Question 26
Let $m \ge n \ge 2$ be integers, and let $A \in \mathcal{M}_{m,n}(\mathbb{C})$ and $B \in \mathcal{M}_{n,m}(\mathbb{C})$, be two matrices such that there exist $k \in \mathbb{N}^*$, and $a_0, a_1, \dots, a_k \in \mathbb{C}$, such that $a_k(AB)^k + a_{k-1}(AB)^{k-1} + \dots + a_1(AB) + a_0I_m = O_m$, and $a_k(BA)^k + a_{k-1}(BA)^{k-1} + \dots + a_1(BA) + a_0I_n \ne O_n$. Prove that $a_0 = 0$.