  A ring $R$ is called strongly clean if every element of $R$ is the sum of a
unit and an idempotent that commute with each other. A recent result of
Borooah, Diesl and Dorsey \cite{BDD05a} completely characterized the
commutative local rings $R$ for which ${\mathbb M}_n(R)$ is strongly clean. For
a general local ring $R$ and $n>1$, however, it is unknown when the matrix ring
${\mathbb M}_n(R)$ is strongly clean. Here we completely determine the local
rings $R$ for which ${\mathbb M}_2(R)$ is strongly clean.
