:04.920","Text":"This is the triangle I\u0027m talking about."},{"Start":"03:07.040 ","End":"03:15.995","Text":"The determinant of this matrix is just the product of the entries along the main diagonal"},{"Start":"03:15.995 ","End":"03:24.289","Text":"so that this gives us k minus 1 to the full for this determinant."},{"Start":"03:24.289 ","End":"03:26.950","Text":"Now remember, it was part of a context"},{"Start":"03:26.950 ","End":"03:33.530","Text":"where in the denominator from the previous exercise, we had this."},{"Start":"03:33.530 ","End":"03:39.875","Text":"The question was to see when x equals 1/2."},{"Start":"03:39.875 ","End":"03:47.000","Text":"We also had to make sure that it\u0027s a unique solution."},{"Start":"03:47.000 ","End":"03:53.210","Text":"For that, we have to also remember that what we found in part a,"},{"Start":"03:53.210 ","End":"03:59.205","Text":"that k must not equal 1 or negative 4."},{"Start":"03:59.205 ","End":"04:02.470","Text":"We have to solve this equation."},{"Start":"04:04.400 ","End":"04:08.840","Text":"This cancels with this."},{"Start":"04:08.840 ","End":"04:14.180","Text":"Of course, at the end, we\u0027re going to have to check that we don\u0027t get 1 or minus 4."},{"Start":"04:14.180 ","End":"04:16.355","Text":"Anyway, at this stage we get,"},{"Start":"04:16.355 ","End":"04:19.055","Text":"there\u0027s a 1 here after we cancel."},{"Start":"04:19.055 ","End":"04:21.995","Text":"1 over k plus 4 equals 1/2."},{"Start":"04:21.995 ","End":"04:28.040","Text":"This is what we get, if you take the reciprocal of both sides,"},{"Start":"04:28.040 ","End":"04:32.200","Text":"k plus 4 equals to bring the 4 over k is minus 2"},{"Start":"04:32.200 ","End":"04:36.940","Text":"and minus 2 is not equal to 1 or minus 4."},{"Start":"04:36.940 ","End":"04:42.545","Text":"We\u0027re okay. This is the answer for this value of k,"},{"Start":"04:42.545 ","End":"04:48.860","Text":"x will equal a half and there will be a unique solution to the system."},{"Start":"04:50.590 ","End":"04:54.150","Text":"We\u0027ll continue on the next clip."}],"ID":9922},{"Watched":false,"Name":"Exercise 4 Part c","Duration":"1m 32s","ChapterTopicVideoID":9576,"CourseChapterTopicPlaylistID":7288,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9576.jpeg","UploadDate":"2017-07-26T08:35:04.9770000","DurationForVideoObject":"PT1M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"We\u0027re continuing from the previous clip."},{"Start":"00:02.730 ","End":"00:08.085","Text":"We just did parts A and B of the exercise and now we\u0027re up to part C,"},{"Start":"00:08.085 ","End":"00:11.250","Text":"which is essentially the same as part B,"},{"Start":"00:11.250 ","End":"00:14.070","Text":"except that in part b,"},{"Start":"00:14.070 ","End":"00:18.075","Text":"we had to answer the question with x equals 1.5."},{"Start":"00:18.075 ","End":"00:21.510","Text":"Here we have x equals 1/5."},{"Start":"00:21.510 ","End":"00:25.350","Text":"I\u0027m just going to copy the line from part B."},{"Start":"00:25.350 ","End":"00:29.370","Text":"But the change here and here where I had a 1/2 before,"},{"Start":"00:29.370 ","End":"00:32.325","Text":"now I have a 1/5."},{"Start":"00:32.325 ","End":"00:37.520","Text":"As before, we cancel this k minus 1 to the 1/4"},{"Start":"00:37.520 ","End":"00:39.655","Text":"with this k minus 1 to the 1/4,"},{"Start":"00:39.655 ","End":"00:42.555","Text":"that leaves a 1 on the numerator."},{"Start":"00:42.555 ","End":"00:47.750","Text":"This gives us that 1 over k plus 4 equals 1 over 5."},{"Start":"00:47.750 ","End":"00:50.780","Text":"k plus 4 is 5."},{"Start":"00:50.780 ","End":"00:55.320","Text":"That would give us k equals 1."},{"Start":"00:55.630 ","End":"00:58.385","Text":"But I\u0027m going to cross this out."},{"Start":"00:58.385 ","End":"00:59.690","Text":"Now, why did I do that?"},{"Start":"00:59.690 ","End":"01:02.060","Text":"Why is k equals 1 bad?"},{"Start":"01:02.060 ","End":"01:04.175","Text":"Well, here\u0027s where it is,"},{"Start":"01:04.175 ","End":"01:07.420","Text":"because we have that k cannot equal 1."},{"Start":"01:07.420 ","End":"01:11.875","Text":"That was one of the conditions for the system having a unique solution."},{"Start":"01:11.875 ","End":"01:16.760","Text":"In fact, for C, there is no value of k,"},{"Start":"01:16.760 ","End":"01:21.230","Text":"no answer to C for no value of k"},{"Start":"01:21.230 ","End":"01:29.460","Text":"and we\u0027re done with part C. Let\u0027s go on to the next part."},{"Start":"01:29.460 ","End":"01:31.600","Text":"I\u0027ll do it in the next clip."}],"ID":9923},{"Watched":false,"Name":"Exercise 4 Part d","Duration":"5m 27s","ChapterTopicVideoID":12069,"CourseChapterTopicPlaylistID":7288,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12069.jpeg","UploadDate":"2018-05-15T18:21:51.3230000","DurationForVideoObject":"PT5M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.800","Text":"This clip continues."},{"Start":"00:01.800 ","End":"00:04.320","Text":"The previous clip was still on the same exercise."},{"Start":"00:04.320 ","End":"00:07.500","Text":"We just finished parts a, b, and c,"},{"Start":"00:07.500 ","End":"00:10.590","Text":"and we\u0027re left with part d,"},{"Start":"00:10.590 ","End":"00:15.645","Text":"still the same system of 5 equations and 5 unknowns."},{"Start":"00:15.645 ","End":"00:19.665","Text":"We have to show that if the system has a unique solution,"},{"Start":"00:19.665 ","End":"00:22.230","Text":"then all the values are equal,"},{"Start":"00:22.230 ","End":"00:28.350","Text":"x equals y equals z equals t equals r. I kept the result"},{"Start":"00:28.350 ","End":"00:35.519","Text":"from the previous part where we have that x was equal to,"},{"Start":"00:35.519 ","End":"00:37.215","Text":"well, let\u0027s start with the denominator."},{"Start":"00:37.215 ","End":"00:43.250","Text":"The denominator is the determinant of the quantities here,"},{"Start":"00:43.250 ","End":"00:45.020","Text":"all ks along the diagonal,"},{"Start":"00:45.020 ","End":"00:50.315","Text":"and 1s here, the restricted matrix or the coefficient matrix,"},{"Start":"00:50.315 ","End":"00:53.955","Text":"its determinant that\u0027s computed to this."},{"Start":"00:53.955 ","End":"00:56.640","Text":"On the numerator for x,"},{"Start":"00:56.640 ","End":"00:59.130","Text":"we replace the first column,"},{"Start":"00:59.130 ","End":"01:05.350","Text":"which was k1111 with just this 11111."},{"Start":"01:05.810 ","End":"01:09.720","Text":"Basically, this k changed into a 1."},{"Start":"01:09.720 ","End":"01:12.705","Text":"Similarly with the other variables,"},{"Start":"01:12.705 ","End":"01:16.245","Text":"but just as with x, we had 1s here,"},{"Start":"01:16.245 ","End":"01:24.870","Text":"for y, we have 1s here and similar situation for the rest of them."},{"Start":"01:24.870 ","End":"01:28.820","Text":"For z, I have the 1s in this column, for t,"},{"Start":"01:28.820 ","End":"01:32.840","Text":"I have the 1s in this column, and last,"},{"Start":"01:32.840 ","End":"01:35.840","Text":"but not least, for r,"},{"Start":"01:35.840 ","End":"01:41.784","Text":"I have all the 1s in this column."},{"Start":"01:41.784 ","End":"01:48.020","Text":"Now, let me just go back and I want to show you"},{"Start":"01:48.020 ","End":"01:55.330","Text":"how we can get from x to y. I\u0027m going to rephrase."},{"Start":"01:55.330 ","End":"01:58.535","Text":"The denominators are the same."},{"Start":"01:58.535 ","End":"02:01.580","Text":"To show that x equals y,"},{"Start":"02:01.580 ","End":"02:05.564","Text":"all I have to do is show that the numerators are the same."},{"Start":"02:05.564 ","End":"02:10.280","Text":"Now, I have the determinant of this matrix and of this matrix,"},{"Start":"02:10.280 ","End":"02:12.010","Text":"if you look at it,"},{"Start":"02:12.010 ","End":"02:18.400","Text":"we can get from here to here using a column swap and a row swap."},{"Start":"02:18.400 ","End":"02:19.720","Text":"If, for example,"},{"Start":"02:19.720 ","End":"02:24.170","Text":"I look at these 2 columns and these 2 rows,"},{"Start":"02:24.250 ","End":"02:30.745","Text":"and then if I swap this row with this row and then this column with this column,"},{"Start":"02:30.745 ","End":"02:34.520","Text":"then I\u0027ll just get this."},{"Start":"02:34.520 ","End":"02:39.505","Text":"You know what? It might just be easier to see if I just focus"},{"Start":"02:39.505 ","End":"02:48.420","Text":"on this part here and this part here because everything else stays the same."},{"Start":"02:48.420 ","End":"02:52.520","Text":"It\u0027s easier to see now that a swap of"},{"Start":"02:52.520 ","End":"02:58.775","Text":"these 2 columns followed by a swap of these 2 rows or vice versa,"},{"Start":"02:58.775 ","End":"03:06.480","Text":"I could even write that R_1 swaps with R_2 in our notation,"},{"Start":"03:06.480 ","End":"03:13.470","Text":"column 2 swaps with column 1 or vice versa."},{"Start":"03:13.470 ","End":"03:17.300","Text":"According to the rules for row operations,"},{"Start":"03:17.300 ","End":"03:22.240","Text":"a swap multiplies the determinant by minus 1."},{"Start":"03:22.240 ","End":"03:28.615","Text":"But if I multiply by minus 1 for the rows and then minus 1 for the column swap,"},{"Start":"03:28.615 ","End":"03:30.300","Text":"then all together,"},{"Start":"03:30.300 ","End":"03:31.980","Text":"minus 1 times minus 1 is 1."},{"Start":"03:31.980 ","End":"03:34.125","Text":"The determinant doesn\u0027t change."},{"Start":"03:34.125 ","End":"03:37.475","Text":"This numerator equals this numerator."},{"Start":"03:37.475 ","End":"03:42.440","Text":"Then similarly, I could show that y equals"},{"Start":"03:42.440 ","End":"03:48.280","Text":"z by focusing on this and this."},{"Start":"03:48.280 ","End":"03:54.560","Text":"Once again, swapping of column 2 and 3 and row 2 and row 3,"},{"Start":"03:54.560 ","End":"03:55.820","Text":"in other words, 1 column swap,"},{"Start":"03:55.820 ","End":"03:59.120","Text":"1 row swap will bring me from here to here."},{"Start":"03:59.120 ","End":"04:01.835","Text":"Again, minus 1 times minus 1 is 1,"},{"Start":"04:01.835 ","End":"04:04.405","Text":"and so y equals z."},{"Start":"04:04.405 ","End":"04:07.850","Text":"Next, if I look here and here,"},{"Start":"04:07.850 ","End":"04:13.900","Text":"we get that z equals t. Finally,"},{"Start":"04:13.900 ","End":"04:16.895","Text":"if we look here and here,"},{"Start":"04:16.895 ","End":"04:22.710","Text":"another row and column swap will show that t equals r,"},{"Start":"04:22.710 ","End":"04:25.690","Text":"so x equals y is y equals z, z equals t,"},{"Start":"04:25.690 ","End":"04:31.170","Text":"t equals r. That shows that they\u0027re all equal."},{"Start":"04:31.170 ","End":"04:36.130","Text":"Now, just to leave you with a few written words of what I just said,"},{"Start":"04:36.130 ","End":"04:43.430","Text":"as we go along from x to y to z to t to r,"},{"Start":"04:44.050 ","End":"04:51.665","Text":"we get the matrix in the numerator from a row swap and a column swap."},{"Start":"04:51.665 ","End":"04:57.325","Text":"Specifically, well, I just wrote it for the first pair,"},{"Start":"04:57.325 ","End":"05:00.120","Text":"C_2 swap for C_1, R_2 with R_1,"},{"Start":"05:00.120 ","End":"05:02.400","Text":"and then, later on,"},{"Start":"05:02.400 ","End":"05:09.255","Text":"3 with 2 and 3 with 2 and then 4 with 3 and then 5 with 4."},{"Start":"05:09.255 ","End":"05:11.220","Text":"We have a minus 1,"},{"Start":"05:11.220 ","End":"05:14.430","Text":"minus 1 equals 1 situation,"},{"Start":"05:14.430 ","End":"05:15.870","Text":"the determinant doesn\u0027t change,"},{"Start":"05:15.870 ","End":"05:17.400","Text":"denominators are identical,"},{"Start":"05:17.400 ","End":"05:24.215","Text":"and so all the 5 variables have identical values and that\u0027s what we had to show."},{"Start":"05:24.215 ","End":"05:27.240","Text":"We are done."}],"ID":12537}],"Thumbnail":null,"ID":7288},{"Name":"The Adjoint Matrix","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Adjoint Matrix - Intro","Duration":"9m 37s","ChapterTopicVideoID":9616,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9616.jpeg","UploadDate":"2017-07-26T08:38:58.1770000","DurationForVideoObject":"PT9M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this clip, I\u0027ll be talking about the concept of an adjoint matrix."},{"Start":"00:05.070 ","End":"00:08.970","Text":"We\u0027re only going to be concerned with square matrices."},{"Start":"00:08.970 ","End":"00:12.360","Text":"Like n by n could be 2 by 2,"},{"Start":"00:12.360 ","End":"00:15.010","Text":"3 by 3, whatever."},{"Start":"00:15.020 ","End":"00:20.080","Text":"I need to say something about notation and terminology."},{"Start":"00:20.080 ","End":"00:24.470","Text":"Because once a long time ago adjoint meant"},{"Start":"00:24.470 ","End":"00:30.060","Text":"1 thing and today the default meaning is something else."},{"Start":"00:30.100 ","End":"00:37.300","Text":"The 1 that I\u0027m going to use is called the classical adjoint matrix to avoid confusion."},{"Start":"00:37.300 ","End":"00:39.710","Text":"Even though I\u0027m just going to be calling it"},{"Start":"00:39.710 ","End":"00:43.610","Text":"the adjoint and we\u0027ll know that I mean the classical adjoint,"},{"Start":"00:43.610 ","End":"00:46.430","Text":"it\u0027s become also fashionable now not to"},{"Start":"00:46.430 ","End":"00:52.085","Text":"use the term classical adjoint for this or even adjoint,"},{"Start":"00:52.085 ","End":"00:57.170","Text":"of course, but to call it the adjugate matrix."},{"Start":"00:57.170 ","End":"01:03.395","Text":"In any event when you see the A-D-J in writing,"},{"Start":"01:03.395 ","End":"01:07.700","Text":"it always means the classical adjoint or the adjugate."},{"Start":"01:07.700 ","End":"01:10.925","Text":"There\u0027s no confusion if you write A-D-J."},{"Start":"01:10.925 ","End":"01:14.540","Text":"I could start with 2 by 2 matrices,"},{"Start":"01:14.540 ","End":"01:21.875","Text":"but actually, it\u0027s better for me to illustrate on a 3 by 3 matrix."},{"Start":"01:21.875 ","End":"01:26.390","Text":"I also want you to go back and look up"},{"Start":"01:26.390 ","End":"01:30.725","Text":"on the lesson on expanding the determinant along the row or column,"},{"Start":"01:30.725 ","End":"01:34.075","Text":"because there\u0027s 2 concepts I need you to know."},{"Start":"01:34.075 ","End":"01:39.120","Text":"The concept of a minor of an element,"},{"Start":"01:39.120 ","End":"01:40.770","Text":"and I\u0027m sure you\u0027ll remember this,"},{"Start":"01:40.770 ","End":"01:45.740","Text":"also the checkerboard pattern of pluses and"},{"Start":"01:45.740 ","End":"01:48.590","Text":"minuses which we used when we expand along"},{"Start":"01:48.590 ","End":"01:52.460","Text":"the row so make sure you are familiar with these terms."},{"Start":"01:52.460 ","End":"01:56.870","Text":"This is the plus or minus and this is what you get when you cross the row and column."},{"Start":"01:56.870 ","End":"02:01.195","Text":"Well, we\u0027ll see. Let\u0027s start with this."