TITLE: Inverse of $2 \times 2$ block matrices
QUESTION [11 upvotes]: Let matrices $A, B, C, D$ be invertible. How can I find the inverse of the following block matrix?
$$\begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}$$
Thank you.

REPLY [13 votes]: As $D$ is invertible, the block matrix is invertible if and only if the Schur complement of $D$ --- i.e. the matrix $S=A-BD^{-1}C$ --- is invertible. In that case, you can see Wikipedia for the block matrix inversion formula.