CIIM 2015 Problem 3

Consider the matrices $$A = \left(\begin{matrix} 1 & 2 0 & 1 \end{matrix}\right) \text{ and } B = \left(\begin{matrix} 1 & 0 2 & 1 \end{matrix}\right).$$ Let $k\geq 1$ an integer. Prove that for any nonzero $i_1,i_2,\dots,i_{k-1},j_1,j_2,\dots,j_k$ and any integers $i_0,i_k$ it holds that $$A^{i_0}B^{j_1}A^{i_1}B^{j_2}\cdots A^{i_{k-1}}B^{i_k}A^{i_k} \not = I.$$