},{"Start":"02:01.195 ","End":"02:09.335","Text":"Let me say that the adjugate of a matrix is also going to be a matrix of the same size."},{"Start":"02:09.335 ","End":"02:11.210","Text":"Here we have a 3 by 3,"},{"Start":"02:11.210 ","End":"02:15.930","Text":"the result is going to be a 3 by 3 matrix."},{"Start":"02:15.930 ","End":"02:19.909","Text":"I\u0027ll show you the recipe of how we get to this adjoint."},{"Start":"02:19.909 ","End":"02:24.169","Text":"Let\u0027s take the top left element."},{"Start":"02:24.169 ","End":"02:31.500","Text":"I cross out the row and column because I want the minor."},{"Start":"02:32.050 ","End":"02:36.175","Text":"This is the minor, it\u0027s what\u0027s left."},{"Start":"02:36.175 ","End":"02:39.695","Text":"The sign for this element is a plus."},{"Start":"02:39.695 ","End":"02:42.485","Text":"You know what? I\u0027ll just remind you."},{"Start":"02:42.485 ","End":"02:47.180","Text":"We always take the top-left as a plus and then you alternate plus,"},{"Start":"02:47.180 ","End":"02:52.240","Text":"minus, plus, minus, plus, minus, plus."},{"Start":"02:52.240 ","End":"02:58.090","Text":"We have a plus and we have this minor."},{"Start":"02:58.090 ","End":"03:02.750","Text":"Here in place of the a and the corresponding position,"},{"Start":"03:02.750 ","End":"03:06.665","Text":"I put the plus with the minor."},{"Start":"03:06.665 ","End":"03:12.230","Text":"Next 1 in this position corresponds to this element."},{"Start":"03:12.230 ","End":"03:15.080","Text":"Once again, I take the minor,"},{"Start":"03:15.080 ","End":"03:17.210","Text":"I\u0027m not going to use all the coloring."},{"Start":"03:17.210 ","End":"03:19.880","Text":"We\u0027re going to get the minor being d,"},{"Start":"03:19.880 ","End":"03:21.715","Text":"f, g i,"},{"Start":"03:21.715 ","End":"03:24.840","Text":"and the sign is going to be a minus."},{"Start":"03:24.840 ","End":"03:27.420","Text":"We get this."},{"Start":"03:27.420 ","End":"03:30.560","Text":"I won\u0027t do all of them, it gets boring after a while."},{"Start":"03:30.560 ","End":"03:34.130","Text":"But let\u0027s suppose I want to see what corresponds to this f,"},{"Start":"03:34.130 ","End":"03:36.110","Text":"in other words, this position here."},{"Start":"03:36.110 ","End":"03:39.395","Text":"Then cross this out and this out."},{"Start":"03:39.395 ","End":"03:42.800","Text":"Then I see that the minor is a, b, g,"},{"Start":"03:42.800 ","End":"03:46.490","Text":"h, and the sign for this is the minus."},{"Start":"03:46.490 ","End":"03:50.909","Text":"Here I end up with this."},{"Start":"03:51.920 ","End":"03:56.155","Text":"Let me just give you all the rest of them."},{"Start":"03:56.155 ","End":"03:58.349","Text":"Here\u0027s the completed picture."},{"Start":"03:58.349 ","End":"04:00.270","Text":"I want to emphasize again,"},{"Start":"04:00.270 ","End":"04:04.225","Text":"because sometimes people forget the sign, the checkerboard."},{"Start":"04:04.225 ","End":"04:10.290","Text":"I\u0027ll just highlight those because sometimes people forget them."},{"Start":"04:10.460 ","End":"04:16.580","Text":"Then of course, once we have a 2 by 2 determinant,"},{"Start":"04:16.580 ","End":"04:18.740","Text":"we know how to compute them."},{"Start":"04:18.740 ","End":"04:22.265","Text":"We just take this times this minus this times this,"},{"Start":"04:22.265 ","End":"04:24.769","Text":"and then we get a numerical result."},{"Start":"04:24.769 ","End":"04:26.870","Text":"Then as an important step,"},{"Start":"04:26.870 ","End":"04:28.855","Text":"you must not forget,"},{"Start":"04:28.855 ","End":"04:31.300","Text":"after we\u0027ve filled in all of these entries,"},{"Start":"04:31.300 ","End":"04:36.610","Text":"then we take the transpose of this matrix."},{"Start":"04:36.610 ","End":"04:40.000","Text":"The transpose means swapping rows and columns."},{"Start":"04:40.000 ","End":"04:43.180","Text":"It\u0027s like a mirror imaging along the diagonal."},{"Start":"04:43.180 ","End":"04:45.175","Text":"This goes here, this goes here,"},{"Start":"04:45.175 ","End":"04:47.305","Text":"this goes here, and so on."},{"Start":"04:47.305 ","End":"04:50.080","Text":"When you see the numerical example,"},{"Start":"04:50.080 ","End":"04:52.340","Text":"it will be much clearer."},{"Start":"04:53.120 ","End":"04:58.500","Text":"Now I want to go back to the 2 by 2 case."},{"Start":"04:58.500 ","End":"05:02.580","Text":"The 2 by 2 case works similarly."},{"Start":"05:02.580 ","End":"05:06.855","Text":"We have also a 2 by 2 matrix."},{"Start":"05:06.855 ","End":"05:09.400","Text":"We\u0027re going to take the transpose at the end."},{"Start":"05:09.400 ","End":"05:11.410","Text":"We do have a checkerboard pattern."},{"Start":"05:11.410 ","End":"05:13.695","Text":"As for what goes here,"},{"Start":"05:13.695 ","End":"05:15.635","Text":"again, we use the minors."},{"Start":"05:15.635 ","End":"05:18.515","Text":"The minor of a, for example,"},{"Start":"05:18.515 ","End":"05:23.180","Text":"would be what I get when I cross out the row and column containing the a."},{"Start":"05:23.180 ","End":"05:27.980","Text":"Then all I\u0027m left with is the determinant of a single element,"},{"Start":"05:27.980 ","End":"05:29.600","Text":"and that\u0027s just the thing itself."},{"Start":"05:29.600 ","End":"05:37.920","Text":"Here I want to put a d and then the next entry,"},{"Start":"05:37.920 ","End":"05:43.935","Text":"when I cross these 2 it\u0027s going to be a c here."},{"Start":"05:43.935 ","End":"05:46.820","Text":"Then corresponding to this,"},{"Start":"05:46.820 ","End":"05:51.710","Text":"we\u0027re going to get the b from here."},{"Start":"05:51.710 ","End":"05:54.335","Text":"Finally, in the lower right,"},{"Start":"05:54.335 ","End":"05:56.990","Text":"we\u0027ll get just this a."},{"Start":"05:56.990 ","End":"06:00.740","Text":"Of course, I put the pluses and minuses in already."},{"Start":"06:00.740 ","End":"06:05.135","Text":"Now we have to take the transpose,"},{"Start":"06:05.135 ","End":"06:11.750","Text":"flipping rows and columns so that here the first row is d minus c,"},{"Start":"06:11.750 ","End":"06:13.655","Text":"so it becomes the first column."},{"Start":"06:13.655 ","End":"06:15.425","Text":"Here the second row,"},{"Start":"06:15.425 ","End":"06:18.250","Text":"minus ba becomes the second column."},{"Start":"06:18.250 ","End":"06:21.290","Text":"Notice that the entries on the diagonal never move when you"},{"Start":"06:21.290 ","End":"06:24.880","Text":"take a transpose, but the others do."},{"Start":"06:24.880 ","End":"06:30.190","Text":"Of course, I also drop the pluses on the d and the a. I don\u0027t need them anymore."},{"Start":"06:30.190 ","End":"06:34.370","Text":"This is actually the adjoint of a 2 by 2."},{"Start":"06:34.370 ","End":"06:38.390","Text":"I often just remember the formula straight from here to here."},{"Start":"06:38.390 ","End":"06:45.620","Text":"By the rule that I just flip the diagonal,"},{"Start":"06:45.620 ","End":"06:50.195","Text":"reverse the positions of these 2 set of ad,"},{"Start":"06:50.195 ","End":"06:55.625","Text":"da and put a minus on the other 2 and leave them where they are."},{"Start":"06:55.625 ","End":"06:59.585","Text":"If this is helpful and use it if not, don\u0027t."},{"Start":"06:59.585 ","End":"07:03.760","Text":"We have a 2 by 2 and a 3 by 3."},{"Start":"07:03.760 ","End":"07:07.070","Text":"As I said, we\u0027ll also be doing some numerical examples,"},{"Start":"07:07.070 ","End":"07:08.495","Text":"so we\u0027ll make it even clearer."},{"Start":"07:08.495 ","End":"07:10.625","Text":"What about 4 by 4?"},{"Start":"07:10.625 ","End":"07:17.290","Text":"Well, it\u0027s the same idea just a bit more tedious. Let\u0027s see."},{"Start":"07:17.290 ","End":"07:21.740","Text":"For 4 by 4 there are 16 entries to compute."},{"Start":"07:21.740 ","End":"07:25.520","Text":"But notice that the same algorithm,"},{"Start":"07:25.520 ","End":"07:30.970","Text":"the same recipe, we have the checkerboard."},{"Start":"07:30.970 ","End":"07:35.085","Text":"I\u0027ll highlight them, that\u0027s important."},{"Start":"07:35.085 ","End":"07:36.860","Text":"We\u0027re going to have plus, minus,"},{"Start":"07:36.860 ","End":"07:39.110","Text":"plus, minus and so on."},{"Start":"07:39.110 ","End":"07:42.605","Text":"Also, at the end,"},{"Start":"07:42.605 ","End":"07:51.095","Text":"not to forget to take the transpose and corresponding to each element,"},{"Start":"07:51.095 ","End":"07:52.835","Text":"we put a minor."},{"Start":"07:52.835 ","End":"07:56.380","Text":"I\u0027ll just illustrate 3 of them."},{"Start":"07:56.380 ","End":"08:00.065","Text":"In the top-left where we had an a,"},{"Start":"08:00.065 ","End":"08:05.720","Text":"we cross out the row and column and just take the determinant of what\u0027s left."},{"Start":"08:05.720 ","End":"08:07.730","Text":"In other words, the minor, that\u0027s the f, g,"},{"Start":"08:07.730 ","End":"08:10.560","Text":"h, j, k, l,"},{"Start":"08:10.560 ","End":"08:12.210","Text":"m, n, o,"},{"Start":"08:12.210 ","End":"08:16.650","Text":"p. Thus the sign that goes along with it."},{"Start":"08:16.650 ","End":"08:20.805","Text":"Then the next 1 is this 1."},{"Start":"08:20.805 ","End":"08:23.015","Text":"If you notice what\u0027s left,"},{"Start":"08:23.015 ","End":"08:26.090","Text":"the e, g, h, and so on, I\u0027ve copied here."},{"Start":"08:26.090 ","End":"08:28.745","Text":"By the checkerboard it has a minus."},{"Start":"08:28.745 ","End":"08:31.249","Text":"Let\u0027s take this element."},{"Start":"08:31.249 ","End":"08:33.270","Text":"The e, well,"},{"Start":"08:33.270 ","End":"08:39.740","Text":"corresponding to it what we get is a minus also from the checkerboard."},{"Start":"08:39.740 ","End":"08:40.940","Text":"Then the b, c,"},{"Start":"08:40.940 ","End":"08:42.770","Text":"d and then j,"},{"Start":"08:42.770 ","End":"08:44.435","Text":"k, l and so on."},{"Start":"08:44.435 ","End":"08:47.010","Text":"I think you get the idea."},{"Start":"08:47.090 ","End":"08:52.130","Text":"The general rule for whatever size it is is as follows."},{"Start":"08:52.130 ","End":"09:01.295","Text":"For each element, you compute its minor by crossing at a row and a column."},{"Start":"09:01.295 ","End":"09:07.390","Text":"You also look up the plus minus by simple counting plus minus,"},{"Start":"09:07.390 ","End":"09:11.125","Text":"put them together in that position."},{"Start":"09:11.125 ","End":"09:13.970","Text":"Then we get a new matrix,"},{"Start":"09:13.970 ","End":"09:16.910","Text":"also n by n, the same size."},{"Start":"09:16.910 ","End":"09:23.850","Text":"At the end we take a transpose and the result of all this is the adjoint."},{"Start":"09:25.130 ","End":"09:29.540","Text":"There will be solved examples with numbers,"},{"Start":"09:29.540 ","End":"09:33.410","Text":"but let me continue with the tutorial by giving you"},{"Start":"09:33.410 ","End":"09:38.340","Text":"some rules of the adjoint, some properties."}],"ID":9924},{"Watched":false,"Name":"The Adjoint Matrix - Rules","Duration":"4m 45s","ChapterTopicVideoID":9617,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9617.jpeg","UploadDate":"2017-07-26T08:39:10.1770000","DurationForVideoObject":"PT4M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"Now let\u0027s talk about some rules or properties for"},{"Start":"00:03.750 ","End":"00:09.285","Text":"the adjoint of a matrix. A is just a typical matrix."},{"Start":"00:09.285 ","End":"00:14.115","Text":"Let\u0027s say that it\u0027s n by n because we\u0027re going to need the dimension"},{"Start":"00:14.115 ","End":"00:19.670","Text":"and the order of the matrix, at least in one of them."},{"Start":"00:19.670 ","End":"00:22.910","Text":"The first one is this."},{"Start":"00:22.910 ","End":"00:27.260","Text":"I want to remind you the adjoint of a matrix A is the same size."},{"Start":"00:27.260 ","End":"00:28.820","Text":"If A is a 3 by 3,"},{"Start":"00:28.820 ","End":"00:34.820","Text":"so is the adjoint square matrix same size so we can multiply them together."},{"Start":"00:34.820 ","End":"00:37.130","Text":"Now it turns out if you multiply the original with"},{"Start":"00:37.130 ","End":"00:39.410","Text":"the adjoint or the adjoint with the original doesn\u0027t"},{"Start":"00:39.410 ","End":"00:44.140","Text":"matter in which order you get a constant multiple of the identity matrix."},{"Start":"00:44.140 ","End":"00:48.205","Text":"That constant is the determinant of A."},{"Start":"00:48.205 ","End":"00:52.590","Text":"But it\u0027s a constant so we can multiply a constant times the matrix."},{"Start":"00:52.810 ","End":"00:55.010","Text":"This is going to help us,"},{"Start":"00:55.010 ","End":"01:03.245","Text":"especially with rule 2 because actually this implies that if have the adjoint,"},{"Start":"01:03.245 ","End":"01:07.400","Text":"then we can have the inverse of the matrix."},{"Start":"01:07.400 ","End":"01:11.030","Text":"The inverse of the matrix is 1 over the determinant."},{"Start":"01:11.030 ","End":"01:15.250","Text":"Determinant is just a constant number times the adjoint."},{"Start":"01:15.250 ","End":"01:21.530","Text":"We compute the adjoint divided by the determinant and we\u0027ve got the inverse."},{"Start":"01:21.530 ","End":"01:27.335","Text":"Now obviously, we\u0027re assuming that the determinant is not 0,"},{"Start":"01:27.335 ","End":"01:32.990","Text":"because we learned that matrix is invertible if and only if the determinant is non 0."},{"Start":"01:32.990 ","End":"01:34.945","Text":"To even talk about this."},{"Start":"01:34.945 ","End":"01:37.050","Text":"Yeah, this is not 0,"},{"Start":"01:37.050 ","End":"01:40.175","Text":"so we divide by it and we\u0027ll see this in the examples."},{"Start":"01:40.175 ","End":"01:44.525","Text":"How we find the inverse by first finding the adjoint."},{"Start":"01:44.525 ","End":"01:52.975","Text":"Next rule is that the adjoint of the identity matrix is just the identity matrix itself."},{"Start":"01:52.975 ","End":"01:55.780","Text":"This is the identity matrix of any order."},{"Start":"01:55.780 ","End":"01:58.315","Text":"You really could put a little subscript n,"},{"Start":"01:58.315 ","End":"02:00.220","Text":"but for any size n,"},{"Start":"02:00.220 ","End":"02:02.485","Text":"the adjoint leaves it untouched."},{"Start":"02:02.485 ","End":"02:06.050","Text":"I mean, the result is the same as what you put in."},{"Start":"02:06.050 ","End":"02:10.200","Text":"This next rule is really 2 in 1."},{"Start":"02:10.200 ","End":"02:16.780","Text":"It tells us that if we take the transpose of a matrix and then take the adjoint,"},{"Start":"02:16.780 ","End":"02:22.165","Text":"or conversely, first take the adjoint and then take a transpose."},{"Start":"02:22.165 ","End":"02:24.475","Text":"It doesn\u0027t matter you\u0027ll get the same thing."},{"Start":"02:24.475 ","End":"02:26.290","Text":"The order doesn\u0027t matter."},{"Start":"02:26.290 ","End":"02:29.325","Text":"Taking transpose and taking adjoint."