Commutators from a hyperplane of matrices

Abstract

Denote by $M_n(K)$ the algebra of $n$ by $n$ matrices with entries in the field $K$. A theorem of Albert and Muckenhoupt states that every trace zero matrix of $M_n(K)$ can be expressed as $AB-BA$ for some pair $(A,B)$ of matrices of $M_n(K)$. Assuming that $n>2$ and that $K$ has more than 3 elements, we prove that the matrices $A$ and $B$ can be required to belong to an arbitrary given hyperplane of $M_n(K)$.