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How do I prove or disprove this matrix identity?

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Let us have two n×n matrices A and B with real entries. Either prove, or disprove by providing a counterexample, that if ABA=0 and B is invertible, then A2=0.

asked Aug 18, 2017 in Math by oli

1 Answer

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Here is a 2×2 counterexample, easily extendable to n×n: Let A orthogonally project onto one axis, and let B rotate the plane by 90. The operation of ABA is to collapse everything down to one axis, then turn that axis, then collapse that axis down to the origin. However, A2=A0

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answered Aug 23, 2017 by davidjohn2121 (140 points)