},{"Start":"02:29.325 ","End":"02:32.779","Text":"The second half is that instead of transpose,"},{"Start":"02:32.779 ","End":"02:34.445","Text":"we can talk about inverse."},{"Start":"02:34.445 ","End":"02:38.780","Text":"We could take the inverse of a matrix and take the adjoint of that,"},{"Start":"02:38.780 ","End":"02:44.630","Text":"or we could first take the adjoint and then do an inverse and we get the same result."},{"Start":"02:44.630 ","End":"02:46.670","Text":"This is what rule 4 says,"},{"Start":"02:46.670 ","End":"02:51.335","Text":"that whether it\u0027s transposed or whether it\u0027s inverse,"},{"Start":"02:51.335 ","End":"02:55.380","Text":"doesn\u0027t matter if you first do the adjoint or afterwards."},{"Start":"02:56.450 ","End":"03:01.325","Text":"This one\u0027s important because you\u0027re liable to get mixed up."},{"Start":"03:01.325 ","End":"03:05.510","Text":"If I have a product of 2 matrices and of course we\u0027re assuming that the same size."},{"Start":"03:05.510 ","End":"03:06.650","Text":"If this is 3 by 3,"},{"Start":"03:06.650 ","End":"03:08.610","Text":"then this is also a 3 by 3."},{"Start":"03:08.610 ","End":"03:12.775","Text":"We multiply the matrices and then we take the adjoint."},{"Start":"03:12.775 ","End":"03:16.360","Text":"Turns out that that\u0027s not quite the same"},{"Start":"03:16.360 ","End":"03:20.035","Text":"as the product of the adjoint of each 1 separately."},{"Start":"03:20.035 ","End":"03:23.350","Text":"It is but in the opposite order."},{"Start":"03:23.350 ","End":"03:25.100","Text":"Here we have AB."},{"Start":"03:25.100 ","End":"03:28.780","Text":"A matrix multiplication is not commutative."},{"Start":"03:28.780 ","End":"03:32.395","Text":"That\u0027s the mathematical way of saying that the order doesn\u0027t matter."},{"Start":"03:32.395 ","End":"03:34.675","Text":"Here, A before B,"},{"Start":"03:34.675 ","End":"03:36.160","Text":"but after the adjoints,"},{"Start":"03:36.160 ","End":"03:37.930","Text":"it\u0027s B before A."},{"Start":"03:37.930 ","End":"03:41.580","Text":"Notice this inversion because if you miss it,"},{"Start":"03:41.580 ","End":"03:43.565","Text":"you\u0027ll get the wrong answer."},{"Start":"03:43.565 ","End":"03:48.910","Text":"Finally, here is a rule and I have the feeling I\u0027ve given it to you before,"},{"Start":"03:48.910 ","End":"03:51.890","Text":"but never mind if we do it again."},{"Start":"03:51.980 ","End":"03:59.605","Text":"If we take the adjoint of a matrix and then take the determinant of that."},{"Start":"03:59.605 ","End":"04:04.930","Text":"We can compute that what it\u0027s going to be by just taking the determinant of"},{"Start":"04:04.930 ","End":"04:10.660","Text":"the original matrix before the adjoint and raising it to the power of n minus 1."},{"Start":"04:10.660 ","End":"04:16.720","Text":"Now this n is the same n as the size of the matrix."},{"Start":"04:16.720 ","End":"04:21.640","Text":"In other words, if you did this for a 4 by 4 matrix,"},{"Start":"04:21.640 ","End":"04:25.135","Text":"you want the determinant of the adjoint of a 4 by 4 matrix."},{"Start":"04:25.135 ","End":"04:29.130","Text":"You take the determinant of the original matrix and raise it to the power of 3."},{"Start":"04:29.130 ","End":"04:32.395","Text":"Why 3? Because it\u0027s 4 minus 1, for example."},{"Start":"04:32.395 ","End":"04:37.580","Text":"Anyway, we\u0027ll see plenty of the use of this and in general how finding"},{"Start":"04:37.580 ","End":"04:43.500","Text":"the adjoint in the solved exercises that follow this tutorial."},{"Start":"04:43.500 ","End":"04:45.850","Text":"We\u0027re done for now."}],"ID":9925},{"Watched":false,"Name":"Exercise 1","Duration":"3m 7s","ChapterTopicVideoID":9611,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9611.jpeg","UploadDate":"2017-07-26T08:37:05.2270000","DurationForVideoObject":"PT3M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, we\u0027re given a matrix 1, 2,"},{"Start":"00:03.510 ","End":"00:07.140","Text":"3, 4, and we have to compute first of all,"},{"Start":"00:07.140 ","End":"00:11.670","Text":"the adjoint of A and then to use that to compute the inverse of"},{"Start":"00:11.670 ","End":"00:17.655","Text":"A. I\u0027ve left here the formula for the adjoint of a 2 by 2,"},{"Start":"00:17.655 ","End":"00:19.680","Text":"general a, b, c,"},{"Start":"00:19.680 ","End":"00:21.945","Text":"d is equal to,"},{"Start":"00:21.945 ","End":"00:24.750","Text":"you can skip the middle step if you want."},{"Start":"00:24.750 ","End":"00:27.000","Text":"It\u0027s just equal to,"},{"Start":"00:27.000 ","End":"00:33.040","Text":"swap the diagonals around and make the other 2 negative."},{"Start":"00:33.580 ","End":"00:35.690","Text":"If you want to,"},{"Start":"00:35.690 ","End":"00:37.100","Text":"you could use the middle step."},{"Start":"00:37.100 ","End":"00:38.480","Text":"If you forget all the formulas,"},{"Start":"00:38.480 ","End":"00:42.560","Text":"you can always say, in the top-left,"},{"Start":"00:42.560 ","End":"00:47.570","Text":"I could cross out this and this and then use plus, minus,"},{"Start":"00:47.570 ","End":"00:54.035","Text":"plus, so it\u0027s plus and the determinant of what\u0027s left is just 4 and so on."},{"Start":"00:54.035 ","End":"00:57.050","Text":"But as I mentioned, I usually recommend that you"},{"Start":"00:57.050 ","End":"01:01.250","Text":"remember the rule for 2 by 2 determinants because it\u0027s"},{"Start":"01:01.250 ","End":"01:07.990","Text":"not that difficult where we swap these diagonals around and you make these 2 minus."},{"Start":"01:07.990 ","End":"01:09.240","Text":"Instead of 1,"},{"Start":"01:09.240 ","End":"01:10.500","Text":"4 here we have 4,"},{"Start":"01:10.500 ","End":"01:11.690","Text":"1 on the diagonal,"},{"Start":"01:11.690 ","End":"01:15.935","Text":"and instead of 2 and 3 we have minus 2 and minus 3."},{"Start":"01:15.935 ","End":"01:18.680","Text":"Now that\u0027s the adjoint."},{"Start":"01:18.680 ","End":"01:22.840","Text":"How do we use this to find the inverse?"},{"Start":"01:22.840 ","End":"01:25.235","Text":"Now from this formula,"},{"Start":"01:25.235 ","End":"01:28.840","Text":"which gives us the inverse in terms of the adjoint,"},{"Start":"01:28.840 ","End":"01:31.325","Text":"what we\u0027re missing is the determinant."},{"Start":"01:31.325 ","End":"01:36.110","Text":"The determinant would be this diagonal minus this diagonal which is"},{"Start":"01:36.110 ","End":"01:41.405","Text":"1 times 4 minus 2 times 3 or 4 minus 6,"},{"Start":"01:41.405 ","End":"01:44.760","Text":"and the adjoint I just copy from here."},{"Start":"01:45.290 ","End":"01:50.515","Text":"That\u0027s the answer, but we need to simplify it a bit or we should."},{"Start":"01:50.515 ","End":"01:56.035","Text":"This constant is just 1 over minus 2 or minus 1/2."},{"Start":"01:56.035 ","End":"01:59.555","Text":"Then I can multiply this by each of the entries,"},{"Start":"01:59.555 ","End":"02:04.930","Text":"and then I get 4 times minus 1/2 is minus 2, and so on."},{"Start":"02:04.930 ","End":"02:09.004","Text":"These 2 come out either with fraction or you can write them in decimal,"},{"Start":"02:09.004 ","End":"02:14.940","Text":"like minus 3 over minus 2 is 1 and 1/2, and so on."},{"Start":"02:14.950 ","End":"02:17.840","Text":"I would also recommend,"},{"Start":"02:17.840 ","End":"02:19.820","Text":"you don\u0027t have to, this is the answer."},{"Start":"02:19.820 ","End":"02:24.050","Text":"But recommend checking that this is indeed the inverse by"},{"Start":"02:24.050 ","End":"02:28.895","Text":"multiplying by the original and the original is 1, 2, 3, 4."},{"Start":"02:28.895 ","End":"02:31.145","Text":"You can multiply on the left or on the right."},{"Start":"02:31.145 ","End":"02:34.430","Text":"If you do the multiplication of matrices like here,"},{"Start":"02:34.430 ","End":"02:39.515","Text":"we have this row here with this column here,"},{"Start":"02:39.515 ","End":"02:49.490","Text":"1 times minus 2 plus twice 1 and 1/2 is minus 2 plus 3,"},{"Start":"02:49.490 ","End":"02:51.455","Text":"which is 1, and so on."},{"Start":"02:51.455 ","End":"02:56.190","Text":"1, 2 times 1 minus a 1/2 is 1,"},{"Start":"02:56.190 ","End":"02:58.800","Text":"minus 1 is 0, and so on."},{"Start":"02:58.800 ","End":"03:02.450","Text":"The check is just to give us"},{"Start":"03:02.450 ","End":"03:08.160","Text":"more confidence that this is the correct answer. We\u0027re done."}],"ID":9926},{"Watched":false,"Name":"Exercise 2","Duration":"4m 35s","ChapterTopicVideoID":9612,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9612.jpeg","UploadDate":"2017-07-26T08:37:20.7000000","DurationForVideoObject":"PT4M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.675","Text":"In this exercise, we\u0027re given a 3 by 3 matrix."},{"Start":"00:03.675 ","End":"00:09.360","Text":"This is the one and we want to compute the adjoint of this."},{"Start":"00:09.360 ","End":"00:13.935","Text":"Then using the adjoint to compute the inverse."},{"Start":"00:13.935 ","End":"00:16.440","Text":"Now there is the formula we had before,"},{"Start":"00:16.440 ","End":"00:18.900","Text":"but of course we don\u0027t do it by the formula,"},{"Start":"00:18.900 ","End":"00:22.565","Text":"just a massive letters,"},{"Start":"00:22.565 ","End":"00:25.470","Text":"just remember the principle."},{"Start":"00:25.520 ","End":"00:30.000","Text":"We have a 3 by 3 we\u0027re expecting to get a 3 by 3."},{"Start":"00:30.000 ","End":"00:38.805","Text":"What we do is the checkerboard pattern is important and I\u0027ll highlight that in 1 color."},{"Start":"00:38.805 ","End":"00:41.174","Text":"At the very end,"},{"Start":"00:41.174 ","End":"00:46.470","Text":"we\u0027re also going to do a transpose of the matrix."},{"Start":"00:46.470 ","End":"00:51.825","Text":"In between I\u0027ll illustrate and position row 3, column 2."},{"Start":"00:51.825 ","End":"00:53.720","Text":"I\u0027m talking about it also here,"},{"Start":"00:53.720 ","End":"00:55.460","Text":"row 3, column 2."},{"Start":"00:55.460 ","End":"01:00.155","Text":"We have the sign already what we do is we"},{"Start":"01:00.155 ","End":"01:06.185","Text":"cross out the row and column of this position to get the minor."},{"Start":"01:06.185 ","End":"01:07.940","Text":"The minor is what\u0027s left,"},{"Start":"01:07.940 ","End":"01:09.020","Text":"it\u0027s a, d,"},{"Start":"01:09.020 ","End":"01:11.810","Text":"c, f. Or if you look at it,"},{"Start":"01:11.810 ","End":"01:13.490","Text":"if you do it in the regular order, a,"},{"Start":"01:13.490 ","End":"01:15.770","Text":"c, d, f and here we have a,"},{"Start":"01:15.770 ","End":"01:17.975","Text":"c, d, f,"},{"Start":"01:17.975 ","End":"01:20.285","Text":"and so on through all of them."},{"Start":"01:20.285 ","End":"01:24.170","Text":"Then we also numerically compute each of these."},{"Start":"01:24.170 ","End":"01:27.020","Text":"For example, here we would have a,"},{"Start":"01:27.020 ","End":"01:28.370","Text":"f minus d,"},{"Start":"01:28.370 ","End":"01:37.105","Text":"c. Let\u0027s go to our particular matrix and do that."},{"Start":"01:37.105 ","End":"01:44.000","Text":"Again, I\u0027ll demonstrate on this position we have this row 3, column 2."},{"Start":"01:44.000 ","End":"01:48.230","Text":"We strike out the row and"},{"Start":"01:48.230 ","End":"01:56.620","Text":"the column and that gives us this and this, which is here."},{"Start":"01:56.620 ","End":"02:01.785","Text":"Then we have the checkerboard sign from here."},{"Start":"02:01.785 ","End":"02:08.435","Text":"This particular relevance is going to be 2 times minus 1 is minus 2,"},{"Start":"02:08.435 ","End":"02:10.605","Text":"less 0, it\u0027s minus 2,"},{"Start":"02:10.605 ","End":"02:14.070","Text":"but times the minus we\u0027ll get a plus 2."},{"Start":"02:14.070 ","End":"02:19.935","Text":"In fact, that is this here."},{"Start":"02:19.935 ","End":"02:21.480","Text":"I left it as minus, minus,"},{"Start":"02:21.480 ","End":"02:22.725","Text":"but the answer is 2."},{"Start":"02:22.725 ","End":"02:24.950","Text":"Similarly for all the others,"},{"Start":"02:24.950 ","End":"02:26.839","Text":"we take the pluses and minuses,"},{"Start":"02:26.839 ","End":"02:32.265","Text":"we compute the 2 by 2 determinants and we get all these values."},{"Start":"02:32.265 ","End":"02:37.970","Text":"Then we still have to do the transpose,"},{"Start":"02:37.970 ","End":"02:40.895","Text":"which means reversing rows and columns,"},{"Start":"02:40.895 ","End":"02:46.025","Text":"like the first column here is 8 minus 1 minus 3."},{"Start":"02:46.025 ","End":"02:50.270","Text":"Here we have the 8 minus 1 minus 3,"},{"Start":"02:50.270 ","End":"02:52.445","Text":"minus 5, 1,"},{"Start":"02:52.445 ","End":"02:54.350","Text":"2, minus 5, 1,"},{"Start":"02:54.350 ","End":"02:59.540","Text":"2 and then this is the answer for the adjoint."},{"Start":"02:59.540 ","End":"03:08.520","Text":"But we\u0027re not done because we have to use the adjoint now to get the inverse matrix."},{"Start":"03:09.770 ","End":"03:13.400","Text":"We use the formula that the inverse is like"},{"Start":"03:13.400 ","End":"03:17.525","Text":"the adjoint except we have to divide by the determinant."},{"Start":"03:17.525 ","End":"03:23.240","Text":"Now, just take it on trust for the moment that the determinant of A is 1,"},{"Start":"03:23.240 ","End":"03:25.895","Text":"I\u0027ll show you the computation in a second."},{"Start":"03:25.895 ","End":"03:31.420","Text":"This gives us 1/1 and here\u0027s the adjoint."},{"Start":"03:31.420 ","End":"03:37.335","Text":"I guess I could just erase the 1/1,"},{"Start":"03:37.335 ","End":"03:38.835","Text":"and say this is the answer."},{"Start":"03:38.835 ","End":"03:44.220","Text":"I\u0027ll just show you why this is equal to 1."},{"Start":"03:44.220 ","End":"03:46.275","Text":"Here\u0027s the computation,"},{"Start":"03:46.275 ","End":"03:53.490","Text":"this is the original matrix A. I\u0027ll expand along the first column,"},{"Start":"03:53.490 ","End":"03:55.544","Text":"so I get 2 entries,"},{"Start":"03:55.544 ","End":"03:57.480","Text":"1 for the 2,"},{"Start":"03:57.480 ","End":"04:02.300","Text":"and then it gives me plus 2 from the checkerboard of plus."},{"Start":"04:02.300 ","End":"04:05.330","Text":"Then the determinant of what\u0027s left,"},{"Start":"04:05.330 ","End":"04:12.820","Text":"2 times 3 minus 2 times negative 1 is 6 minus minus 2."},{"Start":"04:12.820 ","End":"04:17.250","Text":"For the 5, I get also a plus from the checkerboard,"},{"Start":"04:17.250 ","End":"04:19.125","Text":"a 5 from the entry,"},{"Start":"04:19.125 ","End":"04:23.195","Text":"and the minor, which is 1 times minus 1,"},{"Start":"04:23.195 ","End":"04:25.705","Text":"less 2 times 1."},{"Start":"04:25.705 ","End":"04:28.670","Text":"If you do the arithmetic here,"},{"Start":"04:28.670 ","End":"04:30.860","Text":"the answer comes out to be 1,"},{"Start":"04:30.860 ","End":"04:32.540","Text":"which is what we wanted here,"},{"Start":"04:32.540 ","End":"04:35.280","Text":"and so we are done."