0:00:05.140,0:00:09.292 In previous videos, we've seen[br]how to find the determinant of a 0:00:09.292,0:00:13.098 two by two matrix. We've also[br]seen the role that the 0:00:13.098,0:00:15.520 determinant plays in solving[br]simultaneous linear equations, 0:00:15.520,0:00:17.942 and in finding the inverse of a 0:00:17.942,0:00:23.168 matrix. And what we're going to[br]do in this video is we're going 0:00:23.168,0:00:26.658 to look at three by three[br]determinant statements of square 0:00:26.658,0:00:30.497 matrices with three rows and[br]three columns. Now, if you can 0:00:30.497,0:00:33.987 remember the rule for two by[br]two, determinant's was very 0:00:33.987,0:00:36.430 straightforward, but perhaps not[br]very intuitively obvious. 0:00:37.170,0:00:41.097 With three by three matrices, it[br]gets a little bit more 0:00:41.097,0:00:44.310 complicated, and, again not very[br]intuitively obvious and to 0:00:44.310,0:00:48.237 workout the determinant of three[br]by three matrices, we need to 0:00:48.237,0:00:51.807 first of all introduce some[br]other ideas, called minors and 0:00:51.807,0:00:55.377 cofactors, and will start by[br]doing that. And throughout this 0:00:55.377,0:01:00.018 video will always use the same[br]matrix A. So this is the matrix. 0:01:00.018,0:01:05.016 Say that we're going to use and[br]you can see that it's got three 0:01:05.016,0:01:08.943 rows and three columns. We can[br]only find determinants when we 0:01:08.943,0:01:10.371 have a square matrix. 0:01:10.440,0:01:14.064 The matrix that has the same[br]number of rows as it has 0:01:14.064,0:01:19.312 columns. And to find the[br]determinant of this three by 0:01:19.312,0:01:24.570 three matrix, we first of all[br]need to understand what's meant 0:01:24.570,0:01:30.784 by a minor. Every element within[br]our matrix has its own minor, so 0:01:30.784,0:01:36.998 we spell minor like this MINOR,[br]and the minor of an element is 0:01:36.998,0:01:43.212 the value of the determinant we[br]get when we cross out the row 0:01:43.212,0:01:45.124 and the column containing. 0:01:45.140,0:01:49.160 The element we're looking at, so[br]let's look at this element 7. 0:01:49.660,0:01:54.882 Element 7 is in the first row[br]and the first column. So if we 0:01:54.882,0:01:57.120 cross out that throw in that 0:01:57.120,0:02:00.370 column. I'm going to hide them[br]rather than across the Mount. 0:02:02.590,0:02:06.974 You see that we are left with a[br]2 by two matrix and we can find 0:02:06.974,0:02:08.618 the determinant of that two by 0:02:08.618,0:02:15.646 two matrix. So the minor of[br]the element 7 in Matrix A is 0:02:15.646,0:02:21.958 this two by two determinant[br]with entries 3 -- 1 four and 0:02:21.958,0:02:23.010 minus 2. 0:02:24.120,0:02:27.594 Remember the rule for[br]evaluating a two by two 0:02:27.594,0:02:31.068 determinant is to multiply[br]the two elements on the 0:02:31.068,0:02:34.542 leading diagonal and then[br]subtract the product of the 0:02:34.542,0:02:38.402 two elements that aren't on[br]the leading diagonal. So we 0:02:38.402,0:02:40.332 do 3 * -- 2. 0:02:42.130,0:02:45.424 And then we take away 4 * -- 1. 0:02:46.480,0:02:51.637 So that gives us minus 6[br]-- -- 4. 0:02:52.760,0:02:56.780 She's minus 6 + 4, which gives[br]us minus 2. 0:02:57.460,0:03:03.453 So the minor of our elements 7[br]from our original three by three 0:03:03.453,0:03:06.219 Matrix is the value minus 2. 0:03:08.260,0:03:13.622 Now we can do the same thing for[br]any element in the matrix and 0:03:13.622,0:03:18.218 find its minor. So let's pick[br]this four. The four that's in 0:03:18.218,0:03:22.814 the 3rd row and the second[br]column, and we're going to find 0:03:22.814,0:03:24.346 the minor of four. 0:03:27.000,0:03:33.648 So we go[br]through the same 0:03:33.648,0:03:39.556 process. Before is in the bottom[br]row, so we cross that out and 0:03:39.556,0:03:43.086 it's in the second column. So we[br]cross that out. 0:03:44.150,0:03:47.018 And so we see we're left[br]with a 2 by two matrix. 