RO-NMO 2015 Problem 114

Let be three natural numbers $k,m,n$ an $m\times n$ matrix $A,$ an $n\times m$ matrix $B,$ and $k$ complex numbers $a_0,a_1,\ldots ,a_k$ such that the following conditions hold. $\text{(i)} m\ge n\ge 2$ $\text{(ii)} a_0I_m+a_1AB+a_2(AB)^2+\cdots +a_k(AB)^k=O_m$ $\text{(iii)} a_0I_m+a_1BA+a_2(BA)^2+\cdots +a_k(BA)^k\neq O_n$ Prove that $a_0=0.$