}],"ID":9927},{"Watched":false,"Name":"Exercise 3","Duration":"6m ","ChapterTopicVideoID":9613,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9613.jpeg","UploadDate":"2017-07-26T08:37:36.5930000","DurationForVideoObject":"PT6M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.400","Text":"In this exercise, we have to compute the adjoint of a 4 by"},{"Start":"00:05.400 ","End":"00:14.520","Text":"4 matrix A here and then to use this result to find the inverse of A."},{"Start":"00:14.520 ","End":"00:17.310","Text":"Now, the 4 by 4 is a lot of work,"},{"Start":"00:17.310 ","End":"00:19.095","Text":"just look at this,"},{"Start":"00:19.095 ","End":"00:22.275","Text":"if I just write everything here."},{"Start":"00:22.275 ","End":"00:27.510","Text":"Let\u0027s just concentrate on a couple of entries."},{"Start":"00:27.510 ","End":"00:34.695","Text":"Also, notice that I chose lots of 1s and 0s so it\u0027s going to be easy to compute."},{"Start":"00:34.695 ","End":"00:38.415","Text":"Let\u0027s say I want the top left entry,"},{"Start":"00:38.415 ","End":"00:43.235","Text":"so I cross out the row and the column."},{"Start":"00:43.235 ","End":"00:48.755","Text":"What we\u0027re left with is this 3 by 3,"},{"Start":"00:48.755 ","End":"00:51.280","Text":"which I write here."},{"Start":"00:51.280 ","End":"00:53.910","Text":"Then there\u0027s also the sign, the checkerboard,"},{"Start":"00:53.910 ","End":"00:56.885","Text":"it always starts with a plus at the top left."},{"Start":"00:56.885 ","End":"01:01.475","Text":"Then we compute the determinant."},{"Start":"01:01.475 ","End":"01:03.200","Text":"The computation of this,"},{"Start":"01:03.200 ","End":"01:08.270","Text":"I could do by expansion along the first row or column,"},{"Start":"01:08.270 ","End":"01:09.650","Text":"and then we only get this,"},{"Start":"01:09.650 ","End":"01:12.980","Text":"and then we get the minor,"},{"Start":"01:12.980 ","End":"01:16.615","Text":"which is 1 minus 1 is 0."},{"Start":"01:16.615 ","End":"01:19.410","Text":"Here is plus times 1 times 0,"},{"Start":"01:19.410 ","End":"01:22.320","Text":"in any event it\u0027s 0."},{"Start":"01:22.320 ","End":"01:25.985","Text":"As another example, let\u0027s take the 1 here,"},{"Start":"01:25.985 ","End":"01:28.580","Text":"4th row, 3rd column."},{"Start":"01:28.580 ","End":"01:30.500","Text":"This is the entry I mean,"},{"Start":"01:30.500 ","End":"01:35.230","Text":"so erase the column and the row."},{"Start":"01:35.230 ","End":"01:42.735","Text":"We\u0027re left with a minor which is a 3 by 3,"},{"Start":"01:42.735 ","End":"01:44.010","Text":"and that\u0027s this here,"},{"Start":"01:44.010 ","End":"01:46.590","Text":"1, 0, 0, 0,"},{"Start":"01:46.590 ","End":"01:51.555","Text":"1, 0 here, 1, 0, 1 here."},{"Start":"01:51.555 ","End":"01:54.270","Text":"The sign here is the minus."},{"Start":"01:54.270 ","End":"01:59.830","Text":"[inaudible] just highlight this part here."},{"Start":"01:59.890 ","End":"02:04.070","Text":"I could compute this by expanding, let\u0027s say,"},{"Start":"02:04.070 ","End":"02:07.820","Text":"along the third column,"},{"Start":"02:07.820 ","End":"02:09.440","Text":"could have also used the second column."},{"Start":"02:09.440 ","End":"02:11.060","Text":"Let\u0027s take the third column,"},{"Start":"02:11.060 ","End":"02:14.910","Text":"cross out the row and column."},{"Start":"02:14.910 ","End":"02:18.145","Text":"What we\u0027re left with is,"},{"Start":"02:18.145 ","End":"02:29.275","Text":"this part is a plus times the 1 and the determinant of what\u0027s left is 1 minus 0 is 1,"},{"Start":"02:29.275 ","End":"02:31.740","Text":"but there is a minus here,"},{"Start":"02:31.740 ","End":"02:35.790","Text":"so this 1 comes out to be minus 1."},{"Start":"02:35.790 ","End":"02:42.080","Text":"Here is the result of doing all of them."},{"Start":"02:42.080 ","End":"02:44.435","Text":"Notice that here the 2 that we did,"},{"Start":"02:44.435 ","End":"02:47.480","Text":"we did this 0, we did this minus 1."},{"Start":"02:47.480 ","End":"02:49.790","Text":"Here is the other 14,"},{"Start":"02:49.790 ","End":"02:51.665","Text":"so we have 16 altogether."},{"Start":"02:51.665 ","End":"02:55.675","Text":"Then remember the transpose."},{"Start":"02:55.675 ","End":"03:01.175","Text":"The first column becomes the first row and so on, just swap everything."},{"Start":"03:01.175 ","End":"03:06.425","Text":"Or, you could take a mirror image along this diagonal. This is what we get."},{"Start":"03:06.425 ","End":"03:07.970","Text":"That\u0027s the adjoint of A,"},{"Start":"03:07.970 ","End":"03:10.880","Text":"but we\u0027re not done because we have the second part,"},{"Start":"03:10.880 ","End":"03:14.430","Text":"which is to find the inverse of A."},{"Start":"03:14.900 ","End":"03:24.065","Text":"Remember, the formula for the inverse is 1 over the determinant times the adjoint."},{"Start":"03:24.065 ","End":"03:27.210","Text":"We already have the adjoint,"},{"Start":"03:28.060 ","End":"03:34.470","Text":"but we also need to compute the determinant before we can continue here."},{"Start":"03:34.490 ","End":"03:37.980","Text":"I know this is going to come out minus 1,"},{"Start":"03:37.980 ","End":"03:39.390","Text":"I\u0027ll show you in a moment."},{"Start":"03:39.390 ","End":"03:45.930","Text":"Meanwhile, let\u0027s just take it on trust that it\u0027s going to be minus 1."},{"Start":"03:52.700 ","End":"03:56.310","Text":"We have to take this 1 over minus 1, which is minus 1,"},{"Start":"03:56.310 ","End":"03:59.610","Text":"and multiply by the 4 by 4 matrix,"},{"Start":"03:59.610 ","End":"04:02.055","Text":"which means multiplying each entry."},{"Start":"04:02.055 ","End":"04:04.500","Text":"This just means changing all the signs."},{"Start":"04:04.500 ","End":"04:06.180","Text":"The 0s stay 0s,"},{"Start":"04:06.180 ","End":"04:07.820","Text":"but wherever there was a minus 1,"},{"Start":"04:07.820 ","End":"04:09.440","Text":"there\u0027s a 1, and wherever there was a 1,"},{"Start":"04:09.440 ","End":"04:10.850","Text":"it\u0027s a minus 1."},{"Start":"04:10.850 ","End":"04:12.230","Text":"This is the answer,"},{"Start":"04:12.230 ","End":"04:20.465","Text":"this is the inverse matrix of A. I still have to show you how I got to this minus 1."},{"Start":"04:20.465 ","End":"04:24.515","Text":"Here\u0027s the matrix itself."},{"Start":"04:24.515 ","End":"04:30.325","Text":"I\u0027m going to bring this to upper triangular form with row and column swaps."},{"Start":"04:30.325 ","End":"04:34.345","Text":"I might just say row and column ops operations."},{"Start":"04:34.345 ","End":"04:37.370","Text":"For example, to get from here to here,"},{"Start":"04:37.370 ","End":"04:40.250","Text":"I could take row 3,"},{"Start":"04:40.250 ","End":"04:47.420","Text":"which is this, and subtract row 1 and put the result in row 3."},{"Start":"04:47.420 ","End":"04:50.240","Text":"In other words, subtract this row from this row."},{"Start":"04:50.240 ","End":"04:54.200","Text":"This 1 and this 1 make it 0 here and here."},{"Start":"04:54.200 ","End":"04:59.660","Text":"Now, I\u0027m going to swap these 2 rows around."},{"Start":"04:59.660 ","End":"05:01.385","Text":"But remember when you swap,"},{"Start":"05:01.385 ","End":"05:03.945","Text":"you have to put a minus."},{"Start":"05:03.945 ","End":"05:06.290","Text":"Here\u0027s the minus from the swap."},{"Start":"05:06.290 ","End":"05:13.470","Text":"I should have written it as row 3 swaps with row 4."},{"Start":"05:13.470 ","End":"05:14.940","Text":"The 0, 0, 1,"},{"Start":"05:14.940 ","End":"05:17.235","Text":"1 is here and this is over here."},{"Start":"05:17.235 ","End":"05:26.005","Text":"Now, look, this is an upper triangular matrix because below the diagonal it\u0027s all 0s."},{"Start":"05:26.005 ","End":"05:32.930","Text":"All we have to do next is multiply the elements along the diagonal."},{"Start":"05:32.930 ","End":"05:37.370","Text":"1 times 1 times 1 times 1 is 1 together with this minus 1,"},{"Start":"05:37.370 ","End":"05:40.880","Text":"and the answer is minus 1."},{"Start":"05:40.880 ","End":"05:44.450","Text":"That\u0027s the minus 1 that I owed you from here,"},{"Start":"05:44.450 ","End":"05:50.500","Text":"but this is the answer for the inverse matrix."},{"Start":"05:50.900 ","End":"05:55.140","Text":"This was the adjoint. Hope I have them together."},{"Start":"05:55.140 ","End":"05:59.950","Text":"The adjoint and the inverse. We are done."}],"ID":9928},{"Watched":false,"Name":"Exercise 4","Duration":"7m 54s","ChapterTopicVideoID":9614,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9614.jpeg","UploadDate":"2017-07-26T08:38:00.4730000","DurationForVideoObject":"PT7M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.445","Text":"In this exercise, we have a 5 by 5 matrix."},{"Start":"00:05.445 ","End":"00:08.670","Text":"It would be unfair to ask you to compute"},{"Start":"00:08.670 ","End":"00:11.970","Text":"the adjoint of A because it would be too much work,"},{"Start":"00:11.970 ","End":"00:17.190","Text":"but instead, you just have to compute 1 entry in it."},{"Start":"00:17.190 ","End":"00:20.040","Text":"This 1,5 is the notation."},{"Start":"00:20.040 ","End":"00:21.885","Text":"Let me give you an example."},{"Start":"00:21.885 ","End":"00:28.620","Text":"If I was talking about the original matrix and I said A_2,3,"},{"Start":"00:28.620 ","End":"00:35.220","Text":"that would mean that the row number is 2 and the column number is 3,"},{"Start":"00:35.220 ","End":"00:37.410","Text":"so I\u0027d go 2 then 1,"},{"Start":"00:37.410 ","End":"00:40.140","Text":"2, 3, that would be 87."},{"Start":"00:40.140 ","End":"00:42.045","Text":"This is just notation."},{"Start":"00:42.045 ","End":"00:47.895","Text":"This would actually mean the top-right entry,"},{"Start":"00:47.895 ","End":"00:50.640","Text":"because, first row, fifth column,"},{"Start":"00:50.640 ","End":"00:53.900","Text":"so the top right entry of the adjoint of A,"},{"Start":"00:53.900 ","End":"01:01.295","Text":"we have to compute it and also compute the top right entry in the inverse of A."},{"Start":"01:01.295 ","End":"01:04.130","Text":"Then it won\u0027t be so much work."},{"Start":"01:04.130 ","End":"01:08.480","Text":"Now, remember that when we take an adjoint,"},{"Start":"01:08.480 ","End":"01:17.910","Text":"the last step we do is to take the transpose of some computed values."},{"Start":"01:17.910 ","End":"01:21.650","Text":"We compute a matrix and at the end take its transpose."},{"Start":"01:21.650 ","End":"01:26.405","Text":"If I\u0027m interested in the 1,5 position,"},{"Start":"01:26.405 ","End":"01:29.435","Text":"which is the top right,"},{"Start":"01:29.435 ","End":"01:32.525","Text":"then before the transpose,"},{"Start":"01:32.525 ","End":"01:39.260","Text":"this entry would have been at the bottom left in the 5,1 position."},{"Start":"01:39.260 ","End":"01:42.140","Text":"Now just before taking the transpose,"},{"Start":"01:42.140 ","End":"01:48.980","Text":"what we do in each position is take a sign from the checkerboard, and in this case,"},{"Start":"01:48.980 ","End":"01:53.360","Text":"it would be, looking at it from here, plus,"},{"Start":"01:53.360 ","End":"01:56.195","Text":"minus, plus, minus, plus,"},{"Start":"01:56.195 ","End":"01:58.700","Text":"that\u0027s this plus here."},{"Start":"01:58.700 ","End":"02:05.395","Text":"There\u0027s also a minor that we strike out a column and a row."},{"Start":"02:05.395 ","End":"02:11.960","Text":"Then we take a determinant of what\u0027s left and"},{"Start":"02:11.960 ","End":"02:20.975","Text":"that\u0027s also called M_5,1 means the minor."},{"Start":"02:20.975 ","End":"02:28.205","Text":"Let me write it. M is the minors of the entries in A."},{"Start":"02:28.205 ","End":"02:29.690","Text":"In other words, if this is A,"},{"Start":"02:29.690 ","End":"02:34.850","Text":"I replace each value by its minor,"},{"Start":"02:34.850 ","End":"02:37.100","Text":"then that\u0027s what I mean by M_5,1."},{"Start":"02:37.100 ","End":"02:39.200","Text":"That would be this part."},{"Start":"02:39.200 ","End":"02:40.610","Text":"Then this column here,"},{"Start":"02:40.610 ","End":"02:42.870","Text":"that would be this here."},{"Start":"02:43.070 ","End":"02:49.030","Text":"The answer boils down to just computing this minor."},{"Start":"02:49.030 ","End":"02:53.650","Text":"I need to scroll up screen."},{"Start":"02:53.650 ","End":"02:56.500","Text":"But here it is, I copied it."},{"Start":"02:56.500 ","End":"02:59.200","Text":"That 4 by 4 thing."},{"Start":"02:59.200 ","End":"03:03.770","Text":"Now, it\u0027s not exactly"},{"Start":"03:04.260 ","End":"03:12.205","Text":"a triangular matrix because triangular means relative to the main diagonal,"},{"Start":"03:12.205 ","End":"03:19.330","Text":"but it looks triangular if you look at the secondary minor diagonal."},{"Start":"03:19.330 ","End":"03:21.730","Text":"This can easily be fixed."},{"Start":"03:21.730 ","End":"03:25.780","Text":"We\u0027d like to have it to be triangular, upper or lower."},{"Start":"03:25.780 ","End":"03:29.210","Text":"What we could do, for example,"},{"Start":"03:29.210 ","End":"03:34.160","Text":"we could swap this row with this row,"},{"Start":"03:34.160 ","End":"03:36.665","Text":"and this row with this row."},{"Start":"03:36.665 ","End":"03:40.910","Text":"If we did that swap, those 2 swaps,"},{"Start":"03:40.910 ","End":"03:43.130","Text":"then we get this,"},{"Start":"03:43.130 ","End":"03:49.410","Text":"which is a lower triangular matrix."},{"Start":"03:53.050 ","End":"03:57.725","Text":"Above the main diagonal,"},{"Start":"03:57.725 ","End":"03:58.880","Text":"above and to the right,"},{"Start":"03:58.880 ","End":"04:01.955","Text":"it\u0027s all 0s. This is it."},{"Start":"04:01.955 ","End":"04:05.580","Text":"This is a lower triangular matrix."},{"Start":"04:05.580 ","End":"04:13.400","Text":"That means we can get the determinant by multiplying the entries along the diagonal,"},{"Start":"04:13.400 ","End":"04:16.820","Text":"2 times 3 times 4 times 10,"},{"Start":"04:16.820 ","End":"04:20.090","Text":"which comes out to be 240."},{"Start":"04:20.090 ","End":"04:24.290","Text":"That would be the answer for the first part."},{"Start":"04:24.290 ","End":"04:28.335","Text":"Is that the 1,5 position,"},{"Start":"04:28.335 ","End":"04:32.810","Text":"the top right entry in the adjoint,"},{"Start":"04:32.810 ","End":"04:37.340","Text":"is 240 because this is 240,"},{"Start":"04:37.340 ","End":"04:40.495","Text":"and then we transpose and it goes here."},{"Start":"04:40.495 ","End":"04:42.805","Text":"Let\u0027s do the second part,"},{"Start":"04:42.805 ","End":"04:49.670","Text":"which is also the 1,5 position but this time it\u0027s going to"},{"Start":"04:49.670 ","End":"04:59.975","Text":"be not of the adjoint but of the inverse matrix."},{"Start":"04:59.975 ","End":"05:03.250","Text":"We use the same formula as we\u0027ve used"},{"Start":"05:03.250 ","End":"05:07.840","Text":"before that the inverse is 1 over the determinant times"},{"Start":"05:07.840 ","End":"05:15.965","Text":"the adjoint except that we apply it only to the 1,5 position."