Question 152
Let $n \ge 2$ be an integer number. Determine all the values of $m \in \mathbb{N}$ such that there is $A \in \mathcal{M}_n(\mathbb{C})$ with the property that $\text{rank}(A^k) + \text{rank}(A^{n-k}) = m$, for any $k \in \{1, 2, \dots, n-1\}$.