I don't want to beat a dead horse by showing you all of
the different ways to find a determinant, but it might be
useful to beat this dead horse because you'll see it done in
different ways, in different context.
And I thought I would at least show you that what we have
been covering, so far, is very consistent with a way of
determining, or finding determinants, that you might
have been exposed to in your Algebra Two class.
It's called the rule of Sarrus.
Let me just prove it for you.
Let's say we want to find a determinant.
Let's say you want to find this determinant.
So our matrix is a, b, c, d, e, f, g, h, i.
We know how to do this.
This is equal to, let's just go down that first row, a
times the determinant of e, f, h, i minus b times the
determinant of d, g, f, i plus c times the
determinant of d, e, g, h.
And what are these equal to?
This is going to be equal to a, so let me write this, a
times ei minus fh.
And this is going to be minus b times dI minus fg.
This is going to be plus c times dh minus eg and if we
multiplied this out we get this is equal to aei minus afh
minuses bdi plus right, minus times a minus, plus bfg plus
cdh minus ceg.
Now let me group the positive and the negative terms.
So this term is positive, this term is positive and that term
is positive.
So we have this being equal to aei plus bfg plus cdh.
Those are our positive terms.
And then our negative terms are here.
We have that term, that term, and that term.
So we have minus afh minus bdi minus ceg.
So this is a formula for the determinant of this matrix
right here.
Let's see what it actually looks like.
Let me rewrite it.
Let me rewrite our matrix.
We do it in green.
So we have a, b, c, d, e, f, g, h, i.
We wanted to find its determinant.
So let me show you something interesting here.
aei is what?
aei is a product of this guy, this guy, and that guy.
So, you're essentially going along that
diagonal right there.
Now what is bfg?
You're going this guy, this guy, and then you're going all
the way down to this guy.
So it's like if you imagine that when you come out of this
side, you come out of this side.
There's some video games where you go out one end and you end
up showing up on the other end like that.
It would also be a diagonal.
Or even a better way to visualize it, let me redraw
these two columns.
Let me augment this determinant.
It's not official terminology, but I think you'll get what
I'm trying to do.
So if I write these first two columns again. a, d,
g, and b, e, h.
This guy right here bfg, it's this one right here, this
diagonal right there.
And then you might guess what's about to happen.
Where is cdh?
It's this diagonal.
It's that diagonal right there.
So you take this product, add it to this product, add it to
this product.
And then you subtract these guys.
Now what are these guys?
Where is the afh?
That one right there.
So you subtract out afh, and then you subtract out bdi.
bdi is that one right there.
And then you have ceg, which is this one right there.
So the Rule of Sarrus, sounds like something in
The Lord of the Rings.
The Rule of Sarrus is essentially a quick way of
memorizing this little technique.
You write the two columns again, you say, ok, this
product plus this product plus this product, minus this
product minus this product minus that product.
Let's actually do it with the 3 by 3 matrix to make it clear
that the Rule of Sarrus can be useful.
So let's say we have the matrix, we want the
determinant of the matrix, 1, 2, 4, 2, minus 1, 3, and then
we have 4, 0, minus 1.
We want to find that determinant.
So by the Rule of Sarrus, we can rewrite
these first two columns.
So 1, 2, 2, minus 1, 4, 0.
We rewrote those first two columns.
And to figure out this determinant we take this guy.
What is this going to be?
1 times minus 1 times minus 1.
That is just a 1.
Right, the minuses cancel out.
Plus this guy, plus this product right here.
I should draw a little bit neater.
So what is this?
2 times 3 times 4.
2 two times 3 is 6.
6 times 4 is 24, plus 24.
And then we take this guy right here.
4 times 2 times 0, anything times 0 is a 0.
So that's going to be plus 0.
And then we subtract out these guys.
So you have 4 times 4, times minus 1.
That's minus 16.
It's minus 16, but we're going to be on the
minus side of things.
So it's 4 times minus 1 times 4, is minus 16.
But since were going to do a minus on it, it's
going to be plus 16.
So it's 16.
Then you have a 0 times 3 times 1.
That of course is going to be 0.
Would be a minus 0, but we can ignore it.
So we can say plus 0 or minus 0 same thing.
Then you have a minus 1 times 2 times 2.
So that's 4 times minus 1 which is minus 4.
When you go in this direction, from the top right to the
bottom left, you are subtracting.
So this would be a minus 4 but since we're subtracting, this
becomes a plus 4.
So the value of our determinant is equal to, by
the Rule of Sarrus, we're going to have
16 plus 4 is a 20.
20 plus 25 which is equal to 45.
So that actually is, I'd have to say, a faster way of
computing this 3 by 3 derivative.
And I just want to show you this is completely equivalent
to the definition that I introduced you to a couple
videos ago.