Rule of Sarrus of determinants

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.