INVERTIBLE MATRICES OVER SEMIFIELDS

Abstract

A semifield is a commutative semiring (S, +, ·) with zero 0 and iden

tity 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 det A = 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.