1 00:00:05,140 --> 00:00:09,292 In previous videos, we've seen how to find the determinant of a 2 00:00:09,292 --> 00:00:13,098 two by two matrix. We've also seen the role that the 3 00:00:13,098 --> 00:00:15,520 determinant plays in solving simultaneous linear equations, 4 00:00:15,520 --> 00:00:17,942 and in finding the inverse of a 5 00:00:17,942 --> 00:00:23,168 matrix. And what we're going to do in this video is we're going 6 00:00:23,168 --> 00:00:26,658 to look at three by three determinant statements of square 7 00:00:26,658 --> 00:00:30,497 matrices with three rows and three columns. Now, if you can 8 00:00:30,497 --> 00:00:33,987 remember the rule for two by two, determinant's was very 9 00:00:33,987 --> 00:00:36,430 straightforward, but perhaps not very intuitively obvious. 10 00:00:37,170 --> 00:00:41,097 With three by three matrices, it gets a little bit more 11 00:00:41,097 --> 00:00:44,310 complicated, and, again not very intuitively obvious and to 12 00:00:44,310 --> 00:00:48,237 workout the determinant of three by three matrices, we need to 13 00:00:48,237 --> 00:00:51,807 first of all introduce some other ideas, called minors and 14 00:00:51,807 --> 00:00:55,377 cofactors, and will start by doing that. And throughout this 15 00:00:55,377 --> 00:01:00,018 video will always use the same matrix A. So this is the matrix. 16 00:01:00,018 --> 00:01:05,016 Say that we're going to use and you can see that it's got three 17 00:01:05,016 --> 00:01:08,943 rows and three columns. We can only find determinants when we 18 00:01:08,943 --> 00:01:10,371 have a square matrix. 19 00:01:10,440 --> 00:01:14,064 The matrix that has the same number of rows as it has 20 00:01:14,064 --> 00:01:19,312 columns. And to find the determinant of this three by 21 00:01:19,312 --> 00:01:24,570 three matrix, we first of all need to understand what's meant 22 00:01:24,570 --> 00:01:30,784 by a minor. Every element within our matrix has its own minor, so 23 00:01:30,784 --> 00:01:36,998 we spell minor like this MINOR, and the minor of an element is 24 00:01:36,998 --> 00:01:43,212 the value of the determinant we get when we cross out the row 25 00:01:43,212 --> 00:01:45,124 and the column containing. 26 00:01:45,140 --> 00:01:49,160 The element we're looking at, so let's look at this element 7. 27 00:01:49,660 --> 00:01:54,882 Element 7 is in the first row and the first column. So if we 28 00:01:54,882 --> 00:01:57,120 cross out that throw in that 29 00:01:57,120 --> 00:02:00,370 column. I'm going to hide them rather than across the Mount. 30 00:02:02,590 --> 00:02:06,974 You see that we are left with a 2 by two matrix and we can find 31 00:02:06,974 --> 00:02:08,618 the determinant of that two by 32 00:02:08,618 --> 00:02:15,646 two matrix. So the minor of the element 7 in Matrix A is 33 00:02:15,646 --> 00:02:21,958 this two by two determinant with entries 3 -- 1 four and 34 00:02:21,958 --> 00:02:23,010 minus 2. 35 00:02:24,120 --> 00:02:27,594 Remember the rule for evaluating a two by two 36 00:02:27,594 --> 00:02:31,068 determinant is to multiply the two elements on the 37 00:02:31,068 --> 00:02:34,542 leading diagonal and then subtract the product of the 38 00:02:34,542 --> 00:02:38,402 two elements that aren't on the leading diagonal. So we 39 00:02:38,402 --> 00:02:40,332 do 3 * -- 2. 40 00:02:42,130 --> 00:02:45,424 And then we take away 4 * -- 1. 41 00:02:46,480 --> 00:02:51,637 So that gives us minus 6 -- -- 4. 42 00:02:52,760 --> 00:02:56,780 She's minus 6 + 4, which gives us minus 2. 43 00:02:57,460 --> 00:03:03,453 So the minor of our elements 7 from our original three by three 44 00:03:03,453 --> 00:03:06,219 Matrix is the value minus 2. 45 00:03:08,260 --> 00:03:13,622 Now we can do the same thing for any element in the matrix and 46 00:03:13,622 --> 00:03:18,218 find its minor. So let's pick this four. The four that's in 47 00:03:18,218 --> 00:03:22,814 the 3rd row and the second column, and we're going to find 48 00:03:22,814 --> 00:03:24,346 the minor of four. 49 00:03:27,000 --> 00:03:33,648 So we go through the same 50 00:03:33,648 --> 00:03:39,556 process. Before is in the bottom row, so we cross that out and 51 00:03:39,556 --> 00:03:43,086 it's in the second column. So we cross that out. 52 00:03:44,150 --> 00:03:47,018 And so we see we're left with a 2 by two matrix. 