Consider the matrix
[tex]A=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}[/tex]
Multiplying A by itself has the effect of swapping the first and second rows:
[tex]A^2=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}[/tex]
which is diagonal, while A itself is not.
