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)}\quad m\ge n\ge 2 $ $ \text{(ii)}\quad a_0I_m+a_1AB+a_2(AB)^2+\cdots +a_k(AB)^k=O_m $ $ \text{(iii)}\quad a_0I_m+a_1BA+a_2(BA)^2+\cdots +a_k(BA)^k\neq O_n $ Prove that $ a_0=0. $