Necessary and Sufficient Conditions for The Solutions of Linear Equation System

Authors

  • Gregoria Ariyanti Prodi Pendidikan Matematika Universitas Katolik Widya Mandala Surabaya

DOI:

https://doi.org/10.20956/jmsk.v17i1.10352

Keywords:

semiring, linear equations system, matrix

Abstract

A Semiring is an algebraic structure (S,+,x) such that (S,+) is a commutative Semigroup with identity element 0, (S,x) is a Semigroup with identity element 1, distributive property of multiplication over addition, and multiplication by 0 as an absorbent element in S. A linear equations system over a Semiring S is a pair (A,b)  where A is a matrix with entries in S  and b is a vector over S. This paper will be described as necessary or sufficient conditions of the solution of linear equations system over Semiring S viewed by matrix X  that satisfies AXA=A, with A in S.  For a matrix X that satisfies AXA=A, a linear equations system Ax=b has solution x=Xb+(I-XA)h with arbitrary h in S if and only if AXb=b.

References

Anton, H., 2013. Elementary Linear Algebra: Applications Version/ Howard Anton, Chris Rorres, 11th edition. The United States of America.

Ariyanti, G., Suparwanto, A., and Surodjo, B., 2015. Necessary and Sufficient Conditions for The Solution of The Linear Balanced Systems in The Symmetrized Max Plus Algebra. Far East J. Math. Sci. (FJMS), Vol. 97, No. 2, 253-266.

Brewer, J.W., Bunce, J.W., and Van Vleck, F.S., 1986. Linear Systems over Commutative Rings. Marcel Dekker, New York.

Brown, W. C., 1993. Matrices over Commutative Rings. Marcel Dekker, Inc., New York.

Meyer, C. D., 2000. Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia.

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Published

2020-08-24

Issue

Vol. 17 No. 1 (2020): JMSK, SEPTEMBER, 2020

Section

Research Articles

License

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