Invertible matrices in certain commutative subsemirings of full matrix semirings

Abstract

Then every field is a semifield. For a semifield S, we let Dn(S) denote the set of all A ε Mn (S) of the form (Equation Presented) where Mn (S) is the full n × n matrix semiring over S. Then Dn(S) is a maximal commutative subsemiring of the semiring Mn(S). If S is a field, it is known that A ε Dn(S) is invertible if and only if det A ≠ 0. In this paper, invertible matrices in Dn(S) where S is a semifield which is not a field are characterized. It is shown that if S is a semifield which is not a field, then A ε Dn(S) is an invertible matrix over S if and only if (xi = 0 if and only if yi ≠ 0). © 2016 Academic Publications, Ltd.