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 A^{2}=0.

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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 *A**B**A* is to collapse everything down to one axis, then turn that axis, then collapse that axis down to the origin. However, *A*2=*A*≠0

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