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CIIM 2015 Problem 3

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August 9, 2016
CIIM 2015undergraduate

Problem Statement

Consider the matrices A = \left(\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right) \\ \mbox{ and } \\ B = \left(\begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right). Let k≥1k\geq 1 an integer. Prove that for any nonzero i1,i2,…,ik−1,j1,j2,…,jki_1,i_2,\dots,i_{k-1},j_1,j_2,\dots,j_k and any integers i0,iki_0,i_k it holds that Ai0Bj1Ai1Bj2⋯Aik−1BikAik≠I.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.
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