 In this video we're going to provide the solution to question number 3 from the practice exam number 3 from Math 2270 for which we're supposed to let A and B both be 4 by 4 matrices. We know the determinant of A is equal to 2. We know the determinant of B is equal to 3. And we're supposed to compute the determinant of negative A inverse times B transpose. So some things we need to be aware of when it comes to computing determinants here, some properties. Well, one property of determinants that's going to be very useful here is when you take the determinant of a scalar multiple of a matrix. So we have like r where r is some real number, or there's some scalar r times a right here. The determinant has the property that when you pull the scalar out of the determinant, well since the determinant's multilinear, if you take the scalar out of each row individually, that's allowed. So if you take it away from the whole matrix, you're actually going to take r out of the matrix n times. And so you're going to get r to the n times the determinant of A right here. And so what this shows us is that to factor a scalar out of the determinant, you have to take that scalar out, but raise it to the power where the power is going to be the size of the matrix. This is 4 by 4. So if we want to pull that negative 1 out, this will equal or determinant would equal negative 1 to the fourth power times the determinant of A inverse times B transpose like so. In which case you get here negative 1 to the fourth. This will just become a positive one. So the negative 1 actually kind of disappears from consideration. And so I like this example because this is something that causes many people some confusion here. It's like well why did the negative 1 disappear? Why didn't you factor out the negative 1? Why isn't the answer negative? Because we're going to see the answer is one of these numbers, but a lot of students in the past have been very confused why it's not the negative of the answer provided. And that's because again, the determinant is not a linear transformation, it's a multi-linear transformation. We have to take the negative sign out of each of the rows of the matrix. And as there's four rows, we take out four negative ones and an even power of negative 1 actually gives us positive 1. All right, so that's the first property of the determinant useful here. The next property that can be very useful here is if you have the determinant of a product like take the determinant of A times B, this is equal to the determinant of A times the determinant of B. So the determinant is a multiplicative function. You can factor the determinant along the matrix product. So what this tells us here is that if we're taking the determinant of A inverse times B transpose, then this will become the determinant of A inverse times the determinant of B transpose. In which case then we'll finish this up with two more properties to mention that if you take the determinant of an inverse matrix, that is, if you have a non-singular matrix, its determinant of the inverse will be the reciprocal. So you get the determinant or one over the determinant base, you get the reciprocal determinant there. So what that tells us, of course, is that since the determinant of A was 2, the determinant of A inverse is going to be 1 half, that we see. But what about the determinant of B transpose? Well, when it comes to determinant of B transpose, the transpose operation changes rows to columns to columns to rows. But when it comes to determinants, you could use Laplace's cofactor expansion across any row or any column. So if you switch rows to columns, that's still an eligible row or column to expand across. Transpose in a matrix does not affect the determinant. This will just be the determinant of B. Now, if you take the conjugate transpose, then, of course, the transpose won't do anything to the determinant, but the conjugate will. It'll take the conjugate of the determinant. So in this situation, we're just taking the transpose of B. So this will still give you the determinant of B, which is 3. In which case then the correct answer should be 3 halves, which we see is choice C. And like I said, many students have been very tempted by choice B, not realizing that they should take out four negative ones instead of just a single negative one.