  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 ${\rm char\,} K \ge n+1$.
