Inversion of Matrices over Boolean Semirings

N. Sirasuntorn, N. Udomsub

Abstract

It is well-known that a square matrix A over a commutative ring R with identity is invertible over R if and only if detA is a multiplicatively invertible element of R if and only if detA is a multiplicatively invertible element of R. As a consequence, we have that a square matrix A over a Boolean ring R with identity 1 is invertible over R if and only if $\det^+ A+\det^- A = 1$ where $\det^+ A$ and $\det^- A$ are the positive determinant and the negative determinant of A, respectively. This result is generalized to Boolean semirings with identity. By a Boolean semiring we mean a commutative semiring S with zero in which $x^2 = x$ for all $x \in S$. By making use of Reutenauer and Sraubing's work in 1984, we show that an $n\times n$ matrix A over a Boolean semiring S with identity 1 is invertible over matrix A over a Boolean semiring S with identity 1 is invertible over S if and only if $\det^+ A+\det^- A = 1$ and $2A_{ij}A_{ik}=0[2A_{ji}A_{ki}=0]$ for all $i,j,k\in\{1,\cdots,n\}$ such that $j \neq k$.