A Note on the Group Inverses of Block Matrices Over Rings

Abstract

Suppose $R$ is an associative ring with identity 1. In this paper, We give the necessary and sufficient conditions for the existence and explicit representations of the group inverses of the following two classes block matrices.

(i) $M=\left(\begin{array}{cc}AX+YB&A\\B&0\end{array}\right)$, where $A^{\sharp}$ exists,

$XA=AX$, $R(A)\subset R(AX)$ and $R_{r}(A)\subset R_{r}(XA)$;

(ii) $M=\left(\begin{array}{cc}A&B\\C&D\end{array}\right)$, where $A^{\sharp}$ exists, $A^{\pi}B=0$ and $(D-CA^{\sharp}B)^{\sharp}$ exists.\\

The paper's results generalize some relative results of Li et al. (J. Harbin Engineering Univ., 34, 658-661, 2013), Cao et al. (Appl. Math. Comput., 217, 10271-10277, 2011), Bu et al. (Appl. Math. Comput., 215, 132-139, 2009),  Deng et al. (Linear Algebra Appl, 435, 2766-2783, 2011), and Liu et al. (J. Appl. Math. http://dx.doi.org/10.1155

/2013/247028). Some examples are given to illustrate our results.