Problem

Source: Romania National Olympiad 2023

Tags: matrix, linear algebra, Non singular matrices



Let $A,B \in M_{n}(\mathbb{R}).$ Show that $rank(A) = rank(B)$ if and only if there exist nonsingular matrices $X,Y,Z \in M_{n}(\mathbb{R})$ such that \[ AX + YB = AZB. \]