Question

If A and B are nxn and invertible,then A^-1 B^-1 is the inverse of AB.

Answer 1

If A and B are nxn invertible matrices, A^-1 B^-1 is the inverse of AB. This can be confirmed by performing matrix multiplication which shows that the product of AB and A^-1 B^-1 results in the identity matrix, demonstrating the inverse relationship.

Step-by-step explanation:

If A and B are nxn matrices and both are invertible, then A-1 B-1 is indeed the inverse of the matrix product AB. To demonstrate this, we recall that the inverse of a matrix is such that when it is multiplied by the original matrix, the result is the identity matrix. We can check whether A-1 B-1 is the inverse of AB by multiplying it with AB and seeing if we get the identity matrix.

Let's perform the multiplication:

1. (AB)(A-1 B-1)
2. By associative property of matrix multiplication, we can regroup: A(B(A-1B-1))
3. Since B and B-1 are inverses, BB-1 equals the identity matrix, I: A(IA-1)
4. Simplifying further, we get AA-1, which is also the identity matrix, I.
5. Therefore, the end result is I, confirming that A-1 B-1 is indeed the inverse of AB.

In linear algebra, if A and B are both nxn invertible matrices, then the inverse of AB is equal to the product of the inverses of A and B, denoted as (AB)^-1 = B^-1A^-1. To prove this, we can use the fact that if A is invertible, then A^-1A = AA^-1 = I, where I is the identity matrix.