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 $\big( \smallmatrix A & B\\ C & D \endsmallmatrix \big)$ and $\big( \smallmatrix 0 & B\\ C & D \endsmallmatrix \big)$ are obtained without any conditions on sub-blocks.