Consider two invertible matrices A and B. Which one of the following is equal to (AB)^−1 ? (a) A^−1B^−1 (b) A/B (c) B^−1A^−1 (d) 1/AB

The correct answer is (c) B^−1A^−1.

Here's why:

The inverse of a product of two matrices is the product of the inverses of the matrices in reverse order. This is a property of invertible matrices. So, if you have two matrices A and B, and you multiply them to get a matrix AB, the inverse of this matrix (AB)^−1 is equal to B^−1A^−1.

This is a fundamental property in linear algebra.

The correct answer is (c) B^−1A^−1.

Here's why:

The inverse of a product of two matrices is the product of the inverses of the matrices in reverse order. This is a property of invertible matrices. So, if you have two matrices A and B, the inverse of their product (AB)^−1 is equal to B^−1A^−1.

This can be proven by the fact that for any matrix M, M * M^−1 = I (the identity matrix). So, if we let M = AB, then we have (AB)(AB)^−1 = I.

If we assume that (AB)^−1 = B^−1A^−1, then we can substitute and get (AB)(B^−1A^−1) = I.

Because matrix multiplication is associative, we can rearrange this to get A(BB^−1)A^−1 = I.

Since the product of any matrix and its inverse is the identity matrix, we can simplify this to get AIA^−1 = I, which simplifies further to II = I.

Since the identity matrix times any matrix is the original matrix, this proves that (AB)^−1 = B^−1A^−1.