},{"Start":"05:15.965 ","End":"05:22.070","Text":"Now, we already have the 1,5 position for the adjoint of A,"},{"Start":"05:22.070 ","End":"05:26.430","Text":"that was the 240,"},{"Start":"05:26.630 ","End":"05:32.195","Text":"so I just need to divide by the determinant of A."},{"Start":"05:32.195 ","End":"05:35.450","Text":"Now I\u0027m going to give you the answer to the determinant"},{"Start":"05:35.450 ","End":"05:38.480","Text":"just so as not to break the flow at the end."},{"Start":"05:38.480 ","End":"05:43.505","Text":"I\u0027ll compute the determinant of A and show you that this is what it is."},{"Start":"05:43.505 ","End":"05:46.830","Text":"I didn\u0027t even multiply out the factors."},{"Start":"05:46.830 ","End":"05:50.380","Text":"We don\u0027t need to because stuff will cancel."},{"Start":"05:50.570 ","End":"05:55.055","Text":"Everything here is in here except that there\u0027s an extra 2 here,"},{"Start":"05:55.055 ","End":"05:59.110","Text":"so the answer is 1/ 2."},{"Start":"05:59.110 ","End":"06:03.410","Text":"That\u0027s the answer but I still have this debt to show you how I"},{"Start":"06:03.410 ","End":"06:07.925","Text":"got to this product for the determinant of A."},{"Start":"06:07.925 ","End":"06:13.560","Text":"Here\u0027s A, I just copied it from the beginning."},{"Start":"06:13.820 ","End":"06:20.485","Text":"We\u0027re going to use the same trick as we did before with the triangular."},{"Start":"06:20.485 ","End":"06:25.235","Text":"It is triangular-shaped but it\u0027s the wrong diagonal."},{"Start":"06:25.235 ","End":"06:28.190","Text":"We need the main diagonal."},{"Start":"06:28.190 ","End":"06:31.385","Text":"If we do 2 row operations,"},{"Start":"06:31.385 ","End":"06:35.425","Text":"change this with this and this with this,"},{"Start":"06:35.425 ","End":"06:38.905","Text":"which I can also write in the more precise notation,"},{"Start":"06:38.905 ","End":"06:41.290","Text":"row 1 swaps with row 5,"},{"Start":"06:41.290 ","End":"06:44.170","Text":"row 2 with row 4, now,"},{"Start":"06:44.170 ","End":"06:50.725","Text":"each of these causes the determinant to be multiplied by minus 1."},{"Start":"06:50.725 ","End":"06:56.500","Text":"But minus 1 times minus 1 is 1,"},{"Start":"06:56.500 ","End":"07:00.155","Text":"and so the determinant doesn\u0027t change."},{"Start":"07:00.155 ","End":"07:04.615","Text":"I think I didn\u0027t mention this before when we also had 2 swaps"},{"Start":"07:04.615 ","End":"07:09.175","Text":"so I\u0027m retroactively mentioning it now."},{"Start":"07:09.175 ","End":"07:15.480","Text":"What do we get to after we do these 2 swaps? Here it is."},{"Start":"07:15.480 ","End":"07:21.230","Text":"This time it is a triangular matrix, a lower triangular."},{"Start":"07:21.230 ","End":"07:22.850","Text":"We\u0027re used to having upper triangular,"},{"Start":"07:22.850 ","End":"07:24.560","Text":"but it\u0027s lower triangular."},{"Start":"07:24.560 ","End":"07:28.495","Text":"Everything above the main diagonal is 0."},{"Start":"07:28.495 ","End":"07:31.170","Text":"Put a line here."},{"Start":"07:31.170 ","End":"07:33.150","Text":"This is the diagonal,"},{"Start":"07:33.150 ","End":"07:36.705","Text":"and above it to the right, it\u0027s all 0."},{"Start":"07:36.705 ","End":"07:45.145","Text":"In such a case, the determinant is the product of the entries on the diagonal,"},{"Start":"07:45.145 ","End":"07:47.235","Text":"which is just this."},{"Start":"07:47.235 ","End":"07:48.630","Text":"This is what we had here,"},{"Start":"07:48.630 ","End":"07:50.614","Text":"so we\u0027re all okay."},{"Start":"07:50.614 ","End":"07:52.610","Text":"We\u0027ve verified that."},{"Start":"07:52.610 ","End":"07:55.110","Text":"So we\u0027re done."}],"ID":9929},{"Watched":false,"Name":"Exercise 5 Part a","Duration":"5m 5s","ChapterTopicVideoID":9615,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9615.jpeg","UploadDate":"2017-07-26T08:38:16.1770000","DurationForVideoObject":"PT5M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"This exercise has 4 parts, a, b, c, d,"},{"Start":"00:03.630 ","End":"00:06.990","Text":"and we\u0027ll just take each part separately."},{"Start":"00:06.990 ","End":"00:09.270","Text":"For example, in part a,"},{"Start":"00:09.270 ","End":"00:13.005","Text":"we\u0027re given that A is a square matrix."},{"Start":"00:13.005 ","End":"00:17.490","Text":"I\u0027ll just write a few elements, a1,1."},{"Start":"00:17.490 ","End":"00:19.200","Text":"We don\u0027t know its size."},{"Start":"00:19.200 ","End":"00:20.520","Text":"Size n by n,"},{"Start":"00:20.520 ","End":"00:23.640","Text":"let\u0027s say so here a n,1 here a, n,"},{"Start":"00:23.640 ","End":"00:29.175","Text":"n. In general, a typical element will be a i, j."},{"Start":"00:29.175 ","End":"00:32.275","Text":"We are given 2 things."},{"Start":"00:32.275 ","End":"00:38.250","Text":"Well, we\u0027re given that determinant of A is equal to 1."},{"Start":"00:38.250 ","End":"00:44.300","Text":"But when we say prove that if all the elements are integers,"},{"Start":"00:44.300 ","End":"00:49.670","Text":"it\u0027s like we\u0027re given that these are all integers."},{"Start":"00:49.670 ","End":"00:52.580","Text":"Integers means whole numbers,"},{"Start":"00:52.580 ","End":"00:54.485","Text":"but could be negative."},{"Start":"00:54.485 ","End":"01:02.990","Text":"Then what we have to prove is that A inverse is also made up of only integers."},{"Start":"01:02.990 ","End":"01:08.120","Text":"Now the special thing about the determinant being 1 is that"},{"Start":"01:08.120 ","End":"01:13.270","Text":"I claim that the adjoint is the same as the inverse."},{"Start":"01:13.270 ","End":"01:15.360","Text":"I\u0027ll show you what I mean."},{"Start":"01:15.360 ","End":"01:18.760","Text":"A inverse, we know there\u0027s a formula."},{"Start":"01:18.760 ","End":"01:23.210","Text":"It\u0027s 1 over the determinant of A,"},{"Start":"01:23.210 ","End":"01:30.405","Text":"instead of writing in det I wrote it in bars times the adjoint of A."},{"Start":"01:30.405 ","End":"01:34.435","Text":"But if the determinant of A is 1,"},{"Start":"01:34.435 ","End":"01:42.120","Text":"then this is just equal to the adjoint of A."},{"Start":"01:42.120 ","End":"01:44.099","Text":"Because of this equality,"},{"Start":"01:44.099 ","End":"01:51.265","Text":"I could rephrase the question to show that the elements instead of inverse of A,"},{"Start":"01:51.265 ","End":"01:57.910","Text":"I could take the adjoint of A and show that its elements are integers because we\u0027ve just"},{"Start":"01:57.910 ","End":"02:00.640","Text":"shown that the adjoint and the inverse happen to be the"},{"Start":"02:00.640 ","End":"02:05.195","Text":"same because of this special condition that we have."},{"Start":"02:05.195 ","End":"02:08.965","Text":"Now I\u0027m going to remind you how we take an adjoint."},{"Start":"02:08.965 ","End":"02:10.765","Text":"We take an adjoint."},{"Start":"02:10.765 ","End":"02:14.920","Text":"What we do is, there\u0027s actually 3 stages you could think of it."},{"Start":"02:14.920 ","End":"02:17.640","Text":"The first stage is I take each element."},{"Start":"02:17.640 ","End":"02:25.685","Text":"Let\u0027s take this one and replace it or let\u0027s put here, it\u0027s minor."},{"Start":"02:25.685 ","End":"02:28.535","Text":"The minor I\u0027m going to call M, i, j."},{"Start":"02:28.535 ","End":"02:33.230","Text":"The M, i, j is the determinant we get when we"},{"Start":"02:33.230 ","End":"02:38.495","Text":"strike out the row and column containing that element, that\u0027s the minor."},{"Start":"02:38.495 ","End":"02:40.015","Text":"But it\u0027s a determinant."},{"Start":"02:40.015 ","End":"02:44.800","Text":"It\u0027s a determinant of part of the elements from big A."},{"Start":"02:44.800 ","End":"02:47.510","Text":"We do this for all of the elements."},{"Start":"02:47.510 ","End":"02:51.485","Text":"The second step is to apply the checkerboard."},{"Start":"02:51.485 ","End":"02:55.055","Text":"Remember the checkerboard we put plus or minus."},{"Start":"02:55.055 ","End":"02:58.175","Text":"At this stage, it doesn\u0027t matter to me because I\u0027m"},{"Start":"02:58.175 ","End":"03:01.940","Text":"just discussing whether something\u0027s whole number or not."},{"Start":"03:01.940 ","End":"03:04.310","Text":"It\u0027ll either be a plus or minus."},{"Start":"03:04.310 ","End":"03:07.725","Text":"I don\u0027t care at this stage."},{"Start":"03:07.725 ","End":"03:12.860","Text":"The third step is to apply the transpose of the matrix,"},{"Start":"03:12.860 ","End":"03:17.855","Text":"which means to shift everything rows and columns are inverted."},{"Start":"03:17.855 ","End":"03:22.075","Text":"Now, in each of the 3 stages,"},{"Start":"03:22.075 ","End":"03:24.815","Text":"I claim we stick to integers."},{"Start":"03:24.815 ","End":"03:29.510","Text":"The first stage is the most important to realize that"},{"Start":"03:29.510 ","End":"03:36.935","Text":"this M i j is a determinant of parts of this,"},{"Start":"03:36.935 ","End":"03:41.690","Text":"but the determinant of all integers."},{"Start":"03:41.690 ","End":"03:44.240","Text":"Because the entries here,"},{"Start":"03:44.240 ","End":"03:45.800","Text":"are all integers that were given."},{"Start":"03:45.800 ","End":"03:47.630","Text":"M, i, j is a determinant."},{"Start":"03:47.630 ","End":"03:52.370","Text":"It\u0027s actually n minus 1 by n minus 1 doesn\u0027t matter."},{"Start":"03:52.370 ","End":"03:54.560","Text":"If we struck out a row and a column,"},{"Start":"03:54.560 ","End":"03:56.855","Text":"we took a determinant of all integers."},{"Start":"03:56.855 ","End":"04:01.690","Text":"Now, a determinant of all integers is also an integer."},{"Start":"04:01.690 ","End":"04:07.070","Text":"I\u0027m claiming that this is an integer because when we compute determinants,"},{"Start":"04:07.070 ","End":"04:10.700","Text":"we only use 3 operations of arithmetic."},{"Start":"04:10.700 ","End":"04:13.175","Text":"We use plus and minus,"},{"Start":"04:13.175 ","End":"04:15.230","Text":"and we use multiplication,"},{"Start":"04:15.230 ","End":"04:17.629","Text":"but we do not use division,"},{"Start":"04:17.629 ","End":"04:20.480","Text":"so when you start with integers, any combination,"},{"Start":"04:20.480 ","End":"04:24.365","Text":"like we would expand the longer row or do a row operations."},{"Start":"04:24.365 ","End":"04:30.225","Text":"All pluses, additions and subtractions and multiplications,"},{"Start":"04:30.225 ","End":"04:33.300","Text":"so it has to stay in integer."},{"Start":"04:33.300 ","End":"04:35.660","Text":"The M, i, j is an integer,"},{"Start":"04:35.660 ","End":"04:38.375","Text":"and then when I apply a plus or minus,"},{"Start":"04:38.375 ","End":"04:41.365","Text":"that\u0027s going to leave it as an integer."},{"Start":"04:41.365 ","End":"04:43.985","Text":"If I have a matrix of integers,"},{"Start":"04:43.985 ","End":"04:45.380","Text":"I take its transpose,"},{"Start":"04:45.380 ","End":"04:50.410","Text":"I shift rows and columns that\u0027s still going to leave everything called integers."},{"Start":"04:50.410 ","End":"04:57.830","Text":"That\u0027s why the elements of the adjoint of A are all integers."},{"Start":"04:57.830 ","End":"05:03.005","Text":"We\u0027ve already shown that the adjoint and the inverse are the same."},{"Start":"05:03.005 ","End":"05:05.970","Text":"That concludes Part A."}],"ID":9930},{"Watched":false,"Name":"Exercise 5 Part b","Duration":"5m 36s","ChapterTopicVideoID":9607,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9607.jpeg","UploadDate":"2017-07-26T08:36:05.8100000","DurationForVideoObject":"PT5M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.895","Text":"Now we come to Part b."},{"Start":"00:02.895 ","End":"00:05.430","Text":"Just like in Part a,"},{"Start":"00:05.430 ","End":"00:09.900","Text":"we\u0027ll use the relation between the adjoint and"},{"Start":"00:09.900 ","End":"00:17.005","Text":"the inverse in that the inverse of A is 1 over the determinant of A."},{"Start":"00:17.005 ","End":"00:23.950","Text":"This is non-0 because we know that A is invertible,"},{"Start":"00:25.490 ","End":"00:32.505","Text":"times the adjoint of A."},{"Start":"00:32.505 ","End":"00:38.535","Text":"Instead of proving that the inverse of A is lower triangular,"},{"Start":"00:38.535 ","End":"00:40.965","Text":"I can prove instead of this,"},{"Start":"00:40.965 ","End":"00:47.140","Text":"that the adjoint of A is lower triangular."},{"Start":"00:47.140 ","End":"00:52.100","Text":"Let me take an example of a 4 by 4 first,"},{"Start":"00:52.100 ","End":"00:58.410","Text":"suppose that A is a 4 by 4 then,"},{"Start":"00:58.410 ","End":"01:01.940","Text":"because it\u0027s lower triangular,"},{"Start":"01:01.940 ","End":"01:05.180","Text":"it means, well, let me write the diagonal first,"},{"Start":"01:05.180 ","End":"01:07.595","Text":"a_11, a_22,"},{"Start":"01:07.595 ","End":"01:12.000","Text":"a_33, and a_44,"},{"Start":"01:12.000 ","End":"01:15.260","Text":"that everything above the diagonal is 0."},{"Start":"01:15.260 ","End":"01:16.910","Text":"Here I have 0, 0,"},{"Start":"01:16.910 ","End":"01:20.230","Text":"0, 0, 0, 0,"},{"Start":"01:20.230 ","End":"01:29.120","Text":"and here doesn\u0027t matter what well write them in it\u0027s a_21, a_31, a_41."},{"Start":"01:29.120 ","End":"01:39.815","Text":"Then what we do when we take the adjoint is we replace each element with its minor."},{"Start":"01:39.815 ","End":"01:43.160","Text":"I just wrote it out M with the subscripts 11,"},{"Start":"01:43.160 ","End":"01:44.390","Text":"m_2, m_3 and so on."},{"Start":"01:44.390 ","End":"01:46.220","Text":"These are the minors of these."},{"Start":"01:46.220 ","End":"01:48.035","Text":"In other words, what I get,"},{"Start":"01:48.035 ","End":"01:52.760","Text":"for example, here, let\u0027s take this 1 for example."},{"Start":"01:52.760 ","End":"01:56.990","Text":"This would be what I would get if I take the"},{"Start":"01:56.990 ","End":"02:02.645","Text":"corresponding 1 here and crossed out the row and column."},{"Start":"02:02.645 ","End":"02:06.365","Text":"After we do that, then we apply the checkerboard,"},{"Start":"02:06.365 ","End":"02:09.035","Text":"which means like plus here,"},{"Start":"02:09.035 ","End":"02:12.990","Text":"plus, minus, plus, minus."},{"Start":"02:12.990 ","End":"02:14.950","Text":"At the end,"},{"Start":"02:14.950 ","End":"02:18.325","Text":"we do a transpose."},{"Start":"02:18.325 ","End":"02:24.335","Text":"This element ends up finally in this position."},{"Start":"02:24.335 ","End":"02:26.765","Text":"Now the concept of being"},{"Start":"02:26.765 ","End":"02:32.270","Text":"lower triangular means that above the main diagonal above and to the right."},{"Start":"02:32.270 ","End":"02:34.505","Text":"If I put a line here,"},{"Start":"02:34.505 ","End":"02:40.100","Text":"everything above and to the right of this line is 0."},{"Start":"02:40.100 ","End":"02:48.130","Text":"I think it would help if I shade it also the triangle of zeros here."