Methods for computing a 3x3 determinant are important and are used when defining the cross product. Finding a 3x3 determinant is not as computationally heavy as finding the determinant of a larger square matrix. However, finding this determinant is more complicated than finding a 2x2 determinant. Using methods for simplifying determinants through row operations can make finding the 3x3 determinant much simpler.
Computing a 2 by 2 determinant is pretty easy I'm going to assume you already know that. I want to show you how to compute a 3 by 3 determinant. Now this is a little more complicated so the first thing I have to do is talk about the minor of an element so when we look at this determinant all of these numbers are called elements of the determinant. The minor of the element 6, this element, is the determinant that we get when we cross out the row and the column that 6 is is sitting in so the resulting determinant is 1,-1 5,3 okay and that determinant is 3 minus -5 which is 8 so the minor of element 6 is 8. Now we could do that for any other element that we have here we could it do for 10, again we cross out the row and the column that 10 is in and that creates a determinant always a 2 by 2 right when you're starting with the 3 by 3 and the minor this one is going to be 2,-1 -4,3 and we get 6-4 or 2 so this process of finding of calculating the minors of a 3 by 3 determinant is vital to actually calculating a 3 by 3 determinant this is how it's done.
You pick a row, I'll start by doing the first row and you calculate the minors associated with each value in that row so say 6 times the minor associated with 6 which is 1, -1, 5, 3 this determinant and then 10 times the minor associated with 10 is 2, -1, -4, 3 and then finally the last one in this row this top row -5 times and then the minor associated with -5 would be this guy 2, 1, -4, 5.
Now very important, the signs. When you calculate a determinant you have to alternate signs, the first one corresponding with this entry here is going to be positive then negative then positive so you have to put a minus sign here and a plus sign here, that's very important so positive negative positive and then it's 6 times this determinant we already calculated that to be 8 minus 10 times this determinant which we already calculated to be 2 plus -5 is -5 times and now this determinant is 10 minus -4 10+4 14 okay so let's see what we get here, we got 48-20-70 so that's 48-90 -42 so the 3 by 3 determinant is -42.
Again the process is basically, I started with the top row then I took each of these values 6, 10,-5 and I found its minor right? This is the minor for 6 and I add them up only I use opposite signs I start with a plus then I go minus then go plus and I'll give some more specicifics in a future lesson about how you do that when you expand along a row other than the first row.