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