},{"Start":"02:48.130 ","End":"02:56.615","Text":"The thing is in order for the adjoint to come out lower triangular before the transpose,"},{"Start":"02:56.615 ","End":"03:04.095","Text":"we would have to get zeros below the main diagonal."},{"Start":"03:04.095 ","End":"03:07.870","Text":"These would all have to come out zeros."},{"Start":"03:08.690 ","End":"03:14.930","Text":"We should expect that the minus of each of these like this 1,"},{"Start":"03:14.930 ","End":"03:16.715","Text":"should come out 0,"},{"Start":"03:16.715 ","End":"03:19.565","Text":"not the elements themselves."},{"Start":"03:19.565 ","End":"03:22.580","Text":"But if I take the minus of these elements that I\u0027ve"},{"Start":"03:22.580 ","End":"03:28.740","Text":"highlighted below the diagonal, these should come out 0."},{"Start":"03:29.010 ","End":"03:31.270","Text":"Just to make things simpler,"},{"Start":"03:31.270 ","End":"03:37.330","Text":"I put notes and Xs in a copy and what we had was"},{"Start":"03:37.330 ","End":"03:45.010","Text":"the minor of this element, this a_32."},{"Start":"03:46.610 ","End":"03:50.760","Text":"What we get if we cross those out,"},{"Start":"03:50.760 ","End":"03:53.775","Text":"the minor comes out to be,"},{"Start":"03:53.775 ","End":"04:00.480","Text":"let\u0027s see something like this."},{"Start":"04:00.480 ","End":"04:03.580","Text":"Vector comes out to be this."},{"Start":"04:03.920 ","End":"04:08.229","Text":"For example, I could do a row operation by exchanging"},{"Start":"04:08.229 ","End":"04:12.340","Text":"the top and bottom rows and get it to be."},{"Start":"04:12.340 ","End":"04:16.170","Text":"I mean it can be written in block form."},{"Start":"04:16.390 ","End":"04:21.800","Text":"This determinant is this times the determinant of this,"},{"Start":"04:21.800 ","End":"04:24.935","Text":"which is something times 0, which is 0."},{"Start":"04:24.935 ","End":"04:30.725","Text":"Now I have to admit that this is a bit trickier than I had expected."},{"Start":"04:30.725 ","End":"04:33.565","Text":"Let\u0027s just do 1 more,"},{"Start":"04:33.565 ","End":"04:35.745","Text":"instead of this dot,"},{"Start":"04:35.745 ","End":"04:43.310","Text":"now let\u0027s try this entry and this time I would cross out this and this."},{"Start":"04:43.310 ","End":"04:48.360","Text":"Then we see immediately that what we"},{"Start":"04:48.360 ","End":"04:55.950","Text":"get has 3 zeros already we know that it\u0027s going to be 0."},{"Start":"04:55.950 ","End":"04:58.210","Text":"Similarly for the others,"},{"Start":"04:58.210 ","End":"05:03.115","Text":"if you strike out a row and a column, you easily get 0."},{"Start":"05:03.115 ","End":"05:07.510","Text":"It turns out that it\u0027s more difficult to prove in general,"},{"Start":"05:07.510 ","End":"05:09.730","Text":"I just took the case of 4 by 4."},{"Start":"05:09.730 ","End":"05:14.130","Text":"Each of these you can check there\u0027s only 2 more cases really,"},{"Start":"05:14.130 ","End":"05:20.200","Text":"when you go below the diagonal and you cross out a row and a column,"},{"Start":"05:20.200 ","End":"05:25.220","Text":"you get enough zeros that forces the minor to be 0."},{"Start":"05:25.610 ","End":"05:32.200","Text":"Just showing it for the 4 by 4 case and even then not completely."},{"Start":"05:32.200 ","End":"05:36.650","Text":"I think we\u0027ll just leave Part b at that."}],"ID":9931},{"Watched":false,"Name":"Exercise 5 Part c","Duration":"2m 27s","ChapterTopicVideoID":9608,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9608.jpeg","UploadDate":"2017-07-26T08:36:13.0100000","DurationForVideoObject":"PT2M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:05.490","Text":"Now we\u0027re in Part C. We have to show that if A is invertible,"},{"Start":"00:05.490 ","End":"00:09.570","Text":"then so are the adjoint of A and the transpose of"},{"Start":"00:09.570 ","End":"00:15.165","Text":"A. I\u0027m going to start with the easier one, with the transpose."},{"Start":"00:15.165 ","End":"00:18.960","Text":"I want to remind you of an important theorem,"},{"Start":"00:18.960 ","End":"00:21.870","Text":"that for a matrix to be invertible,"},{"Start":"00:21.870 ","End":"00:26.130","Text":"it\u0027s the same thing as to say that its determinant is non-zero."},{"Start":"00:26.130 ","End":"00:27.690","Text":"It\u0027s an if and only if."},{"Start":"00:27.690 ","End":"00:29.340","Text":"A is invertible,"},{"Start":"00:29.340 ","End":"00:32.970","Text":"if and only if its determinant is non-zero."},{"Start":"00:32.970 ","End":"00:36.610","Text":"If A is invertible,"},{"Start":"00:36.830 ","End":"00:46.350","Text":"then that implies that the determinant of A is not equal to 0."},{"Start":"00:46.350 ","End":"00:54.065","Text":"But the determinant of A is the same as the determinant of A transpose,"},{"Start":"00:54.065 ","End":"00:56.690","Text":"one of the theorems, one of the rules."},{"Start":"00:56.690 ","End":"00:58.850","Text":"That\u0027s also not 0."},{"Start":"00:58.850 ","End":"01:01.445","Text":"Now if the determinant of this is not 0"},{"Start":"01:01.445 ","End":"01:04.790","Text":"because of the if and only if that I mentioned earlier,"},{"Start":"01:04.790 ","End":"01:10.120","Text":"this means that A transpose is invertible."},{"Start":"01:10.120 ","End":"01:14.800","Text":"That concludes Part 1, this part."},{"Start":"01:14.800 ","End":"01:17.775","Text":"Now let\u0027s go in the other part for the adjoint."},{"Start":"01:17.775 ","End":"01:25.315","Text":"Now here we can use a theorem that the determinant of adjoint"},{"Start":"01:25.315 ","End":"01:34.965","Text":"of A is equal to the determinant of A^n minus 1,"},{"Start":"01:34.965 ","End":"01:37.290","Text":"where let\u0027s say that A is"},{"Start":"01:37.290 ","End":"01:46.750","Text":"an n by n matrix."},{"Start":"01:47.630 ","End":"01:52.790","Text":"Because A is invertible,"},{"Start":"01:52.790 ","End":"01:56.465","Text":"this is not 0."},{"Start":"01:56.465 ","End":"02:04.080","Text":"If it\u0027s not 0, then to the power of n minus 1 is also not 0."},{"Start":"02:05.180 ","End":"02:08.420","Text":"This is not equal to 0,"},{"Start":"02:08.420 ","End":"02:18.315","Text":"which means that the adjoint of A is invertible."},{"Start":"02:18.315 ","End":"02:22.380","Text":"That was simple once we had this rule."},{"Start":"02:23.430 ","End":"02:27.680","Text":"Next, Part D."}],"ID":9932},{"Watched":false,"Name":"Exercise 5 Part d","Duration":"8m 28s","ChapterTopicVideoID":9609,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9609.jpeg","UploadDate":"2017-07-26T08:36:38.7800000","DurationForVideoObject":"PT8M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:03.525","Text":"In this part D,"},{"Start":"00:03.525 ","End":"00:06.175","Text":"we\u0027re given 4 matrices, A and B,"},{"Start":"00:06.175 ","End":"00:11.615","Text":"which are invertible, and C and D which are not invertible."},{"Start":"00:11.615 ","End":"00:15.240","Text":"Then we\u0027re given 5 other combinations,"},{"Start":"00:15.240 ","End":"00:21.095","Text":"and we have to say for each which is or isn\u0027t invertible."},{"Start":"00:21.095 ","End":"00:25.480","Text":"In all of these, I\u0027m going to heavily use the theorem,"},{"Start":"00:25.480 ","End":"00:30.220","Text":"I want to remind you that for a matrix,"},{"Start":"00:30.220 ","End":"00:31.695","Text":"let\u0027s call it M,"},{"Start":"00:31.695 ","End":"00:33.285","Text":"I don\u0027t want to use one of these."},{"Start":"00:33.285 ","End":"00:39.670","Text":"M is invertible if and only if the determinant of"},{"Start":"00:39.670 ","End":"00:47.140","Text":"M is non-zero for m just empty matrix,"},{"Start":"00:47.140 ","End":"00:51.515","Text":"square matrix otherwise we can talk about determinants."},{"Start":"00:51.515 ","End":"00:54.635","Text":"Anyway, let\u0027s start,"},{"Start":"00:54.635 ","End":"01:05.645","Text":"and the first one asks about the sum of 2 matrices which are not invertible."},{"Start":"01:05.645 ","End":"01:08.210","Text":"I claim that in number 1,"},{"Start":"01:08.210 ","End":"01:14.990","Text":"we can\u0027t say by which I mean,"},{"Start":"01:14.990 ","End":"01:16.760","Text":"it could be either,"},{"Start":"01:16.760 ","End":"01:19.880","Text":"that C plus D,"},{"Start":"01:19.880 ","End":"01:23.150","Text":"I could find an example where it is invertible,"},{"Start":"01:23.150 ","End":"01:25.580","Text":"an example where it isn\u0027t,"},{"Start":"01:25.580 ","End":"01:29.900","Text":"so I\u0027ll give you an example of each."},{"Start":"01:29.900 ","End":"01:33.325","Text":"If we have that C is equal,"},{"Start":"01:33.325 ","End":"01:38.880","Text":"I\u0027ll take a 2 by 2 matrix example 1, 0,0,0,"},{"Start":"01:38.880 ","End":"01:46.120","Text":"then certainly C is not invertible because its determinant is 0."},{"Start":"01:46.120 ","End":"01:49.040","Text":"Either by direct computation or you can see"},{"Start":"01:49.040 ","End":"01:56.075","Text":"that product of the elements on its diagonal matrix, whatever."},{"Start":"01:56.075 ","End":"02:01.400","Text":"Let\u0027s take D is equal to 0, 0, 0,"},{"Start":"02:01.400 ","End":"02:08.570","Text":"1, then these are both not invertible."},{"Start":"02:08.570 ","End":"02:10.730","Text":"But C plus D,"},{"Start":"02:10.730 ","End":"02:17.900","Text":"which is 1,0,0,1 is the identity matrix and it\u0027s certainly is invertible."},{"Start":"02:17.900 ","End":"02:21.270","Text":"Here we have an example of,"},{"Start":"02:21.550 ","End":"02:26.585","Text":"I\u0027ll just say like x for not invertible,"},{"Start":"02:26.585 ","End":"02:29.980","Text":"not invertible is invertible,"},{"Start":"02:29.980 ","End":"02:34.970","Text":"and now I will give an example the other way."},{"Start":"02:34.970 ","End":"02:39.005","Text":"Simplest thing is to take c equals 0,"},{"Start":"02:39.005 ","End":"02:45.270","Text":"0 matrix say in 2 by 2 and D equals also 0,"},{"Start":"02:45.270 ","End":"02:46.860","Text":"0, 0, 0."},{"Start":"02:46.860 ","End":"02:51.060","Text":"Then we have that C plus D equals 0,"},{"Start":"02:51.060 ","End":"02:52.675","Text":"0, 0, 0."},{"Start":"02:52.675 ","End":"02:56.405","Text":"In this case, we have a not invertible,"},{"Start":"02:56.405 ","End":"02:58.580","Text":"plus a not invertible,"},{"Start":"02:58.580 ","End":"03:01.115","Text":"also is not invertible,"},{"Start":"03:01.115 ","End":"03:03.730","Text":"so we really can\u0027t say."},{"Start":"03:03.730 ","End":"03:06.275","Text":"Part 1 could be either."},{"Start":"03:06.275 ","End":"03:13.860","Text":"Now let\u0027s go on to Part 2 where we have the sum of invertible matrices,"},{"Start":"03:13.860 ","End":"03:18.830","Text":"and I am claiming that the same fate here"},{"Start":"03:18.830 ","End":"03:25.485","Text":"also is that we just can\u0027t say."},{"Start":"03:25.485 ","End":"03:31.610","Text":"I\u0027m going to give you an example where A plus B is Invertible,"},{"Start":"03:31.610 ","End":"03:33.835","Text":"and an example where it isn\u0027t."},{"Start":"03:33.835 ","End":"03:38.100","Text":"I could take A to be 1,"},{"Start":"03:38.100 ","End":"03:39.450","Text":"2, 3, 4,"},{"Start":"03:39.450 ","End":"03:40.590","Text":"we\u0027ve seen this already,"},{"Start":"03:40.590 ","End":"03:49.050","Text":"it\u0027s invertible, mentally you can check the determinant is 4 minus 6 is not 0."},{"Start":"03:49.210 ","End":"03:55.615","Text":"We could take B to B,"},{"Start":"03:55.615 ","End":"03:59.010","Text":"the same thing, 1, 2, 3,"},{"Start":"03:59.010 ","End":"04:05.445","Text":"4 is also invertible because same thing and"},{"Start":"04:05.445 ","End":"04:12.285","Text":"A plus B is just twice this or I could multiply it out,"},{"Start":"04:12.285 ","End":"04:14.895","Text":"2, 4, 6, 8,"},{"Start":"04:14.895 ","End":"04:18.225","Text":"and this is invertible,"},{"Start":"04:18.225 ","End":"04:21.695","Text":"this is invertible, and this is invertible."},{"Start":"04:21.695 ","End":"04:30.060","Text":"Now I\u0027m going to give you an example where we get the sum not being invertible,"},{"Start":"04:30.060 ","End":"04:32.780","Text":"so how about if I take the same A,"},{"Start":"04:32.780 ","End":"04:35.330","Text":"1, 2, 3, 4,"},{"Start":"04:35.330 ","End":"04:40.860","Text":"but this time for B I\u0027ll take the negative of this minus 1,"},{"Start":"04:40.860 ","End":"04:42.645","Text":"minus 2, minus 3,"},{"Start":"04:42.645 ","End":"04:45.360","Text":"minus 4, in this case,"},{"Start":"04:45.360 ","End":"04:48.570","Text":"we\u0027d get that A plus B is just 0,"},{"Start":"04:48.570 ","End":"04:51.240","Text":"0, 0, 0."},{"Start":"04:51.240 ","End":"04:54.150","Text":"Now, this is invertible,"},{"Start":"04:54.150 ","End":"04:56.830","Text":"this is invertible,"},{"Start":"04:57.110 ","End":"05:01.320","Text":"and this is not invertible,"},{"Start":"05:01.320 ","End":"05:03.420","Text":"so in part 2,"},{"Start":"05:03.420 ","End":"05:05.920","Text":"we also can\u0027t say."},{"Start":"05:08.390 ","End":"05:11.715","Text":"Now on to part 3,"},{"Start":"05:11.715 ","End":"05:14.875","Text":"so part 3,"},{"Start":"05:14.875 ","End":"05:22.610","Text":"which is this, where we have an invertible matrix, times a non-invertible."},{"Start":"05:22.610 ","End":"05:26.135","Text":"Why don\u0027t I just bring this back in site,"},{"Start":"05:26.135 ","End":"05:29.075","Text":"like so will do."},{"Start":"05:29.075 ","End":"05:35.010","Text":"In this case, I\u0027m claiming there is a specific answer,"},{"Start":"05:35.010 ","End":"05:38.130","Text":"it\u0027s not that we can\u0027t say, we can say,"},{"Start":"05:38.130 ","End":"05:43.190","Text":"and the answer is that it is not invertible."},{"Start":"05:43.190 ","End":"05:48.605","Text":"A is invertible, which means that the determinant of A is not 0,"},{"Start":"05:48.605 ","End":"05:53.810","Text":"and D isn\u0027t, which means that the determinant of D is 0."},{"Start":"05:53.810 ","End":"06:01.000","Text":"Now, the determinant of AD is equal to,"},{"Start":"06:01.000 ","End":"06:02.990","Text":"the determinant of a product is the product of"},{"Start":"06:02.990 ","End":"06:05.450","Text":"the determinant which is the determinant of A,"},{"Start":"06:05.450 ","End":"06:10.310","Text":"times determinant of D, which is, well,"},{"Start":"06:10.310 ","End":"06:11.660","Text":"we don\u0027t know what this is,"},{"Start":"06:11.660 ","End":"06:13.520","Text":"but we know that this is 0,"},{"Start":"06:13.520 ","End":"06:16.055","Text":"so anything times 0 is 0,"},{"Start":"06:16.055 ","End":"06:19.910","Text":"and if the determinant of AD is 0,"},{"Start":"06:19.910 ","End":"06:27.520","Text":"then AD is not invertible."},{"Start":"06:28.520 ","End":"06:31.500","Text":"Let\u0027s see. Done 3,"},{"Start":"06:31.500 ","End":"06:33.720","Text":"now we\u0027re up to 4."},{"Start":"06:33.720 ","End":"06:41.055","Text":"In 4 we have C and D are both not invertible,"},{"Start":"06:41.055 ","End":"06:42.810","Text":"very similar to this."