Group inverse for the block matrices with an invertible subblock q Changjiang Bu * , Min Li, Kuize Zhang, Lan Zheng Dept. of Applied Math., College of Science, Harbin Engineering University, Nantong Street, Harbin 150001, PR China article info Keywords: Skew field Group inverse Invertible subblock {1}-Inverse abstract Let M ¼ A B C D   (A is square) be a square block matrix with an invertible subblock over a skew field K. In this paper, we give the necessary and sufficient conditions for the existence as well as the expressions of the group inverse for M under some conditions. Published by Elsevier Inc. 1. Introduction The group inverses of block matrices have numerous applications in many areas, such as singular differential and differ- ence equations, Markov chains, iterative methods, cryptography and so on (see [1–6]). In 1979, Campbell and Meyer proposed an open problem to find an explicit representation for the Drazin inverse of a 2  2 block matrix A B C D   , where the blocks A and D are supposed to be square matrices but their sizes need not be the same (see [1]). Until now, this problem has not been solved completely, and there is even no known expression for the Drazin (group) inverse of A B C 0   which was posed by Campbell in 1983 in [3]. However, there are many literatures about the existence and the representation of the Drazin (group) inverse for the block matrix M ¼ A B C D   under some conditions (see [7–19]). For example, in [7], the exis- tence of the group inverse of M over a field F was investigated under the condition: A and I þ CA 2 B are invertible. The representations of the Drazin inverse and the group inverse of M over the complex number field C were given in [12], when A ¼ B ¼ I n and D ¼ 0. In [13,14,16], the authors studied the existence and representations of the group inverse of M over skew fields under different conditions: in [13], A is square, A ¼ B; A 2 ¼ A and D ¼ 0. In [14], A ¼ I n ; D ¼ 0 and rankðCBÞ 2 ¼ rankðBÞ¼ rankðCÞ. And in [16], A is square and C ¼ 0. Let K mn be the set of all m  n matrices over a skew field K, I n the n  n identity matrix over K and rankðAÞ the rank of A 2 K mn . For a square matrix A, the Drazin inverse of A is the matrix A D satisfying A lþ1 A D ¼ A l ; A D AA D ¼ A D ; AA D ¼ A D A for all integers l P k; ð1:1Þ where k ¼ IndðAÞ is the index of A, the smallest nonnegative integer such that rankðA k Þ¼ rankðA kþ1 Þ. It is known that A D is existent and unique (see [20]). For A 2 K nn , if there exists a matrix X 2 K nn satisfying the matrix equation AXA ¼ A; ð1:2Þ then we say that X is a {1}-inverse of A and it is denoted by A ð1Þ . Denote the set of all {1}-inverses of A by Af1g (see [20]). Furthermore, if the matrix X also satisfies the matrix equations 0096-3003/$ - see front matter Published by Elsevier Inc. doi:10.1016/j.amc.2009.04.054 q Supported by Natural Science Foundation of the Heilongjiang Province, No. 159110120002. * Corresponding author. E-mail addresses: buchangjiang@hrbeu.edu.cn (C. Bu), limin1983_2004@yahoo.com.cn (M. Li), zkz0017@163.com (K. Zhang), zhenglan000@163.com (L. Zheng). Applied Mathematics and Computation 215 (2009) 132–139 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc