Fields over Which Matrices Can Be Represented As the Sum of Potent and Nilpotent Matrices

We study fields over which every matrix can be represented as a sum of two potent matrices and a nilpotent matrix. In particular, it is shown that over a field $$P$$ every matrix can be represented as a sum of two idempotent matrices and a nilpotent matrix exactly when either $$P \cong {{\mathbb{F}}_{2}},$$ or $$P \cong {{\mathbb{F}}_{3}}$$ .