0:03:48.230,0:03:51.090 And so we workout the[br]determinant of that two by 0:03:51.090,0:03:51.662 two matrix. 0:03:53.710,0:03:57.450 These are two by two Matrix 7,[br]one, 0 -- 1. 0:03:58.000,0:04:03.952 So we do 7 * -- 1, which is[br]minus 7 and then we take away 0:04:03.952,0:04:07.672 nought times one we're taking[br]away nought. And so we're 0:04:07.672,0:04:09.904 getting an answer of minus 7. 0:04:10.660,0:04:15.535 So the minor of the element for[br]in this matrix is minus 7. 0:04:19.570,0:04:24.211 Now we could, but I'm not going[br]to now. We could workout the 0:04:24.211,0:04:29.209 minor of every single element in[br]our matrix A and we get a value 0:04:29.209,0:04:33.850 of the minor. So we get 9[br]values, one for each element of 0:04:33.850,0:04:34.921 the matrix A. 0:04:35.460,0:04:38.892 Now you can see now what the[br]process is to work those apps. 0:04:39.510,0:04:43.254 I want to go now from minus to 0:04:43.254,0:04:46.214 cofactors. And to get[br]from minors cofactors, 0:04:46.214,0:04:49.503 I need the idea of what[br]we call a place sign. 0:04:50.740,0:04:54.788 Every place within the Matrix[br]has a sign associated with it. 0:04:55.800,0:04:59.830 We always start with a plus in[br]the top left hand corner and 0:04:59.830,0:05:03.860 then we alternate the signs. So[br]as we go across the road, we're 0:05:03.860,0:05:05.720 going to go plus minus plus. 0:05:07.060,0:05:12.871 As we go down the column, we're[br]going to go plus, minus plus. 0:05:14.040,0:05:18.254 You can see that if we fill this[br]in, it will look like this. 0:05:19.050,0:05:23.857 This is the matrix of place[br]signs. So for instance, the 0:05:23.857,0:05:28.664 element 4 we've looked at[br]before has a place sign of 0:05:28.664,0:05:33.471 minus, whereas the seven that[br]we've looked at as a place 0:05:33.471,0:05:38.715 sign of plus and we used to[br]play signs to convert minors 0:05:38.715,0:05:40.900 into what we call cofactors. 0:05:42.400,0:05:50.224 So the cofactor of the element[br]7 is equal to the place 0:05:50.224,0:05:53.166 sign. Times 0:05:53.166,0:05:58.790 the minor.[br]That's true for all the 0:05:58.790,0:06:03.650 elements. Every element has a[br]cofactor which we find by doing 0:06:03.650,0:06:08.210 the product of its place sign[br]with the value of its minor. 0:06:08.720,0:06:12.910 So if the element 7, the place[br]sign was plus. 0:06:13.530,0:06:19.074 And then at 7 the minor was[br]minus two. So we get plus minus 0:06:19.074,0:06:21.450 two, which is just equal to 0:06:21.450,0:06:28.108 minus 2. To the cofactor of the[br]element 7 in this matrix A is 0:06:28.108,0:06:33.954 minus 2. If you take the[br]element 4, the cofactor. 0:06:34.530,0:06:40.946 A full Is equal to its place[br]sign times it's minor, so the 0:06:40.946,0:06:46.224 minor of four we saw was minus[br]seven. The place sign is a 0:06:46.224,0:06:51.096 minus, so we don't play sign[br]minus times the value of the 0:06:51.096,0:06:55.562 minor minus seven, so minus[br]minus seven is plus seven. So 0:06:55.562,0:07:00.840 the cofactor of Element 4 in[br]this matrix is 7 when they are 0:07:00.840,0:07:04.900 going to calculate the[br]determinant of our matrix A by 0:07:04.900,0:07:06.118 using the cofactors. 0:07:06.650,0:07:10.322 So if we look at what we've[br]written down here, what I've 0:07:10.322,0:07:14.606 done is I've taken our matrix A[br]and I've worked out the 9 minus. 0:07:14.606,0:07:18.278 So the minor of seven. We've[br]already seen his minus two and 0:07:18.278,0:07:22.256 the minor of four we've seen as[br]minus seven. I've worked out all 0:07:22.256,0:07:25.928 the other minors and I've put[br]them into this matrix here that 0:07:25.928,0:07:30.132 I've called M. Remember that we[br]have a place sign matrix that 0:07:30.132,0:07:33.708 starts with a plus in the top[br]left hand corner and alternate 0:07:33.708,0:07:35.794 sign as you go down or across. 