 Now, the determinant is a multilinear function, but it's also a function that preserves multiplication. Linear maps remember they preserve vector addition and they preserve scalar multiplication. Multilinear means it's linear in every factor in every variable. Now, when it comes to linear maps, we don't make any requirement that a linear map preserves multiplication. Because for a typical vector space, there is no notion of vector multiplication. But for matrices, we can multiply them. It does actually beg the question, can the determinant preserve multiplication? The answer to that question is yes. The determinant is a map that preserves matrix multiplication. The determinant of A times B is equal to the determinant of A times the determinant of B. I want to give you an example of such a thing. Consider the following. If we take the determinant of A, 6, 1, 3, 2, this is just a 2 by 2. We're going to get 6 times 2. Remember how we do this one. You're going to take the product of this diagonal and subtract it from the product of this diagonal. You get 6 and 2, which is 12, minus 1 and 3, which is 3. That gives you 9. If we do the next one, 4, 3, 1, and 2. Again, for 2 by 2, you should take the diagonals 4, 2, 3, 1. The ones that go to the right, you're going to add the ones you go to the left, you're going to subtract. And so you get 4 times 2, which is 8, minus 3 times 1, which is 2. 8 minus 2 equals 6, all right? Actually, I made a goof somewhere, didn't I? We had 4, 3, 1, 2. So, oh, where did the 2 come from? I'm sorry. So 4 times 2 is 8. Then you get 3 times 1, which is a 3. Make that fix. No one's going to notice. And therefore, 8 minus 3 equals 5. There we go. Now let's multiply together A and B, like so. So we get 6, 1, 3, 2. And times that by the matrix 4, 3, 1, 2. When we multiply those together, first row, first column, we are going to get 6 times 4, which is 24 plus 1, so that's a 25. Then this first row, second column, 6 times 3 is 18, plus 1 times 2, which is 2, should that give us a 20? Right there. Second row, first column, 3 and 4 is 12. 2 and 1 is 2, so 12 plus 2 is 14. And then lastly, the second row, second column, 3 and 3 is 9, 2 and 2 is 4. And 9 plus 4 is 13. So if we calculate the determinant of their product, 25, 20, 14, 13, this one will be a little bit more difficult, but we can do it. 25 times 13, that is equals to 325. And then we're going to get 20 times 14, that's 280. And then 325 minus 280 equals 45. And 45, notice, is 9 times 5. So the product of the matrices has the determinant, which is the product as well. And there are tons and tons of consequences of this multiplication-preserving property of the determinant that I want to talk about in this lecture right here. So for example, a matrix is non-singular. It has an inverse if and only if the determinant is not zero. And in that case, the determinant of a inverse is just equal to the reciprocal of the matrix. And the idea is the following. The proof is actually pretty short and slick right here. If a matrix is non-singular, that means there exists an inverse so that its product equals the identity. If you take the determinant of both sides, you get the determinant of a inverse and the determinant of one. Well, because of the factorization property, the left-hand side becomes the determinant of a times the determinant of a inverse. And then what's the determinant of the identity matrix? Well, like we talked about in the previous lecture, if you have a triangular matrix, which diagonal matrices are triangular, the determinant is just the product of the diagonals. Well, the identity matrix is diagonal with ones along the diagonal. So the determinant of the identity is always equal to one. And so the only way a product of two things can equal one is only if the factors can't be zero. And in fact, if you solve for the determinant of a inverse, you end up with one over the determinant of a. So invertible matrices have reciprocal determinants. And in particular, a matrix will be non-singular exactly when it has a non-zero determinant.