Chapter 2.1, Problem 61E

Guide Proof Prove that if $A$ and $B$ are diagonal matrices (of the same size), then $AB=BA.$

Getting Started: To prove that the matrices $AB$ and $BA$ are equal, you need to show that their corresponding entries are equal.

(i) Begin your proof by letting $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ be two diagonal $n\times n$ matrices.

(ii) The $ij$ th entry of the product $AB$ is

$c_{ij}=\underset{k=1}{\overset{n}{\sum }}{a}_{ik}{b}_{kj}.$

(iii) Evaluate the entries $c_{ij}$ for the two cases $i\ne j$ and $i=j$.

(iv) Repeat this analysis for the product $BA.$