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) $\text{If }A\text{ and }B\text{are }2\times 2$ upper triangular matrices, then $A+B$ is a $2\times 2$ upper triangular matrix.

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

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