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A^m=B^n implies [A,B]=0

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November 25, 2019
linear algebra

Problem Statement

Let be two 2×2 2\times 2 real matrices A,BA,B having the property that all their natural powers are not real multiples of the identity. Prove that if some natural power of A A is equal to some natural power of B, B, then, A,B A,B commute. Is the converse statement true?
Cosmin Nitu
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