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In this video, we're going to[br]workout the determinant of a

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given three by three matrix.

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So here's a matrix B with three[br]rows, three columns. We're going

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to workout its determinant. Now[br]remember that when we're working

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out a determinant, we just pick[br]a row or a column and we need to

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workout the cofactors of the[br]elements in that row or column.

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So what I'm going to do is I'm[br]going to pick the third column

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and then I know that the result[br]is the determinant of B is equal

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to the elements that column are[br]three 10, so we need to take

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three and multiplied by the[br]cofactor at three.

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I'm going to add on one times[br]the cofactor of 1, and then I'm

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going to add on nought times the[br]cofactor of 0.

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Now this looks over to workout[br]three cofactors, but if I look

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at it more closely, I see that[br]this term here is zero times the

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cofactor of 0, so I don't[br]actually need to workout the

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value of the Co factor of 0[br]because it's going to get

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multiplied by zero, so I only[br]need to workout the cofactor of

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three and the cofactor of 1.

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So we'll start with the cofactor[br]of three. Now remember that to

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find the cofactor, you first got[br]to find the minor.

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So the minor.

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Of three.

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So we look at the element. We[br]cross out its row and its column

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and get left by it with a 2 by[br]two matrix. Five 2 -- 2 seven.

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So I've crossed out the first[br]row and the third column and I

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have to find the determinant of[br]that matrix. So 5 * 7 is 35

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takeaway 2 * -- 2, So takeaway[br]minus four. So 35 + 4 is 39, so

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that's the minor of three.

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But we need the cofactor, so we[br]have to think what's the place,

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sign or remember play signs go[br]plus minus plus minus plus,

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minus, plus, minus plus.

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So the three is in[br]the top right, so the

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place sign is a plus.

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So what it means is that the[br]cofactor of three is plus

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display sign times. It's minor[br]times 39, which is 39.

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So that's when the cofactor[br]of three to find the

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cofactor of one. We start by[br]finding the minor.

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The miner of 1.

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So the ones here, so we're[br]crossing out the 3rd row in the

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second column, so we're left[br]with 4 -- 1 -- 2 seven find the

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determinant of that.

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4 * 7 is 28 takeaway minus 1[br]* -- 2, so that's plus two, so

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we're taking away two, so 28[br]takeaway two is 26.

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So that's the minor of 1.[br]The play sign of 1 we can

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see here is minus. So the[br]cofactor of one is equal to

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minus the minor minus 26.

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So we've found the two cofactors[br]that we needed in order to

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workout the determinant of B.

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Determined to be.

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Is 3 times the cofactor of[br]three, so that's 3 * 39.

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Plus one times the cofactor of[br]1, so that's 1 * -- 26.

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And then whatever the cofactor[br]of zero was, it gets multiplied

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by zero, so it's plus zero.

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So 3 * 39 is 100 and[br]seventeen 1 * -- 26 is

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minus 26, and so when we[br]work that out we get 91.

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And so the determinant of this[br]matrix is 91.

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Now of course, I could have[br]chosen a different row or

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column, and in fact I could[br]quite as easily have chosen the

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3rd row, because if we choose[br]the 3rd row again, it's got the

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O in it, so we could have a -- 2[br]* A cofactor of minus 2 + 7

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times the cofactor of seven. And[br]if you do that for yourself,

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you'll see that the value still[br]comes out to be 91.

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And indeed, you could have[br]chosen any row or column and you

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have still got the answer 91 but[br]you have to do a little bit more

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work if you didn't choose the[br]row or the row or the column

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that's got the zero in it.