Chapter A.1, Problem 7P

Let $\text{A=diag}\left(a_{11},\mathrm{.....}{a}_{nn}\right)$ be a diagonal matrix. Prove that ${\text{A}}^{\text{k}}=\text{diag}\left(a_{11}{}^k,\mathrm{......}{a}_{nn}{}^k\right)$, where

${\text{A}}^{\text{2}}{\text{=AA,A}}^{\text{3}}\text{=AAA,}\mathrm{....}$