On the Waring Problem for Matrices over Finite Fields

We prove that if $k$ is a positive integer then for every finite field $\mathbb{F}$ of cardinality $q\mathrm{\ne }2$ and for every positive integer $n$ such that $q^n>(k-1{)}^4$, every $n\times n$ matrix over $\mathbb{F}$ can be expressed as a sum of three $k$-th powers. Moreover, if $n\ge 7$ and $k<q$, every $n\times n$ matrix over $\mathbb{F}$ can be written as a sum of two $k$-th powers.