- Hi. It's Allison from East Cobb Tutoring Center. We're going to find the determinant of this 3-by-3 matrix using a method called augmented matrices. So, first, let me get rid of the matrix symbols and replace that with the bars that signify we're going to do a determinant, straight up and down lines. When we do an augmented matrix, we take the first two columns and write them over again after the matrix, like this. 

All right. So there's our augmented matrix. And now what we're going to do is multiply the diagonals, just like we do when we have a 2-by-2 determinant. So I'm going to multiply this guy. 1 times 0 times 4 is 0. I'm going to multiply this guy. 2 times 5 times 0 is 0. I'm adding these. And then one more time-- 3 times negative 1 times 2 is negative 6. 

All right. So let's hold that thought for a moment. Now I'm going to go the other direction. 3 times 0 times 0 is 0. 1 times 5 times 2 is 10. So I'm adding these purple ones up as well. One more time-- 2 times negative 1 times 4 is negative 8. What we do with these guys is subtract them. So I've got negative 6 minus-- let's see. 10 minus 8 is 2. So negative 6 minus 2 is negative 8. So my determinant is negative 8. 

Let's see one more example. Here's my matrix. I'm going to remove the matrix symbol and replace that with the symbol that tells us we're going to do a determinant. It's an augmented matrix. So we're going to write the first two columns over again. 1, 2, and 1. 

And now I'm going to multiply the diagonals, this guy. 3 times 2 times negative 1 is negative 6. This guy, that's going to give me 0. And 1 times negative 1 times 1 is negative 1. So that's the first set. 

Now I'm going to go in the other direction. Let's see. This guy-- 4 times 2 times 1 is 8. Next one is a 0. Next 1, negative 1, negative 1 and 1 will be positive 1. So we subtract these two quantities. So I have negative 7-- whoops, that's a very long 7-- negative 7 minus 9. So my determinant is negative 16.