The expression for the group inverse of the anti-triangular block matrix

Abstract

In this paper, we present the explicit expression for the group inverse of the sum of two matrices.
As an application, the explicit expression for the group inverse of the anti-triangular block matrix
$\begin{pmatrix}A&B\\C&0\end{pmatrix}$ and $\begin{pmatrix}0&B\\C&D\end{pmatrix}$ are obtained without any conditions on sub-blocks