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∃n:A^n=-I_2 implies A^2=-I_2 or A^3=-I_2

Source: SEEMOUS 2013 P4

June 8, 2021
matrixlinear algebra

Problem Statement

Let A∈M2(Q)A\in M_2(\mathbb Q) such that there is n∈N,n≠0n\in\mathbb N,n\ne0, with An=−I2A^n=-I_2. Prove that either A2=−I2A^2=-I_2 or A3=−I2A^3=-I_2.
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