0:07:37.010,0:07:40.810 And then we calculate the[br]cofactors by taking the minor 0:07:40.810,0:07:44.610 and multiplying by its place[br]site. So I've assembled into 0:07:44.610,0:07:48.790 the Matrix, see the cofactors[br]of all the elements of our 0:07:48.790,0:07:53.730 matrix A and you can see that[br]you get the cofactors from the 0:07:53.730,0:07:56.770 minors simply by multiplying[br]by the place site. 0:07:57.960,0:07:59.760 Now to workout the determinant 0:07:59.760,0:08:04.720 of a. What we have to do is we[br]can pick any row or any column. 0:08:05.770,0:08:09.520 So let's say we were to pick the[br]first row and you see that the 0:08:09.520,0:08:12.520 row has three elements in it[br]would pick the column that would 0:08:12.520,0:08:14.270 have had three elements in it as 0:08:14.270,0:08:19.098 well. What we do is we pick[br]our row. We've got three 0:08:19.098,0:08:21.882 elements and they've each got[br]their own cofactor. 0:08:23.020,0:08:27.154 So we're going to do is going to[br]form the product of element 0:08:27.154,0:08:31.606 times its cofactor, so we'll be[br]doing 7 * -- 2 two times, three 0:08:31.606,0:08:32.878 under 1 * 9. 0:08:33.920,0:08:36.602 I've just explained what the[br]steps are for calculating 0:08:36.602,0:08:38.092 determinant, so let's now write 0:08:38.092,0:08:42.096 it all down. We've seen that the[br]only things that really matter 0:08:42.096,0:08:45.504 are the matrix we started with[br]and its cofactors, so that's all 0:08:45.504,0:08:47.208 we've got on on the sheet. 0:08:48.380,0:08:49.880 So here's our matrix A. 0:08:50.800,0:08:52.915 And here's the matrix of 0:08:52.915,0:08:56.711 cofactors. Remember what we need[br]to do is we were picking the 0:08:56.711,0:08:59.974 first row. We could pick any row[br]or column, but we've chosen to 0:08:59.974,0:09:00.978 pick the first row. 0:09:01.570,0:09:03.100 And workout the determinant of 0:09:03.100,0:09:10.376 a. Or write Det the determinant,[br]the T brackets a meaning of the 0:09:10.376,0:09:11.990 determinant of a. 0:09:12.520,0:09:16.075 And sometimes you'll see this[br]written with vertical lines. 0:09:16.590,0:09:19.260 To mean the determinant of a. 0:09:20.530,0:09:25.750 And what we do, because we've[br]chosen the first row, we write 0:09:25.750,0:09:30.535 down the elements of the row[br]7211, and we multiply each 0:09:30.535,0:09:34.885 element by its cofactor. So[br]introduce 7 * -- 2. 0:09:36.240,0:09:38.494 We're going to do 2 * 3. 0:09:40.130,0:09:42.466 And we're going to do 1 * 9. 0:09:43.120,0:09:47.608 So we can get these three[br]products and then we add them 0:09:47.608,0:09:54.790 up. So 7 * -- 2 is minus[br]14. Two times three is 6 of 1 0:09:54.790,0:10:02.022 * 9 is 9, so we workout minus[br]14 + 6 + 9. We get the 0:10:02.022,0:10:07.408 answer one. So the determinant[br]of a is equal to 1. 0:10:08.370,0:10:11.540 Now you're probably thinking,[br]well, what if I chosen a 0:10:11.540,0:10:15.027 different row or a different[br]column? Surely I would have got 0:10:15.027,0:10:18.197 a different answer, so let's[br]choose a different row, a 0:10:18.197,0:10:21.050 different column, see what[br]happens. Let's choose the second 0:10:21.050,0:10:27.306 column. So if we're going to use[br]the second column to workout the 0:10:27.306,0:10:31.959 determinant of a, we've got[br]elements 2, three and four, 2, 0:10:31.959,0:10:33.228 three, and four. 0:10:33.770,0:10:38.610 And we multiply these by their[br]cofactors. 2 gets multiplied by 0:10:38.610,0:10:43.010 three, 3 gets multiplied by[br]minus 11, and four gets 0:10:43.010,0:10:44.330 multiplied by 7. 