Question 151
Let $n$ be an integer, $n \geq 2$, and let $\mathbf{A}$ be a matrix in $\mathcal{M}_{n}(\mathbb{C})$ such that $\mathbf{A}$ and $\mathbf{A}^{2}$ have different ranks. Prove that there exists a nonzero matrix $\mathbf{B}$ in $\mathcal{M}_{n}(\mathbb{C})$ such that $\mathrm{AB}=\mathrm{BA}=\mathrm{B}^{2}=\mathrm{O}_{n}$.