Let $L$ be a symmetric block matrix written as
$$L = \left[\begin{array}{ccc} \alpha & \beta e_{n}' & \gamma e_n'\\ \beta e_{n} & A & B \\ \gamma e_n & B' & C \end{array} \right]$$
Is there any way to find the Moore-Penrose inverse (pseudoinverse) $L^\dagger$ of $L$ as a block matrix such that the blocks of $L^\dagger$ are expressed in terms of blocks of $L$?
Here, $\alpha$, $\beta$, $\gamma$ are scalars, $A$, $C$ are square symmetric matrices and $e_n$ denotes the $n \times 1$ vector of all ones. Further, $'$ is used to denote the transpose of a matrix.
Any suggestions/hints will be beneficial. Thanks in advance!