If A and B are square matrices of the same order such that `A B = B A`, then proveby induction that `A B^n=B^n A`. Further, prove that `(A B)^n=A^n B^n`for all `n in N`.

Explanation

To prove the statements given in the question, we will use the principle of mathematical induction. We will break down the proof into two parts.

### Part 1: Prove that \( AB^n = B^n A \) for all \( n \in \mathbb{N} \)

**Base Case (n = 1):**
For \( n = 1 \):
\[
AB^1 = AB = BA
...