},{"Start":"06:42.810 ","End":"06:48.110","Text":"The product is not going to be invertible because the determinant of C equals 0,"},{"Start":"06:48.110 ","End":"06:50.675","Text":"determinant of D equals 0,"},{"Start":"06:50.675 ","End":"06:52.520","Text":"and so the determinant of C,"},{"Start":"06:52.520 ","End":"06:55.280","Text":"D is the determinant of C times"},{"Start":"06:55.280 ","End":"06:59.570","Text":"the determinant of D. In this case it\u0027s also something times 0,"},{"Start":"06:59.570 ","End":"07:01.325","Text":"it\u0027s like 0 times 0,"},{"Start":"07:01.325 ","End":"07:03.160","Text":"which is 0,"},{"Start":"07:03.160 ","End":"07:07.055","Text":"which means that the same conclusion,"},{"Start":"07:07.055 ","End":"07:10.085","Text":"not invertible, but not AD,"},{"Start":"07:10.085 ","End":"07:12.830","Text":"CD is not invertible."},{"Start":"07:12.830 ","End":"07:20.050","Text":"Finally, Roman 5, AB,"},{"Start":"07:20.050 ","End":"07:24.140","Text":"here also the answer is that it is invertible"},{"Start":"07:24.140 ","End":"07:30.320","Text":"because the determinant of A is not 0 because A is invertible,"},{"Start":"07:30.320 ","End":"07:34.880","Text":"then the determinant of B is also not 0 because B is invertible,"},{"Start":"07:34.880 ","End":"07:40.190","Text":"the determinant of AB again is the product of the determinants."},{"Start":"07:40.190 ","End":"07:42.965","Text":"Determinant of A times determinant of B."},{"Start":"07:42.965 ","End":"07:49.775","Text":"This is not equal to 0 because we have the property of numbers that"},{"Start":"07:49.775 ","End":"07:59.325","Text":"non-zero times non-zero is non-zero,"},{"Start":"07:59.325 ","End":"08:02.660","Text":"you can\u0027t get 2 non-zero numbers that"},{"Start":"08:02.660 ","End":"08:07.530","Text":"multiply together to give 0 the property of the real numbers."},{"Start":"08:08.200 ","End":"08:16.430","Text":"This means that AB is invertible,"},{"Start":"08:16.430 ","End":"08:23.160","Text":"and that\u0027s part 5 and we\u0027re done."},{"Start":"08:23.360 ","End":"08:26.160","Text":"That\u0027s Part 5 of Part D,"},{"Start":"08:26.160 ","End":"08:28.740","Text":"so yeah, we finished this exercise."}],"ID":9933},{"Watched":false,"Name":"Exercise 6","Duration":"4m 47s","ChapterTopicVideoID":9610,"CourseChapterTopicPlaylistID":7289,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9610.jpeg","UploadDate":"2017-07-26T08:36:54.9930000","DurationForVideoObject":"PT4M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In this exercise, we have a 5 by"},{"Start":"00:02.310 ","End":"00:07.260","Text":"5 matrix which contains the parameter k in various places."},{"Start":"00:07.260 ","End":"00:09.465","Text":"We want the values of k,"},{"Start":"00:09.465 ","End":"00:13.395","Text":"which make the matrix non-invertible."},{"Start":"00:13.395 ","End":"00:17.835","Text":"Remember that there\u0027s a theorem"},{"Start":"00:17.835 ","End":"00:22.965","Text":"that a matrix is invertible if and only if its determinant is non-zero."},{"Start":"00:22.965 ","End":"00:26.055","Text":"Now, if we\u0027re interested in non-invertible,"},{"Start":"00:26.055 ","End":"00:30.300","Text":"then that would be equivalent to the determinant being equal to 0."},{"Start":"00:30.300 ","End":"00:33.480","Text":"Let\u0027s take the determinant of this matrix."},{"Start":"00:33.480 ","End":"00:35.580","Text":"After looking at it for a few seconds,"},{"Start":"00:35.580 ","End":"00:37.770","Text":"I see that there\u0027s a lot of 0\u0027s in"},{"Start":"00:37.770 ","End":"00:43.330","Text":"the second column so we\u0027re going to expand by the second column,"},{"Start":"00:43.330 ","End":"00:45.465","Text":"along the second column."},{"Start":"00:45.465 ","End":"00:51.830","Text":"The only entry that\u0027s going to contribute anything which is non-zero is the 2."},{"Start":"00:51.830 ","End":"00:56.460","Text":"We\u0027re going to start by crossing out the row and the column."},{"Start":"00:56.460 ","End":"01:01.275","Text":"Now what we have to do is multiply 3 things;"},{"Start":"01:01.275 ","End":"01:04.834","Text":"a sign, the entry, and the minor."},{"Start":"01:04.834 ","End":"01:08.090","Text":"As for the sign and its the checkerboard thing is thoughts of being"},{"Start":"01:08.090 ","End":"01:11.960","Text":"plus, minus, plus, minus."},{"Start":"01:11.960 ","End":"01:15.970","Text":"Or we could go plus, minus, plus, minus."},{"Start":"01:15.970 ","End":"01:18.030","Text":"We need a minus,"},{"Start":"01:18.030 ","End":"01:20.595","Text":"and then we need the entry,"},{"Start":"01:20.595 ","End":"01:25.850","Text":"the 2, and then we need the determinant of what\u0027s left, which is this."},{"Start":"01:25.850 ","End":"01:27.695","Text":"The sign, the entry,"},{"Start":"01:27.695 ","End":"01:29.660","Text":"and then the determinant of what\u0027s left,"},{"Start":"01:29.660 ","End":"01:31.355","Text":"which is the minor."},{"Start":"01:31.355 ","End":"01:35.650","Text":"Now I\u0027m looking at this and seeing we have a 4 by 4 determinant."},{"Start":"01:35.650 ","End":"01:40.395","Text":"I look for a row or column with a lot of 0\u0027s."},{"Start":"01:40.395 ","End":"01:44.445","Text":"I see the second row."},{"Start":"01:44.445 ","End":"01:46.970","Text":"This has plenty of 0\u0027s."},{"Start":"01:46.970 ","End":"01:48.950","Text":"The only non-zero is this,"},{"Start":"01:48.950 ","End":"01:56.450","Text":"so we only need the contribution from this element entry."},{"Start":"01:56.450 ","End":"02:03.350","Text":"Once again, figured out that this is a plus you could do plus, minus plus."},{"Start":"02:03.350 ","End":"02:05.600","Text":"You can remember all the diagonals."},{"Start":"02:05.600 ","End":"02:14.560","Text":"Main diagonal is always a plus so this plus with the 3k is here."},{"Start":"02:14.560 ","End":"02:17.630","Text":"The minus 2 was there from before,"},{"Start":"02:17.630 ","End":"02:19.430","Text":"and the minor is 4,"},{"Start":"02:19.430 ","End":"02:22.085","Text":"5, 0, and so on is this."},{"Start":"02:22.085 ","End":"02:25.890","Text":"Now we\u0027re down to a 3 by 3 determinant."},{"Start":"02:25.890 ","End":"02:28.725","Text":"I look for zeros."},{"Start":"02:28.725 ","End":"02:31.990","Text":"I think I can do better."},{"Start":"02:33.160 ","End":"02:36.710","Text":"I could get a 0 here with a row operation."},{"Start":"02:36.710 ","End":"02:40.025","Text":"If I add twice this row to this row,"},{"Start":"02:40.025 ","End":"02:44.275","Text":"it won\u0027t change the determinant and I\u0027ll get an extra 0."},{"Start":"02:44.275 ","End":"02:48.530","Text":"The minus 6k was from before,"},{"Start":"02:48.530 ","End":"02:53.810","Text":"and then copied this row and this row and twice this plus this,"},{"Start":"02:53.810 ","End":"03:02.515","Text":"twice 3 minus 5 is 1 and 1 here twice this plus this is 0."},{"Start":"03:02.515 ","End":"03:05.180","Text":"If you like the precise notation,"},{"Start":"03:05.180 ","End":"03:14.175","Text":"it\u0027s twice row 2 plus row 3 into row 3."},{"Start":"03:14.175 ","End":"03:17.070","Text":"You don\u0027t have to write this."},{"Start":"03:17.070 ","End":"03:20.490","Text":"Now we have two 0\u0027s. This is very good."},{"Start":"03:20.490 ","End":"03:26.340","Text":"Because we\u0027re going to expand along the third column."},{"Start":"03:26.340 ","End":"03:32.100","Text":"The only non-zero is here so only this will contribute."},{"Start":"03:32.100 ","End":"03:34.859","Text":"Let\u0027s see, plus, minus,"},{"Start":"03:34.859 ","End":"03:37.940","Text":"plus, minus, it\u0027s a minus."},{"Start":"03:37.940 ","End":"03:40.010","Text":"Well, there\u0027s already is a minus there,"},{"Start":"03:40.010 ","End":"03:42.690","Text":"so minus minus 1."},{"Start":"03:42.890 ","End":"03:46.875","Text":"That\u0027s this bit here from here,"},{"Start":"03:46.875 ","End":"03:52.920","Text":"and then the minor is the 4,"},{"Start":"03:52.920 ","End":"03:56.335","Text":"5, 1,1 Now a 2 by 2 determinant."},{"Start":"03:56.335 ","End":"03:58.535","Text":"We just do it in our heads."},{"Start":"03:58.535 ","End":"04:05.260","Text":"This diagonal product is 4 minus 5, it\u0027s minus 1."},{"Start":"04:05.260 ","End":"04:07.725","Text":"If this is minus 1,"},{"Start":"04:07.725 ","End":"04:09.840","Text":"this and this is plus 1,"},{"Start":"04:09.840 ","End":"04:11.520","Text":"and with the minus 6k,"},{"Start":"04:11.520 ","End":"04:17.180","Text":"we get 6k as the determinant for the original matrix."},{"Start":"04:17.180 ","End":"04:22.580","Text":"Now, for this to be equal to 0 is"},{"Start":"04:22.580 ","End":"04:29.475","Text":"the same thing as to say that k is 0 because 6 is non-zero, we can know it."},{"Start":"04:29.475 ","End":"04:34.610","Text":"The condition for it to be non-invertible is that 6k is 0,"},{"Start":"04:34.610 ","End":"04:37.760","Text":"which is the same condition that k is 0."},{"Start":"04:37.760 ","End":"04:41.720","Text":"Then we might want to write that in words and then give a full answer."},{"Start":"04:41.720 ","End":"04:47.880","Text":"The matrix is not invertible if and only if k is 0, and we\u0027re done."}],"ID":9934}],"Thumbnail":null,"ID":7289},{"Name":"Geometrical Applications of Determinants","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercises 1","Duration":"14m 2s","ChapterTopicVideoID":9582,"CourseChapterTopicPlaylistID":7290,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9582.jpeg","UploadDate":"2017-07-26T08:37:13.2570000","DurationForVideoObject":"PT14M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"This exercise, which has several parts,"},{"Start":"00:04.065 ","End":"00:10.350","Text":"is concerned with applications of determinants to analytical geometry."},{"Start":"00:10.350 ","End":"00:14.475","Text":"There are 2 concerned with computing areas,"},{"Start":"00:14.475 ","End":"00:20.010","Text":"1 concerned with a volume and 1 with the equation of a plane and so on."},{"Start":"00:20.010 ","End":"00:26.250","Text":"It turns out that many of these kinds of problems are easily solvable with determinants."},{"Start":"00:26.250 ","End":"00:28.395","Text":"Let\u0027s just take them 1 at a time."},{"Start":"00:28.395 ","End":"00:31.485","Text":"In part A, well,"},{"Start":"00:31.485 ","End":"00:34.740","Text":"it\u0027s actually subdivided into 2 parts."},{"Start":"00:34.740 ","End":"00:42.100","Text":"We have to compute the area of parallelograms given their vertices."},{"Start":"00:42.100 ","End":"00:44.660","Text":"I\u0027ll give you the formula in a moment."},{"Start":"00:44.660 ","End":"00:52.749","Text":"You might be wondering why did I write these in gray?"},{"Start":"00:52.749 ","End":"00:54.805","Text":"This 4th point."},{"Start":"00:54.805 ","End":"00:58.340","Text":"Turns out that for a parallelogram you really only need 3 points."},{"Start":"00:58.340 ","End":"01:00.110","Text":"I mean, if you think about it,"},{"Start":"01:00.110 ","End":"01:05.555","Text":"if I\u0027m given this point and this point and this point,"},{"Start":"01:05.555 ","End":"01:09.920","Text":"that completely determines the parallelogram because the 4th point has to"},{"Start":"01:09.920 ","End":"01:15.360","Text":"go in such a way that this line is parallel to this and this parallel to this."},{"Start":"01:15.360 ","End":"01:18.905","Text":"We don\u0027t really need the 4th,"},{"Start":"01:18.905 ","End":"01:25.785","Text":"but it was given so I just made it in different shade."},{"Start":"01:25.785 ","End":"01:32.475","Text":"Now the formula given 3 points, here it is."},{"Start":"01:32.475 ","End":"01:37.435","Text":"The 1st column are the x coordinates of the 3 points,"},{"Start":"01:37.435 ","End":"01:39.380","Text":"A, B, and C. Well,"},{"Start":"01:39.380 ","End":"01:41.720","Text":"I haven\u0027t labeled them in a moment."},{"Start":"01:41.720 ","End":"01:45.350","Text":"Then the 2nd column of the y coordinates of A,"},{"Start":"01:45.350 ","End":"01:47.960","Text":"B, and C in such a way that they\u0027re in order."},{"Start":"01:47.960 ","End":"01:50.580","Text":"I mean, A here,"},{"Start":"01:50.580 ","End":"01:51.890","Text":"B here, C here,"},{"Start":"01:51.890 ","End":"01:54.005","Text":"as long as we\u0027re consistent."},{"Start":"01:54.005 ","End":"01:59.945","Text":"I will mention something small and technical and you can ignore this if it\u0027s confusing."},{"Start":"01:59.945 ","End":"02:01.640","Text":"But just to be precise,"},{"Start":"02:01.640 ","End":"02:03.980","Text":"you might say I don\u0027t know which is A, which is B,"},{"Start":"02:03.980 ","End":"02:09.410","Text":"and which is C. In a way we don\u0027t and actually if we have A,"},{"Start":"02:09.410 ","End":"02:11.855","Text":"B, C, it really forms a triangle."},{"Start":"02:11.855 ","End":"02:15.725","Text":"There\u0027s really 2 places we could put D. D could be here,"},{"Start":"02:15.725 ","End":"02:17.000","Text":"but it could also,"},{"Start":"02:17.000 ","End":"02:20.435","Text":"be if we make this parallel and this parallel,"},{"Start":"02:20.435 ","End":"02:22.850","Text":"D could also be here."},{"Start":"02:22.850 ","End":"02:26.659","Text":"We might be getting this parallelogram,"},{"Start":"02:26.659 ","End":"02:30.590","Text":"but they both have the same area because their both twice the triangle,"},{"Start":"02:30.590 ","End":"02:32.165","Text":"so it doesn\u0027t really matter."},{"Start":"02:32.165 ","End":"02:38.120","Text":"I just mentioned this to be technically correct, watertight."},{"Start":"02:38.120 ","End":"02:40.160","Text":"But don\u0027t worry about it."},{"Start":"02:40.160 ","End":"02:42.860","Text":"You just put the 3 points that you know,"},{"Start":"02:42.860 ","End":"02:44.540","Text":"and that\u0027s the formula."},{"Start":"02:44.540 ","End":"02:46.945","Text":"Now in our case,"},{"Start":"02:46.945 ","End":"02:50.280","Text":"I\u0027m first going to substitute 0,0, 5,2,"},{"Start":"02:50.280 ","End":"02:58.030","Text":"6,5, and then these 3 are going to disappear off screen so should have noted them."},{"Start":"02:59.720 ","End":"03:02.820","Text":"Here are the 3 points 0,0,"},{"Start":"03:02.820 ","End":"03:05.270","Text":"5,2, 6,5 and here all 1s."},{"Start":"03:05.270 ","End":"03:09.460","Text":"Oh, and I forgot to mention about the plus-minus."},{"Start":"03:09.460 ","End":"03:11.565","Text":"S is the area, of course."},{"Start":"03:11.565 ","End":"03:16.745","Text":"The plus minus is just in case this comes out to be negative,"},{"Start":"03:16.745 ","End":"03:19.490","Text":"then we take a minus here to make it come out"},{"Start":"03:19.490 ","End":"03:23.840","Text":"positive because we want the areas all to come out positive."},{"Start":"03:23.840 ","End":"03:25.850","Text":"I could have said absolute value,"},{"Start":"03:25.850 ","End":"03:29.975","Text":"but it\u0027s just hard to write absolute value around the determinant."},{"Start":"03:29.975 ","End":"03:32.210","Text":"It\u0027s also 2 vertical bars."},{"Start":"03:32.210 ","End":"03:36.930","Text":"Put plus or minus means take the plus of this answer."