Exercise 5.3.23

The equality is usually not true, although it’s true when $A$ and $B$ are diagonal matrices. For example, we may pick $A=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$ and $B=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$. By some calculation we have

and

and

However, we may also calculate that