Chapter 4.4, Problem 63E

A square matrix is a diagonal matrix if all elements not on the principal diagonal are zero. So a $2\times 2$ diagonal matrix has the form

$A=\left[\begin{array}{cc}a& 0\\ 0& d\end{array}\right]$

where a and $d$ are real numbers. Discuss the validity of each of the following statements. If the statement is always true, explain why. If not, give examples.

(A) If $A$ and $B$ are $2\times 2$ diagonal matrices, then $A+B$ is a $2\times 2$ diagonal matrix.

(B) If $A$ and $B$ are $2\times 2$ diagonal matrices, then $A+B=B+A.$

(C) If $A$ and $B$ are $2\times 2$ diagonal matrices, then $AB$ is a $2\times 2$ diagonal matrix.

(D) If $A$ and $B$ are $2\times 2$ diagonal matrices, then $AB=BA.$