MathNet 2018-part-05 — 181 problems (EN–VI) Question 26

Let $n$ be a positive integer and let $A$ be a matrix from $M_n(\mathbb{C})$. Prove that $A^2 = O_n$ if and only if there exist two matrices $B$ and $C$ in $M_n(\mathbb{C})$ so that $A = BC$ and $CB = O_n$.