Question

Find two 2 x 2 upper triangular matrices C,D such that C and D are both
not diagonal, but CD is diagonal.

Answer 1

For example:

$C=\begin{bmatrix} 1 &1 \\0&1 \end{bmatrix}, \; D=\begin{bmatrix} 1 &-1 \\0 &1 \end{bmatrix}, \; C D=\begin{bmatrix} 1 &0\\0 &1 \end{bmatrix}$

You can find other examples:

$C=\begin{bmatrix} a &b \\0&c \end{bmatrix}, \; D=\begin{bmatrix} d &-b \\0 &a \end{bmatrix} , \; CD=\begin{bmatrix} ad &0 \\0&ac \end{bmatrix}$ , where a,b,c,d≠0