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matrices that satisfty A^2+B^2=AB

Source: romanian nmo 2007, grade 11, problem 1

April 15, 2007
linear algebramatrixalgebrapolynomialvectorlinear algebra unsolved

Problem Statement

Let A,B∈M2(R)A,B\in\mathcal{M}_{2}(\mathbb{R}) (real 2×22\times 2 matrices), that satisfy A2+B2=ABA^{2}+B^{2}=AB. Prove that (AB−BA)2=O2(AB-BA)^{2}=O_{2}.
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