Prove that matrix addition is associative ie that abc abc

Question

Problem

1. Give a pair of square matrices A and B such that:

(a) AB = BA (it commutes).

(b) AB BA (does not commute).

In general, matrix multiplication is not commutative.

2. Prove that matrix addition is associative, i.e. that (A+B)+C = A+(B+C) for compatible matrices A, B and C.

3. Prove that matrix multiplication is associative, i.e. that (AB)C = A(BC) for compatible matrices A, B and C.