Does there exist a $4\times4$ matrix $A$ such that $A$ and $A^2$ are not normal but $A^3$ is normal?

Question

I'm asked to provide a proof or a counterexample for this claim:

Does there exist a $4\times4$ matrix $A$ such that $A$, $A^2$ is not normal, but $A^3$ is normal.

Any ideas are welcome Thanks in advance.

Answer 1

Here is a counterexample where $A$ is nonsingular. Let $C=\pmatrix{0&-1\\ 1&-1}$ be the companion matrix of the characteristic polynomial $x^2+x+1=0$. One can verify that $C$ and $C^2$ are not normal but $C^3=I_2$. Now take $A=\pmatrix{C\\ &I_2}$ or $A=\pmatrix{C\\ &C}$.

Answer 2

Note: An upper triangular matrix is normal iff it is diagonal.

This suggests $$\begin{pmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0\end{pmatrix}. $$