53 00:03:48,230 --> 00:03:51,090 And so we workout the determinant of that two by 54 00:03:51,090 --> 00:03:51,662 two matrix. 55 00:03:53,710 --> 00:03:57,450 These are two by two Matrix 7, one, 0 -- 1. 56 00:03:58,000 --> 00:04:03,952 So we do 7 * -- 1, which is minus 7 and then we take away 57 00:04:03,952 --> 00:04:07,672 nought times one we're taking away nought. And so we're 58 00:04:07,672 --> 00:04:09,904 getting an answer of minus 7. 59 00:04:10,660 --> 00:04:15,535 So the minor of the element for in this matrix is minus 7. 60 00:04:19,570 --> 00:04:24,211 Now we could, but I'm not going to now. We could workout the 61 00:04:24,211 --> 00:04:29,209 minor of every single element in our matrix A and we get a value 62 00:04:29,209 --> 00:04:33,850 of the minor. So we get 9 values, one for each element of 63 00:04:33,850 --> 00:04:34,921 the matrix A. 64 00:04:35,460 --> 00:04:38,892 Now you can see now what the process is to work those apps. 65 00:04:39,510 --> 00:04:43,254 I want to go now from minus to 66 00:04:43,254 --> 00:04:46,214 cofactors. And to get from minors cofactors, 67 00:04:46,214 --> 00:04:49,503 I need the idea of what we call a place sign. 68 00:04:50,740 --> 00:04:54,788 Every place within the Matrix has a sign associated with it. 69 00:04:55,800 --> 00:04:59,830 We always start with a plus in the top left hand corner and 70 00:04:59,830 --> 00:05:03,860 then we alternate the signs. So as we go across the road, we're 71 00:05:03,860 --> 00:05:05,720 going to go plus minus plus. 72 00:05:07,060 --> 00:05:12,871 As we go down the column, we're going to go plus, minus plus. 73 00:05:14,040 --> 00:05:18,254 You can see that if we fill this in, it will look like this. 74 00:05:19,050 --> 00:05:23,857 This is the matrix of place signs. So for instance, the 75 00:05:23,857 --> 00:05:28,664 element 4 we've looked at before has a place sign of 76 00:05:28,664 --> 00:05:33,471 minus, whereas the seven that we've looked at as a place 77 00:05:33,471 --> 00:05:38,715 sign of plus and we used to play signs to convert minors 78 00:05:38,715 --> 00:05:40,900 into what we call cofactors. 79 00:05:42,400 --> 00:05:50,224 So the cofactor of the element 7 is equal to the place 80 00:05:50,224 --> 00:05:53,166 sign. Times 81 00:05:53,166 --> 00:05:58,790 the minor. That's true for all the 82 00:05:58,790 --> 00:06:03,650 elements. Every element has a cofactor which we find by doing 83 00:06:03,650 --> 00:06:08,210 the product of its place sign with the value of its minor. 84 00:06:08,720 --> 00:06:12,910 So if the element 7, the place sign was plus. 85 00:06:13,530 --> 00:06:19,074 And then at 7 the minor was minus two. So we get plus minus 86 00:06:19,074 --> 00:06:21,450 two, which is just equal to 87 00:06:21,450 --> 00:06:28,108 minus 2. To the cofactor of the element 7 in this matrix A is 88 00:06:28,108 --> 00:06:33,954 minus 2. If you take the element 4, the cofactor. 89 00:06:34,530 --> 00:06:40,946 A full Is equal to its place sign times it's minor, so the 90 00:06:40,946 --> 00:06:46,224 minor of four we saw was minus seven. The place sign is a 91 00:06:46,224 --> 00:06:51,096 minus, so we don't play sign minus times the value of the 92 00:06:51,096 --> 00:06:55,562 minor minus seven, so minus minus seven is plus seven. So 93 00:06:55,562 --> 00:07:00,840 the cofactor of Element 4 in this matrix is 7 when they are 94 00:07:00,840 --> 00:07:04,900 going to calculate the determinant of our matrix A by 95 00:07:04,900 --> 00:07:06,118 using the cofactors. 96 00:07:06,650 --> 00:07:10,322 So if we look at what we've written down here, what I've 97 00:07:10,322 --> 00:07:14,606 done is I've taken our matrix A and I've worked out the 9 minus. 98 00:07:14,606 --> 00:07:18,278 So the minor of seven. We've already seen his minus two and 99 00:07:18,278 --> 00:07:22,256 the minor of four we've seen as minus seven. I've worked out all 100 00:07:22,256 --> 00:07:25,928 the other minors and I've put them into this matrix here that 101 00:07:25,928 --> 00:07:30,132 I've called M. Remember that we have a place sign matrix that 102 00:07:30,132 --> 00:07:33,708 starts with a plus in the top left hand corner and alternate 103 00:07:33,708 --> 00:07:35,794 sign as you go down or across. 