The Properties Of Invertible An Semi-invertibhitv Matrices Over Zero-sum-free Semirings

This paper discussed the properties of invertible matrices and semi-invertibility matrices and their applications for solving matrix equations over zero-sum-freesemirings. It first got a necessary and sufcient condition that a matrix is invertible, andshowed the structure of an invertible matrix, then proved that the invertible matrix after thepower of limited times can get an invertible diagonal matrix, obtained that the invertible ma-trix product the transpose of itself can be expressed by some permutation matrices multipliedtheir transpose matrices. It also discussed the bideterminant of matrix and the sufcient con-dition of matrix which is indivertible. After that, it gave an example which shows that matrixequations AX=B and X+A1B=A2B do not always have the same solutions, where A is aknown n×n semi-invertible matrix and B is an unknown n-dimensions column vector, A1andA2satisfy that I+AA1=AA2and I+A1A=A2A. Finally, it presented some conditions thatboth two systems as mentioned have the same solutions, and investigated some properties ofsemi-invertible of matrices.