It is known the connection between the best approximation and Moore-Penrose invertibility. In particular, the best approximation pair of two linear varieties is related to generalized invertibility of a block matrix of the form \( \begin{bmatrix} I_n & A\\ B & I_m \end{bmatrix} \)\( \).
In this paper, we address Moore-Penrose and group invertibility of a matrix of the form \( \begin{bmatrix} I_n & A\\ B & D \end{bmatrix} \).