104 00:07:37,010 --> 00:07:40,810 And then we calculate the cofactors by taking the minor 105 00:07:40,810 --> 00:07:44,610 and multiplying by its place site. So I've assembled into 106 00:07:44,610 --> 00:07:48,790 the Matrix, see the cofactors of all the elements of our 107 00:07:48,790 --> 00:07:53,730 matrix A and you can see that you get the cofactors from the 108 00:07:53,730 --> 00:07:56,770 minors simply by multiplying by the place site. 109 00:07:57,960 --> 00:07:59,760 Now to workout the determinant 110 00:07:59,760 --> 00:08:04,720 of a. What we have to do is we can pick any row or any column. 111 00:08:05,770 --> 00:08:09,520 So let's say we were to pick the first row and you see that the 112 00:08:09,520 --> 00:08:12,520 row has three elements in it would pick the column that would 113 00:08:12,520 --> 00:08:14,270 have had three elements in it as 114 00:08:14,270 --> 00:08:19,098 well. What we do is we pick our row. We've got three 115 00:08:19,098 --> 00:08:21,882 elements and they've each got their own cofactor. 116 00:08:23,020 --> 00:08:27,154 So we're going to do is going to form the product of element 117 00:08:27,154 --> 00:08:31,606 times its cofactor, so we'll be doing 7 * -- 2 two times, three 118 00:08:31,606 --> 00:08:32,878 under 1 * 9. 119 00:08:33,920 --> 00:08:36,602 I've just explained what the steps are for calculating 120 00:08:36,602 --> 00:08:38,092 determinant, so let's now write 121 00:08:38,092 --> 00:08:42,096 it all down. We've seen that the only things that really matter 122 00:08:42,096 --> 00:08:45,504 are the matrix we started with and its cofactors, so that's all 123 00:08:45,504 --> 00:08:47,208 we've got on on the sheet. 124 00:08:48,380 --> 00:08:49,880 So here's our matrix A. 125 00:08:50,800 --> 00:08:52,915 And here's the matrix of 126 00:08:52,915 --> 00:08:56,711 cofactors. Remember what we need to do is we were picking the 127 00:08:56,711 --> 00:08:59,974 first row. We could pick any row or column, but we've chosen to 128 00:08:59,974 --> 00:09:00,978 pick the first row. 129 00:09:01,570 --> 00:09:03,100 And workout the determinant of 130 00:09:03,100 --> 00:09:10,376 a. Or write Det the determinant, the T brackets a meaning of the 131 00:09:10,376 --> 00:09:11,990 determinant of a. 132 00:09:12,520 --> 00:09:16,075 And sometimes you'll see this written with vertical lines. 133 00:09:16,590 --> 00:09:19,260 To mean the determinant of a. 134 00:09:20,530 --> 00:09:25,750 And what we do, because we've chosen the first row, we write 135 00:09:25,750 --> 00:09:30,535 down the elements of the row 7211, and we multiply each 136 00:09:30,535 --> 00:09:34,885 element by its cofactor. So introduce 7 * -- 2. 137 00:09:36,240 --> 00:09:38,494 We're going to do 2 * 3. 138 00:09:40,130 --> 00:09:42,466 And we're going to do 1 * 9. 139 00:09:43,120 --> 00:09:47,608 So we can get these three products and then we add them 140 00:09:47,608 --> 00:09:54,790 up. So 7 * -- 2 is minus 14. Two times three is 6 of 1 141 00:09:54,790 --> 00:10:02,022 * 9 is 9, so we workout minus 14 + 6 + 9. We get the 142 00:10:02,022 --> 00:10:07,408 answer one. So the determinant of a is equal to 1. 143 00:10:08,370 --> 00:10:11,540 Now you're probably thinking, well, what if I chosen a 144 00:10:11,540 --> 00:10:15,027 different row or a different column? Surely I would have got 145 00:10:15,027 --> 00:10:18,197 a different answer, so let's choose a different row, a 146 00:10:18,197 --> 00:10:21,050 different column, see what happens. Let's choose the second 147 00:10:21,050 --> 00:10:27,306 column. So if we're going to use the second column to workout the 148 00:10:27,306 --> 00:10:31,959 determinant of a, we've got elements 2, three and four, 2, 149 00:10:31,959 --> 00:10:33,228 three, and four. 150 00:10:33,770 --> 00:10:38,610 And we multiply these by their cofactors. 2 gets multiplied by 151 00:10:38,610 --> 00:10:43,010 three, 3 gets multiplied by minus 11, and four gets 152 00:10:43,010 --> 00:10:44,330 multiplied by 7. 