Question 179
Show that if $k \ge 1$ and $n \ge 2$ are integers such that there exists $A, B \in \mathcal{M}_n(\mathbb{Z})$ with the property that $A^3 = O_n$ and $A^k B + BA = I_n$, then $k = 1$ and $n$ is even. Conversely, show that if $n \ge 2$ is even, then there exists $A, B \in \mathcal{M}_n(\mathbb{Z})$ such that $A^3 = O_n$ and $AB + BA = I_n$.