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=A≠0
How do I prove or disprove this matrix identity?
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.