Show that if AB is invertible, so is A. Hint: there is a matrix W such that ABW = I.

Question

Show that if AB is invertible, so is A. Hint: there is a matrix W such that ABW = I.

Answer 1

The hint they give you is already a big help, but here are the other ingredients you need:

1. How do you show a matrix is invertible? The most straightforward way to show A is invertible is to provide another matrix C and show that AC = I.
2. Recall that matrix multiplication is associative, meaning that for any three matrices A, B, and C, you have the following: (AB)C = A(BC) (In other words, as long as you keep A to the left and C to the right, it doesn't matter whether you multiply A and B together first, or B and C together first.)

Do you see how to combine these with the hint to get your answer?

Answer 2

Following the hint, there is a matrix W such that (AB)W = I, since AB is invertible. Thus A(BW) = I, which implies A is invertible (with inverse BW).