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# Rule of Sarrus of determinants

## Video transcript

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.