A matrix ring with commuting graph of maximal diameter

The commuting graph of a semigroup is the set of non-central elements; the edges are defined as pairs $(u,v)$ satisfying $uv=vu$. We provide an example of a field $F$ and an integer $n$ such that the commuting graph of ${\mathrm{Mat}}_n(F)$ has maximal possible diameter, equal to six.