# INVERTIBLE MATRICES OVER SEMIFIELDS

Keywords:
Semiﬁeld, invertible matrix

### 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.

Published

2020-02-25

Section

Articles