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)(u,v) satisfying uv=vuuv=vu. We provide an example of a field FF and an integer n   such that the commuting graph of Matn(F)Matn(F) has maximal possible diameter, equal to six.