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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 contexts and I thought I would at least show you that what we've 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 2 class it's called the rule of Sarrus and essentially well let me just prove it for you so let's say we want to find the determinant so we 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 time's the determinant of EF h i.e f h i minus B minus B times the determinant of D G Fi d g f I plus C times the determinant of deg h d e g h and what are these equal to this is going to be equal to a so let me write this two this is going to be equal to a times e AI e i- f h minus f h and this is going to be minus b times di di - FG - FG and this is going to be plus c times d h d h - e g - e g and if we multiply this out we get this as being equal to AE i - a f h- b di + right - times a minus so plus b f g + c d h - c e g now let me group the positive and the negative term so i have this term is positive this term is positive and that term is positive so we have this being equal to AE i + b FG + c d h those are our positive terms and that our negative terms are here we have that term that term and that term so we have minus a f h- b di minus c EG so this is a formula for the determinant of this matrix right here let's see what it actually looks like so let me rewrite it let me rewrite our matrix so if we do it in green so we have a b c d e f g h i and 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 in that guy so you're essentially going along that diagonal right there now what is b FG b FG 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 and there's some video games where you know one end 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 draw let me redraw these two columns I'm kind of augment this determinant it's not a 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 b FG it's this one right here it's this diagonal right there and then you might guess what's about to happen where is C D H it's this diagonal it's that diagonal right there so you take this product add it to this product add to this product and then you subtract these guys now what are these guys where is the a FH a FH it's that one right there so you subtract out a FH and then you subtract out B di be di is that one right there and then you have ceg which is ceg is this one right there so the rule of Sarrus sounds like something Lord of the Rings the rule of Sarrus the rule of Sarrus is essentially the it's a quick way or a mem or a way of memorizing this little technique where you write the two columns again and you say okay this product Plus this product for this product - this product - this product - that product let's actually do it with a three by three matrix to make it clear that the rule of Sarrus can be useful it can be useful so let's say we have the matrix we want the determinant of the matrix 1 2 1 2 4 2 minus 1 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 will rewrote those first two columns and to figure out this determinant we take so we take this guy so what is this going to be 1 times minus 1 times minus 1 that is just a 1 right the minus is canceled out Plus this guy Plus this product right here I should draw it a little bit neater so what is this 2 times 3 times 4 2 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 0 so it'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 going to this is on the minus side of things so it's the 4 times minus 1 times 4 is minus 16 but since we're going to do a a minus sign 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 a 0 would be a minus 0 but we can ignore it so we could say plus 0 or minus 0 same thing and then you have a minus 1 minus 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 four but since we're subtracting this becomes a plus four so the value of our determinant is equal to by the rule of Sarrus so these guys we're going to be sixteen plus four is a 20 plus plus 21 plus 20 25 which is equal to 45 so that actually is I guys you know I'd have to say a faster way of computing this 3x3 derivative and I just want to show you it's completely equivalent to the definition that I introduced you to a couple of videos ago