Chapter 1.5, Problem 66E

A $\left(3\times 3\right)$ matrix $T=\left(t_{ij}\right)$ is called an upper triangular matrix if $T$ has the form

$T=\left[\begin{array}{ccc}{t}_{11}& t_{12}& t_{13}\\ 0& t_{22}& t_{23}\\ 0& 0& t_{33}\end{array}\right]$

Formally, $T$ is upper triangular if $t_{ij}=0$ whenever $i>j$. If $A$ and $B$ are upper triangular $\left(3\times 3\right)$ matrices, verify that the product $AB$ is also upper triangular.