I have a problem where I have to calculate the inverse of a specific block matrix: $$ \begin{pmatrix}M^{-1}+B\Lambda^{-1}B^T & B\Lambda^{-1}A^T\\ A\Lambda^{-1} B^T & L^{-1} + A\Lambda^{-1}A^T \end{pmatrix} $$ All three matrices $M$, $L$ and $\Lambda$ are symmetric but $A$ and $B$ are not square.
The general formula for computing the inverse of a block matrix requires either the inverse of $M^{-1} + B\Lambda^{-1}B^T$ or the inverse of $L^{-1} + A\Lambda^{-1}A^T$. Since that is the inverse of a sum of two matrices it can't be solved so I was hoping that there is another approach that might work.
If it might help, this is part of a larger covariance matrix: $$ \begin{pmatrix}\Lambda^{-1} & \Lambda^{-1}B^T & \Lambda^{-1}A^T \\ B\Lambda^{-1} & M^{-1}+B\Lambda^{-1}B^T & B\Lambda^{-1}A^T\\ A\Lambda^{-1} & A\Lambda^{-1} B^T & L^{-1} + A\Lambda^{-1}A^T \end{pmatrix} $$