A note on Products of Nilpotent Matrices

Abstract

Let $\mathscr{F}$ be a field. A matrix $A$ of order $n \times n$ over $\mathscr{F}$ is a product of two nilpotent matrices if and only if it is singular, except if $A$ is a nonzero nilpotent matrix of order $2 \times 2$. This result was proved independently by Sourour and Laffey. While these results remain true and the general strategies and principles of the proofs correct, there are certain problematic details in the original proofs which are resolved in this article. A detailed and rigorous proof of the result based on Laffey's original proof is provided, and a problematic detail in the Proposition which is the main device underpinning the proof by Sourour is resolved.