Mathieu subspaces of codimension less than n of Mat_n(K)

We classify all Mathieu subspaces of ${\rm Mat}_n(K)$ of codimension less than $n$, under the assumption that ${\rm char\,} K = 0$ or ${\rm char\,} K \ge n$. More precisely, we show that any proper Mathieu subspace of ${\rm Mat}_n(K)$ of codimension less than $n$ is a subspace of $\{M \in {\rm Mat}_n(K) \mid {\rm tr\,} M = 0\}$ if ${\rm char\,} K = 0$ or ${\rm char\,} K \ge n$. On the other hand, we show that every subspace of $\{M \in {\rm Mat}_n(K) \mid {\rm tr\,} M = 0\}$ of codimension less than $n$ in ${\rm Mat}_n(K)$ is a Mathieu subspace of ${\rm Mat}_n(K)$ if ${\rm char\,} K = 0$ or ${\rm char\,} K \ge n+1$.