>> Here's another situation that occasionally comes up. I want to show you what happens if we have kind of a triangulation of 0's. Now notice that here's the main diagonal and we have 0's completely above it; and what I'm going to show you works in situations where we would have 0's completely below the main diagonal. But let's just suppose, we don't know that there's any pattern associated with this, so we would probably elect then to involve the 3 0's maybe at the top of the matrix. So we'll have 4 times, let's see the determinant associated with 4 or the co-factor, has a sign and the sign is plus because of this position. But then the determinant part, take out row and column and we have this. So it's 4 times all of this. Now for the determinant of this rascal, once again we see the nice 0's across the top, oh let's use this top row. So for the determinant of this matrix, then we have minus 4 and now the sign here from the sign pattern is plus again so that's not a real factor here--take out row and column. We have the negative 1, 3, 0, negative 1--negative 1, 3, 0, negative 1 so we have this. Now in calculating this determinant since it's a 2 by 2, it's just this product minus that one and notice that it's minus 1 times minus 1; negative 1 times negative 1, and then minus 3 times 0 which goes out. Now I'm going to emphasize that these two items are the factors involved in making the calculations simply because look at the 4 negative 4, negative 1, negative 1. Those are precisely the items that are along this main diagonal. You see that's the idea, that's the shortcut. That instead of going through all of this business, that when we have this triangulation of 0's either above or below the main diagonal, all we need to do is multiply the entries that are along the main diagonal. So multiplying those we find this determinant to be negative 16.