153 00:10:45,270 --> 00:10:46,920 We got the three products. 154 00:10:47,480 --> 00:10:48,670 And we add them up. 155 00:10:49,840 --> 00:10:57,244 So 2 * 3 is six 3 * -- 11 is minus 156 00:10:57,244 --> 00:11:00,946 33 and 4 * 7 is 157 00:11:00,946 --> 00:11:04,498 28. Work this out. We get one. 158 00:11:05,050 --> 00:11:08,410 So we get the same value. 159 00:11:09,190 --> 00:11:13,258 And we will always get the same value whichever row or column 160 00:11:13,258 --> 00:11:14,953 that we choose to use. 161 00:11:15,480 --> 00:11:19,920 Doesn't matter, it's completely our choice. So what this means 162 00:11:19,920 --> 00:11:25,248 is that if we're working out the determinant of a matrix, we 163 00:11:25,248 --> 00:11:28,800 actually don't need to workout all nine cofactors. 164 00:11:30,360 --> 00:11:35,417 You see, when we use the first row, we only needed these three 165 00:11:35,417 --> 00:11:39,307 cofactors. When we use the second column, we only needed 166 00:11:39,307 --> 00:11:40,474 those three cofactors. 167 00:11:41,040 --> 00:11:46,470 So before we go ahead and work out all nine, if all we want to 168 00:11:46,470 --> 00:11:51,176 do is workout the determinant of a, we should decide which row or 169 00:11:51,176 --> 00:11:55,158 column we're going to use first, and then only workout the 170 00:11:55,158 --> 00:11:59,140 cofactors that correspond to the row or the column that we've 171 00:11:59,140 --> 00:12:03,960 chosen. And in general it's handy idea. Good idea to keep 172 00:12:03,960 --> 00:12:08,848 your workload down is if you've got zeros in a row or column. 173 00:12:08,848 --> 00:12:13,360 That's a good row or column to choose, because you don't need 174 00:12:13,360 --> 00:12:17,496 to actually workout the cofactor that goes with the zero element, 175 00:12:17,496 --> 00:12:22,008 because it's going to get end up being multiplied by that zero 176 00:12:22,008 --> 00:12:24,640 element. So if we use the first 177 00:12:24,640 --> 00:12:30,090 column. The first value is 7, the next value is zero and the 178 00:12:30,090 --> 00:12:31,865 last value is minus 3. 179 00:12:32,880 --> 00:12:38,320 We then have to multiply these elements by their corresponding 180 00:12:38,320 --> 00:12:41,040 cofactors 7. * -- 2. 181 00:12:41,570 --> 00:12:44,986 Nought Times 8 and 3 * -- 5. 182 00:12:48,410 --> 00:12:55,386 Have them up. We get 7 * -- 2 is minus 14 nought. Times 8 is 183 00:12:55,386 --> 00:12:58,874 nought minus 3 * -- 5 is plus 184 00:12:58,874 --> 00:13:04,302 15. An minus 14 plus not plus 15 is equal to 1. The same answers 185 00:13:04,302 --> 00:13:08,754 before we knew it was going to come out to be the same answers 186 00:13:08,754 --> 00:13:12,252 before. The important thing is that we really didn't need to 187 00:13:12,252 --> 00:13:16,704 know what the code factor of 0 was. So once we've chosen to use 188 00:13:16,704 --> 00:13:20,520 this column, we actually need to find the cofactors of seven and 189 00:13:20,520 --> 00:13:24,336 minus three because the Co factor of nought is going to get 190 00:13:24,336 --> 00:13:28,392 multiplied by normal. So what we've seen then is how to 191 00:13:28,392 --> 00:13:31,736 workout the determinant of a three by three matrix. What we 192 00:13:31,736 --> 00:13:35,688 do is we pick Aurora column and keep our work as low as 193 00:13:35,688 --> 00:13:39,336 possible. We choose the row or column with the most zeros, and 194 00:13:39,336 --> 00:13:42,984 then what we do is we workout the cofactors of every element 195 00:13:42,984 --> 00:13:46,936 in the row or column that we've chosen, but we don't have to 196 00:13:46,936 --> 00:13:48,456 workout the cofactors of zero 197 00:13:48,456 --> 00:13:52,689 elements. Because what we do then is we multiply each 198 00:13:52,689 --> 00:13:56,901 element by its cofactor, and then we add up the products of 199 00:13:56,901 --> 00:14:00,411 element times, code factor, and that's how we workout the 200 00:14:00,411 --> 00:14:03,570 determinant. Now this process works for any size square 201 00:14:03,570 --> 00:14:07,080 matrix, although once you get above 3, the amount of 202 00:14:07,080 --> 00:14:09,537 arithmetic involve gets to be very large.