https:/.../determinantof3x3matrixexamplef61mb-aspect.mp4

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

given three by three matrix.

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

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

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

workout the cofactors of the
elements in that row or column.

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

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

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

three and multiplied by the
cofactor at three.

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

going to add on nought times the
cofactor of 0.

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

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

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

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

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

three and the cofactor of 1.

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

find the cofactor, you first got
to find the minor.

So the minor.

Of three.

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

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

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

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

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

that's the minor of three.

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

sign or remember play signs go
plus minus plus minus plus,

minus, plus, minus plus.

So the three is in
the top right, so the

place sign is a plus.

So what it means is that the
cofactor of three is plus

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

So that's when the cofactor
of three to find the

cofactor of one. We start by
finding the minor.

The miner of 1.

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

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

determinant of that.

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

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

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

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

minus the minor minus 26.

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

workout the determinant of B.

Determined to be.

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

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

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

by zero, so it's plus zero.

So 3 * 39 is 100 and
seventeen 1 * -- 26 is

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

And so the determinant of this
matrix is 91.

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

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

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

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

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

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

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

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

work if you didn't choose the
row or the row or the column

that's got the zero in it.