Inverse of $2 \times 2$ block matrices

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.

It is not necessarily invertible at all -- certainly not if, for example, $A=B$ and $C=D$.

Conversely, the block matrix can be invertible even if neither of the blocks are. In cases such as $$ \left[\begin{array}{cc|cc}1&0&0&0\\0&0&0&1\\ \hline 0&1&0&0\\0&0&1&0\end{array}\right] \quad\text{or}\quad \left[\begin{array}{cc|cc}1&0&0&1\\0&2&1&0\\ \hline 0&2&2&0\\2&0&0&1\end{array}\right] $$ you can even have $AD-BC=0$, so attempting to extend determinant-based methods to block matrices seems to be doomed to failure.