In my lecture notes is this method for inverting a matrix
I do not recall seeing this method of splitting the matrix up and finding the inverses of the quadrants before. Does it have a name? Is it dependent on the top right and bottom left being empty, or when else is it valid?
See the block matrix inverse formula. Note when $B=C=O$, it simplifies to
$$\begin{pmatrix} A & O \\ O & D\end{pmatrix}^{-1}=\begin{pmatrix}A^{-1} & O \\ O & D^{-1} \end{pmatrix}.$$