On Group Inverses of Matrices with Order and Rank Perturbations

Abstract

Group inverse of a square matrix A exists if and only if rank of A is equal to rank of A². Group inverses have many applications, prominent among them is in the analysis of finite Markov chains discussed by Meyer (1982). In this note necessary and sufficient conditions for the existence of group inverses of bordered matrix, $B=\left[{}_{{\beta }^{\prime }}^A\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\hspace{0.17em}}_c^a\right]\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}C=A+a{\beta }^{\prime }$ are obtained and expressions for the group inverses in terms of group inverse of A are given, whenever they exist. Also necessary and sufficient condition for the existence of group inverse of A in terms of group inverse of B and C are given. An application to perturbation in Markov chains is illustrated.