Question 191
The matrices $A \in \mathcal{M}_{m, n}(\mathbb{C})$ and $B \in \mathcal{M}_{n, m}(\mathbb{C})$, where $m \geq n \geq 2$, have the property that there exist $k \in \mathbb{N}^{*}$ and $a_{0}, a_{1}, \ldots, a_{k} \in \mathbb{C}$ such that
$$
a_{k}(A B)^{k}+a_{k-1}(A B)^{k-1}+\cdots+a_{1}(A B)+a_{0} I_{m}=O_{m}
$$
while
$$
a_{k}(B A)^{k}+a_{k-1}(B A)^{k-1}+\cdots+a_{1}(B A)+a_{0} I_{n} \neq O_{n}.
$$
Prove that $a_{0}=0$.
$$
a_{k}(A B)^{k}+a_{k-1}(A B)^{k-1}+\cdots+a_{1}(A B)+a_{0} I_{m}=O_{m}
$$
while
$$
a_{k}(B A)^{k}+a_{k-1}(B A)^{k-1}+\cdots+a_{1}(B A)+a_{0} I_{n} \neq O_{n}.
$$
Prove that $a_{0}=0$.