Question 140
Let $n$ be a natural number with $n \ge 2$ and let $\mathbf{A}$ be a matrix in $M_n(\mathbb{C})$, such that $\mathbf{A}$ and $\mathbf{A}^2$ have different ranks. Prove that there exists a non-zero matrix $\mathbf{B}$ in $M_n(\mathbb{C})$, such that $\mathbf{A}\mathbf{B} = \mathbf{B}\mathbf{A} = \mathbf{B}^2 = \mathbf{O}_n$.