Some Research On Generalized Inverse Of Matrices And Matrix Equations Over Commutative Semirings

This thesis mainly focuses on the generalized inverses of matrices and the solvability of a class of matrix equations over commutative semirings.Firstly,we discuss the relationship between the generalized inverses and the determinant rank of matrices,and use the bideterminant to obtain the existence conditions of the {1}-inverse of a matrix with determinant rank r.Secondly,we study the solvability of a class of matrix equations by using the generalized inverses and multiplicative regular complements of matrices.We consider the solvability of the matrix equation AXB=C and the system of matrix equations moreover,we give the expression of general solution of the system of matrix equations on additive cancellable commutative semirings.Then,we give some necessary and sufficient conditions that the equation AXB+CYD=E is solvable.Finally,we discuss the solvable conditions of the matrix equation AX+YB=C,and give a partial expression of the {1,2}-inverse of a block matrix.