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Romania NMO 2023 Grade 11 P2

Source: Romania National Olympiad 2023

April 14, 2023
matrixlinear algebraNon singular matrices

Problem Statement

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