  An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it
can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that
commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of
matrices over projective-free rings. These extend many known results on
strongly clean matrices over commutative local rings.
