A characterization of clean matrices in M3(Z)

    An n x n matrix over a commutative ring with identity is clean if it is the sum of an idempotent matrix and a unit. In 2009, Rajeswari and Aziz gave necessary and sufficient criteria for a matrix in M₃(Z) to be clean and discussed the involved Diophantine equations. In this paper, we extend those results to a larger set, M₃(Z). We characterize when a matrix  

is clean. As its application, we discuss the relation between clean matrices and the existence of non-trivial solution of certain types of Diophantine equations.