0:10:45.270,0:10:46.920 We got the three products. 0:10:47.480,0:10:48.670 And we add them up. 0:10:49.840,0:10:57.244 So 2 * 3 is six[br]3 * -- 11 is minus 0:10:57.244,0:11:00.946 33 and 4 * 7 is 0:11:00.946,0:11:04.498 28. Work this out. We get one. 0:11:05.050,0:11:08.410 So we get the same value. 0:11:09.190,0:11:13.258 And we will always get the same[br]value whichever row or column 0:11:13.258,0:11:14.953 that we choose to use. 0:11:15.480,0:11:19.920 Doesn't matter, it's completely[br]our choice. So what this means 0:11:19.920,0:11:25.248 is that if we're working out the[br]determinant of a matrix, we 0:11:25.248,0:11:28.800 actually don't need to workout[br]all nine cofactors. 0:11:30.360,0:11:35.417 You see, when we use the first[br]row, we only needed these three 0:11:35.417,0:11:39.307 cofactors. When we use the[br]second column, we only needed 0:11:39.307,0:11:40.474 those three cofactors. 0:11:41.040,0:11:46.470 So before we go ahead and work[br]out all nine, if all we want to 0:11:46.470,0:11:51.176 do is workout the determinant of[br]a, we should decide which row or 0:11:51.176,0:11:55.158 column we're going to use first,[br]and then only workout the 0:11:55.158,0:11:59.140 cofactors that correspond to the[br]row or the column that we've 0:11:59.140,0:12:03.960 chosen. And in general it's[br]handy idea. Good idea to keep 0:12:03.960,0:12:08.848 your workload down is if you've[br]got zeros in a row or column. 0:12:08.848,0:12:13.360 That's a good row or column to[br]choose, because you don't need 0:12:13.360,0:12:17.496 to actually workout the cofactor[br]that goes with the zero element, 0:12:17.496,0:12:22.008 because it's going to get end up[br]being multiplied by that zero 0:12:22.008,0:12:24.640 element. So if we use the first 0:12:24.640,0:12:30.090 column. The first value is 7,[br]the next value is zero and the 0:12:30.090,0:12:31.865 last value is minus 3. 0:12:32.880,0:12:38.320 We then have to multiply these[br]elements by their corresponding 0:12:38.320,0:12:41.040 cofactors 7. * -- 2. 0:12:41.570,0:12:44.986 Nought Times 8 and 3 * -- 5. 0:12:48.410,0:12:55.386 Have them up. We get 7 * -- 2[br]is minus 14 nought. Times 8 is 0:12:55.386,0:12:58.874 nought minus 3 * -- 5 is plus 0:12:58.874,0:13:04.302 15. An minus 14 plus not plus 15[br]is equal to 1. The same answers 0:13:04.302,0:13:08.754 before we knew it was going to[br]come out to be the same answers 0:13:08.754,0:13:12.252 before. The important thing is[br]that we really didn't need to 0:13:12.252,0:13:16.704 know what the code factor of 0[br]was. So once we've chosen to use 0:13:16.704,0:13:20.520 this column, we actually need to[br]find the cofactors of seven and 0:13:20.520,0:13:24.336 minus three because the Co[br]factor of nought is going to get 0:13:24.336,0:13:28.392 multiplied by normal. So what[br]we've seen then is how to 0:13:28.392,0:13:31.736 workout the determinant of a[br]three by three matrix. What we 0:13:31.736,0:13:35.688 do is we pick Aurora column and[br]keep our work as low as 0:13:35.688,0:13:39.336 possible. We choose the row or[br]column with the most zeros, and 0:13:39.336,0:13:42.984 then what we do is we workout[br]the cofactors of every element 0:13:42.984,0:13:46.936 in the row or column that we've[br]chosen, but we don't have to 0:13:46.936,0:13:48.456 workout the cofactors of zero 0:13:48.456,0:13:52.689 elements. Because what we do[br]then is we multiply each 0:13:52.689,0:13:56.901 element by its cofactor, and[br]then we add up the products of 0:13:56.901,0:14:00.411 element times, code factor, and[br]that's how we workout the 0:14:00.411,0:14:03.570 determinant. Now this process[br]works for any size square 0:14:03.570,0:14:07.080 matrix, although once you get[br]above 3, the amount of 0:14:07.080,0:14:09.537 arithmetic involve gets to be[br]very large.