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.
Khurana, D. & Lam, T. Y. (2004). Clean matrices and unit-regular matrices. Journal of Algebra, 280, 683-698.
Rajeswari, K. N. & Aziz, R. (2009). A note on clean matrices in . International Journal of Algebra, 3(5), 241- 248.
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