Hello, I need the answer to this question please.

1. (30 pts) For all the calculations and proofs you are required to do, state clearly the properties (if any) and laws (if any) used. Note that parts (a), (b), (c) and (d) are not related. (a) Given that A and B are square matrices, show that if AB - A and BA - B, then A and B are both idempotent. (b) Given that A is an m x n matrix, show that both AA' and A'A are symmetric. What are the dimensions of AA' and A'A? (c) Given that A is a square matrix such that A' - 34, show that there exists a number a such that aA - I is the inverse of A - I. (d) Given that A and B are two square matrices that satisfy AB = B'A, show that ( AB)? = B6 A2