Problem

Source: Romania National Olympiad 2022

Tags: linear algebra, Matrices, romania



Let $A,B\in\mathcal{M}_n(\mathbb{C})$ such that $A^2+B^2=2AB.$ Prove that for any complex number $x$\[\det(A-xI_n)=\det(B-xI_n).\]Mihai Opincariu and Vasile Pop