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Research articles

ScienceAsia 37 (2011): 130-135 |doi: 10.2306/scienceasia1513-1874.2011.37.130

Some results concerning invertible matrices over semirings

Surachai Sombatboriboona, Winita Morab, Yupaporn Kemprasita,*

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. Additively inverse commutative semirings with zero 0 and identity 1 are a generalization of commutative rings with identity. In this paper, we generalize the above known result as follows. An n×n matrix A over an additively inverse commutative semiring S=(S,+,⋅) with 0, 1 is invertible over S if and only if det+A+(detA)′ is multiplicatively invertible in S and AijAik[AjiAki] is additively invertible in S for all i, j, k∈{1,...,n} with jk where det+A and detA are the positive determinant and the negative determinant of A, respectively, and (detA)′ is the unique inverse of detA in the inverse semigroup (S,+).

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a Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
b Department of Mathematics, Faculty of Science, Prince of Songkla University, Songkhla 90112, Thailand

* Corresponding author, E-mail: yupaporn.k@chula.ac.th

Received 20 Jan 2011, Accepted 11 Apr 2011