Outer inverses: Jacobi type identities and nullities of submatrices Original Research Article

According to the Jacobi identity, if A is an invertible matrix then any minor of A−1 equals, up to a sign, the determinant of A−1 times the complementary minor in the transpose of A. The identity is extended to any outer inverse, thereby generalizing several results in the literature for special generalized inverses. A permanental analog of the Jacobi identity is proved. Bounds are obtained for the difference between the nullity of a submatrix of A and that of the complementary submatrix in any generalized inverse or an outer inverse of A. The result extends earlier work of Fiedler, Markham and Gustafson for the inverse and of Robinson for the Moore–Penrose inverse.