Question

Write a 3×3 matrix A which A^2 is diagonal while A is not diagonal ?

Answer 1

Consider the matrix

$A=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}$

Multiplying A by itself has the effect of swapping the first and second rows:

$A^2=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$

which is diagonal, while A itself is not.