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Strange problem about matrices

Source: Romanian National Olympiad 2015, grade xi, p.4

August 23, 2019
linear algebramatrixcomplex numbers

Problem Statement

Let be three natural numbers k,m,n k,m,n an m×n m\times n matrix A, A, an n×m n\times m matrix B, B, and k k complex numbers a0,a1,…,ak 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 a0=0. a_0=0.
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