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+(det−A)′ is multiplicatively invertible in S and AijAik[AjiAki] is additively invertible in S for all i, j, k∈{1,...,n} with j≠k where det+A and det−A are the positive determinant and the negative determinant of A, respectively, and (det−A)′ is the unique inverse of det−A in the inverse semigroup (S,+).
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Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand |
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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
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