},{"Start":"03:38.240 ","End":"03:45.810","Text":"Here we\u0027re going to expand by the top row"},{"Start":"03:45.810 ","End":"03:53.120","Text":"obviously and then we only have the single non-0 entry and this is a plus, minus, plus."},{"Start":"03:53.120 ","End":"03:54.650","Text":"It\u0027s just the determinant,"},{"Start":"03:54.650 ","End":"03:58.630","Text":"5 times 5 minus 6 times 2,"},{"Start":"03:58.630 ","End":"04:01.970","Text":"which is 25, minus 12 is 13."},{"Start":"04:01.970 ","End":"04:04.460","Text":"The plus or minus we took the plus,"},{"Start":"04:04.460 ","End":"04:08.090","Text":"it\u0027s 13 and that\u0027s the answer to the 1st part and"},{"Start":"04:08.090 ","End":"04:14.135","Text":"the same formula with the other set of points."},{"Start":"04:14.135 ","End":"04:17.690","Text":"This time I\u0027ll expand along the 1st column."},{"Start":"04:17.690 ","End":"04:20.900","Text":"Then we\u0027ll have 2 non-0 entries."},{"Start":"04:20.900 ","End":"04:23.945","Text":"I have this and I have this."},{"Start":"04:23.945 ","End":"04:26.240","Text":"For the minus 1,"},{"Start":"04:26.240 ","End":"04:30.085","Text":"the determinant is 5 times 1,"},{"Start":"04:30.085 ","End":"04:33.735","Text":"minus minus 4 times 1,"},{"Start":"04:33.735 ","End":"04:37.485","Text":"it\u0027s 5 plus 4 times the minus 1."},{"Start":"04:37.485 ","End":"04:42.060","Text":"For the 1, this is multiplied by the minor,"},{"Start":"04:42.060 ","End":"04:46.300","Text":"which is 0 minus 5."},{"Start":"04:46.300 ","End":"04:48.995","Text":"What we get here,"},{"Start":"04:48.995 ","End":"04:50.330","Text":"this time it\u0027s going to be negative."},{"Start":"04:50.330 ","End":"04:51.650","Text":"We have minus 9,"},{"Start":"04:51.650 ","End":"04:53.960","Text":"minus 5, which is minus 14."},{"Start":"04:53.960 ","End":"04:58.970","Text":"This time we pick the minus and we make it plus 14."},{"Start":"04:58.970 ","End":"05:00.755","Text":"That\u0027s part A."},{"Start":"05:00.755 ","End":"05:02.905","Text":"Now on to part B."},{"Start":"05:02.905 ","End":"05:07.760","Text":"In part B, I\u0027m going to give you a formula for the volume of the tetrahedron."},{"Start":"05:07.760 ","End":"05:11.975","Text":"But I just thought maybe you don\u0027t know what a tetrahedron is or you\u0027ve forgotten."},{"Start":"05:11.975 ","End":"05:15.050","Text":"It doesn\u0027t really matter because we can just use the formula,"},{"Start":"05:15.050 ","End":"05:19.910","Text":"but a tetrahedron is also called a triangular pyramid."},{"Start":"05:19.910 ","End":"05:24.620","Text":"The tetra comes from 4 and the hedron is from faces."},{"Start":"05:24.620 ","End":"05:25.955","Text":"It has 4 faces."},{"Start":"05:25.955 ","End":"05:27.920","Text":"If we take the points A,"},{"Start":"05:27.920 ","End":"05:31.810","Text":"B, C, and D,"},{"Start":"05:31.810 ","End":"05:33.500","Text":"then it might look"},{"Start":"05:33.500 ","End":"05:43.035","Text":"like this and the hidden edge"},{"Start":"05:43.035 ","End":"05:46.520","Text":"here and there\u0027s edge here."},{"Start":"05:46.520 ","End":"05:50.750","Text":"It\u0027s a triangular pyramid and the A, B, C,"},{"Start":"05:50.750 ","End":"05:59.140","Text":"D doesn\u0027t matter in what order the 4 vertices of this thing."},{"Start":"06:00.380 ","End":"06:03.175","Text":"The general formula for the volume,"},{"Start":"06:03.175 ","End":"06:05.930","Text":"V is volume is,"},{"Start":"06:05.930 ","End":"06:08.330","Text":"as in the previous exercise,"},{"Start":"06:08.330 ","End":"06:09.500","Text":"we have a plus or minus,"},{"Start":"06:09.500 ","End":"06:11.240","Text":"which just means that we want the answer to be"},{"Start":"06:11.240 ","End":"06:14.060","Text":"positive and if this determinant comes out negative,"},{"Start":"06:14.060 ","End":"06:18.440","Text":"then we apply the minus part to make it positive."},{"Start":"06:18.440 ","End":"06:23.220","Text":"The points and there\u0027s 4 of them,"},{"Start":"06:23.220 ","End":"06:25.070","Text":"these are the coordinates,"},{"Start":"06:25.070 ","End":"06:26.900","Text":"the XYZ of A,"},{"Start":"06:26.900 ","End":"06:28.340","Text":"the XYZ of B,"},{"Start":"06:28.340 ","End":"06:30.065","Text":"the XYZ of C,"},{"Start":"06:30.065 ","End":"06:31.985","Text":"the XYZ of D,"},{"Start":"06:31.985 ","End":"06:36.690","Text":"and the 4th column is always just 1s."},{"Start":"06:37.400 ","End":"06:42.544","Text":"If you go back and look at the numbers that we had in our case,"},{"Start":"06:42.544 ","End":"06:44.705","Text":"they\u0027d be given as 0,0,0."},{"Start":"06:44.705 ","End":"06:48.930","Text":"This was the 2nd, this was the 3rd and this was the 4th."},{"Start":"06:49.110 ","End":"06:53.725","Text":"Now we just have to compute a 4 by 4 determinant."},{"Start":"06:53.725 ","End":"06:56.600","Text":"Let\u0027s see what will be good."},{"Start":"06:56.600 ","End":"07:05.525","Text":"I think expansion along the first row where there\u0027s only a single non-zero element."},{"Start":"07:05.525 ","End":"07:09.870","Text":"The only thing that will contribute anything will be this."},{"Start":"07:10.200 ","End":"07:15.940","Text":"Although we don\u0027t really need the sign but just to be correct,"},{"Start":"07:15.940 ","End":"07:19.225","Text":"we start plus, minus, plus, minus."},{"Start":"07:19.225 ","End":"07:25.400","Text":"It\u0027s a minus with the 1 and with the minor,"},{"Start":"07:25.400 ","End":"07:28.175","Text":"which is a 3 by 3."},{"Start":"07:28.175 ","End":"07:30.950","Text":"The plus or minus just copied from here,"},{"Start":"07:30.950 ","End":"07:34.790","Text":"the minus with the 1 from here."},{"Start":"07:34.790 ","End":"07:39.020","Text":"Here\u0027s the minor determinant."},{"Start":"07:39.020 ","End":"07:43.400","Text":"This 1 we can also expand along"},{"Start":"07:43.400 ","End":"07:50.105","Text":"the first row and we\u0027ll get 2 entries that are significant."},{"Start":"07:50.105 ","End":"07:51.800","Text":"This 1 is non-zero,"},{"Start":"07:51.800 ","End":"07:57.080","Text":"and this 1 will be non-zero, and the checkerboard."},{"Start":"07:57.080 ","End":"07:58.370","Text":"They\u0027ll both be plus."},{"Start":"07:58.370 ","End":"08:01.595","Text":"I\u0027ll get a plus 1 times this."},{"Start":"08:01.595 ","End":"08:11.050","Text":"Then well, a minus 2 times this ensure this bit is this 1 here."},{"Start":"08:11.050 ","End":"08:13.615","Text":"The minor is,"},{"Start":"08:13.615 ","End":"08:15.790","Text":"without this being crossed out,"},{"Start":"08:15.790 ","End":"08:18.605","Text":"it\u0027s the 2, 4, 1, 0."},{"Start":"08:18.605 ","End":"08:21.545","Text":"It\u0027s 0 minus 4."},{"Start":"08:21.545 ","End":"08:24.410","Text":"For the other 1 we have this bit."},{"Start":"08:24.410 ","End":"08:28.055","Text":"The plus with the minus 2 is this."},{"Start":"08:28.055 ","End":"08:29.930","Text":"Then we have 1, 2, 7, 1."},{"Start":"08:29.930 ","End":"08:34.025","Text":"It\u0027s 1 minus 14."},{"Start":"08:34.025 ","End":"08:38.810","Text":"If we compute this part is minus 4,"},{"Start":"08:38.810 ","End":"08:41.615","Text":"this part is plus 26."},{"Start":"08:41.615 ","End":"08:45.365","Text":"This part is 22, minus 22."},{"Start":"08:45.365 ","End":"08:47.720","Text":"But the plus or minus means fix it,"},{"Start":"08:47.720 ","End":"08:50.135","Text":"so it\u0027s plus 22."},{"Start":"08:50.135 ","End":"08:56.915","Text":"This is our answer and we\u0027re done with this part."},{"Start":"08:56.915 ","End":"09:03.095","Text":"In part c, here we were given the coordinates of 3 points."},{"Start":"09:03.095 ","End":"09:08.254","Text":"This is the general equation of the plane passing through 3 points."},{"Start":"09:08.254 ","End":"09:15.200","Text":"Then I went back to the beginning and copied the 3 points that were given were these."},{"Start":"09:15.200 ","End":"09:17.975","Text":"These will be our x_1,"},{"Start":"09:17.975 ","End":"09:19.775","Text":"y_1, z_1, and so on."},{"Start":"09:19.775 ","End":"09:22.130","Text":"If you plug that into here,"},{"Start":"09:22.130 ","End":"09:24.605","Text":"then this is the equation we get."},{"Start":"09:24.605 ","End":"09:28.010","Text":"When we\u0027ve expanded it will get an equation in x, y, and z."},{"Start":"09:28.010 ","End":"09:29.720","Text":"Of course, we don\u0027t know what x, y,"},{"Start":"09:29.720 ","End":"09:32.945","Text":"and z are, I mean they are the variables of the plane."},{"Start":"09:32.945 ","End":"09:37.175","Text":"Let\u0027s see how we should expand this."},{"Start":"09:37.175 ","End":"09:41.160","Text":"How to tackle this particular determinant."},{"Start":"09:41.980 ","End":"09:45.620","Text":"I suggest expanding by the second row."},{"Start":"09:45.620 ","End":"09:49.025","Text":"It has a 0 in it and there\u0027s no variables in it."},{"Start":"09:49.025 ","End":"09:54.830","Text":"We\u0027ll have 2 non 0 entries."},{"Start":"09:54.830 ","End":"09:57.005","Text":"We\u0027ll get an entry for this,"},{"Start":"09:57.005 ","End":"09:59.000","Text":"and then later for this."},{"Start":"09:59.000 ","End":"10:01.025","Text":"Lets do 1 I wanted to time."},{"Start":"10:01.025 ","End":"10:07.670","Text":"We\u0027ll need the checkerboard side we\u0027ll start from a plus here and then we alternate."},{"Start":"10:07.670 ","End":"10:11.390","Text":"This is plus, so this would be minus."},{"Start":"10:11.390 ","End":"10:13.925","Text":"We also need this 1 for later."},{"Start":"10:13.925 ","End":"10:18.300","Text":"Plus minus plus, this will also be a minus."},{"Start":"10:20.710 ","End":"10:23.645","Text":"Here\u0027s the contribution for this."},{"Start":"10:23.645 ","End":"10:26.220","Text":"It\u0027s a minus."},{"Start":"10:28.450 ","End":"10:30.680","Text":"I forgot the minus."},{"Start":"10:30.680 ","End":"10:32.945","Text":"never mind, just put a minus there."},{"Start":"10:32.945 ","End":"10:39.440","Text":"The second part is when I take the last column and,"},{"Start":"10:39.440 ","End":"10:42.605","Text":"here we will get minus 3."},{"Start":"10:42.605 ","End":"10:44.600","Text":"Then the minor this,"},{"Start":"10:44.600 ","End":"10:46.115","Text":"this, this and this,"},{"Start":"10:46.115 ","End":"10:51.860","Text":"and the equation is that this is equal to 0 minus with"},{"Start":"10:51.860 ","End":"10:59.105","Text":"a minus gives plus this times this is y minus 3, minus twice this."},{"Start":"10:59.105 ","End":"11:03.665","Text":"It\u0027s plus 2, and then here,"},{"Start":"11:03.665 ","End":"11:11.300","Text":"this is what we get and simplify this curly brace and it\u0027s 4 minus 3 times,"},{"Start":"11:11.300 ","End":"11:14.960","Text":"simplify this, and then combine,"},{"Start":"11:14.960 ","End":"11:17.525","Text":"and this is the answer."},{"Start":"11:17.525 ","End":"11:21.470","Text":"Well, almost I like to cancel,"},{"Start":"11:21.470 ","End":"11:24.935","Text":"take factors out and noticed that everything here is even."},{"Start":"11:24.935 ","End":"11:26.675","Text":"Let\u0027s divide by 2,"},{"Start":"11:26.675 ","End":"11:31.225","Text":"make it neater and give this as our final answer."},{"Start":"11:31.225 ","End":"11:33.360","Text":"Onto the next part."},{"Start":"11:33.360 ","End":"11:35.480","Text":"For the final part, d,"},{"Start":"11:35.480 ","End":"11:40.025","Text":"we need the area of a triangle with given vertices,"},{"Start":"11:40.025 ","End":"11:43.265","Text":"and the formula is this."},{"Start":"11:43.265 ","End":"11:45.290","Text":"Just by the way,"},{"Start":"11:45.290 ","End":"11:47.270","Text":"if you go back to part a,"},{"Start":"11:47.270 ","End":"11:51.635","Text":"is the same formula as for the parallelogram,"},{"Start":"11:51.635 ","End":"11:53.990","Text":"except that here there\u0027s a half."},{"Start":"11:53.990 ","End":"11:58.670","Text":"The reason is that the triangle is half, the parallelogram."},{"Start":"11:58.670 ","End":"12:00.815","Text":"Very similar formulas."},{"Start":"12:00.815 ","End":"12:03.905","Text":"If that doesn\u0027t interest you just forget what I said."},{"Start":"12:03.905 ","End":"12:06.665","Text":"Let\u0027s continue any way."},{"Start":"12:06.665 ","End":"12:12.635","Text":"In our case, the 3 points we were given were this,"},{"Start":"12:12.635 ","End":"12:16.475","Text":"this and this just filled them in."},{"Start":"12:16.475 ","End":"12:19.490","Text":"I guess I should have mentioned that like before"},{"Start":"12:19.490 ","End":"12:22.295","Text":"with the parallelogram and with the volume,"},{"Start":"12:22.295 ","End":"12:25.850","Text":"the plus or minus means that we\u0027re expecting areas and"},{"Start":"12:25.850 ","End":"12:29.315","Text":"volumes and so on to be plus or if we get a minus,"},{"Start":"12:29.315 ","End":"12:33.720","Text":"we just choose the minus here to fix it."},{"Start":"12:34.240 ","End":"12:38.660","Text":"How are we going to do this determinant?"},{"Start":"12:38.660 ","End":"12:41.300","Text":"We don\u0027t have any zeros here."},{"Start":"12:41.300 ","End":"12:46.265","Text":"I think we could do a couple of row operations and get this in better shape."},{"Start":"12:46.265 ","End":"12:49.910","Text":"For example, I notice that here I have 1, 1, 1."},{"Start":"12:49.910 ","End":"12:55.130","Text":"If I subtract the first row from both the second and from the third,"},{"Start":"12:55.130 ","End":"12:56.630","Text":"then I\u0027ll get more zeros."},{"Start":"12:56.630 ","End":"12:59.510","Text":"Let me just write out what I\u0027m going do."},{"Start":"12:59.510 ","End":"13:03.185","Text":"Row 2, I\u0027m going to subtract row 1,"},{"Start":"13:03.185 ","End":"13:05.210","Text":"and that goes into row 2."},{"Start":"13:05.210 ","End":"13:11.300","Text":"Then row 3 minus row 1 goes into row 3."},{"Start":"13:11.300 ","End":"13:13.864","Text":"Then we get this,"},{"Start":"13:13.864 ","End":"13:19.534","Text":"which seems the natural choice would be to expand"},{"Start":"13:19.534 ","End":"13:25.235","Text":"along the third column because there\u0027s only the 1 that\u0027s non-zero."},{"Start":"13:25.235 ","End":"13:27.110","Text":"I cross this and this out,"},{"Start":"13:27.110 ","End":"13:31.010","Text":"and the sign is plus minus, it\u0027s a plus."},{"Start":"13:31.010 ","End":"13:37.325","Text":"We have a plus 1 times the determinant of what\u0027s left."},{"Start":"13:37.325 ","End":"13:40.685","Text":"This half was here from before."},{"Start":"13:40.685 ","End":"13:42.950","Text":"This plus 1 is here,"},{"Start":"13:42.950 ","End":"13:45.560","Text":"and this minor,"},{"Start":"13:45.560 ","End":"13:47.600","Text":"2 times 6 is 12,"},{"Start":"13:47.600 ","End":"13:51.455","Text":"minus 4 times 2 is 8,"},{"Start":"13:51.455 ","End":"13:56.990","Text":"and 12 minus 8 is 4 times a half is 2."},{"Start":"13:56.990 ","End":"14:02.910","Text":"We don\u0027t need the plus or minus 2 is the answer and we are done."}],"ID":9935}],"Thumbnail":null,"ID":7290}]

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1.1

1

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