Chapter 4.4, Problem 64E

A square matrix is an upper triangular matrix if all elements below the principal diagonal are zero. So a $2\times 2$ upper triangular matrix has the form

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

where $a,b,\text{ and }d$ 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 $2\times 2$ upper triangular matrices, then $A+B\text{ is a }2\times 2$ upper triangular matrix.

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

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

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