MathNet 2023-part-05 — 191 problems (EN–VI) Question 59

Let $A$ and $B$ be two matrices from $M_n(\mathbb{R})$. Show that $\text{rank}(A) = \text{rank}(B)$ if and only if there are three invertible matrices $X, Y, Z \in M_n(\mathbb{R})$ such that $AX + YB = AZB$.