The Invertible Matrices And Rank Of Matrices Over Commutative Semirings

Abstract This paper mainly focuses on the invertible matrices and determinantal rank of matrices over commutative semirings. First, we shall establish the extended semiring of a given semiring. and study the properties of elements in extended semirings. Then, an equivalent description of the invertible matrix and a method for finding the inverse of an invertible matrix without dependence on any auxiliary function will be given. Finally, we shall show some properties about the determinantal rank of matrices and their applications on the solvability of linear system over commutative semirings.