INVERTIBLE MATRICES OVER SEMIFIELDS

Abstract

A semifield is a commutative semiring (S,+,·) with zero 0 and identity 1 such that (S{0},·) is a group. Then every field is a semifield. It is known that a square matrix A over a field F is an invertible matrix over F if and only if detA = 0. In this paper, invertible matrices over 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 square matrix A over S is an invertible matrix over S if and only if every row and every column of A contains exactly one nonzero element.