| This paper investigates the characterizations of a basis in semilinear spaceof n-dimensional vectors over semirings, some necessary and sufcient conditions for matricesto be invertible and the factor rank of matrices. First, it gives some sufcient conditionsthat each basis has the same number of elements. Then it presents the matrix is invertibleif and only if the column vectors of the matrix are standard orthogonal. In the following, weobtain some sufcient conditions that the marix which the factor rank is1are idempotent andthe matrix’s factor rank is equal to the factor rank of its square. At last, the relationshipsbetween generalized inverse of matrix A over semirings and linear systems AX=b are discussed.Furthermore, the necessary and sufcient conditions that matrix equation AXB=C is solvableare shown. |
