A note on Products of Nilpotent Matrices

Abstract

Let F be a field. A matrix A ∈ M n (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 2×2. This result was proved independently by Sourour [6] and Laffey [4]. 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 [4] is provided, and a problematic detail in the Proposition which is the main device underpinning the proof by Sourour is resolved.