Problem

Source: Romanian National Olympiad 2000, Grade XI, Problem 1

Tags: Matrices, linear algebra, complex numbers



Let $ \mathcal{M} =\left\{ A\in M_2\left( \mathbb{C}\right)\big| \det\left( A-zI_2\right) =0\implies |z| < 1\right\} . $ Prove that: $$ X,Y\in\mathcal{M}\wedge X\cdot Y=Y\cdot X\implies X\cdot Y\in\mathcal{M} . $$