Determinant of matrix composition Considering a matrix given by composition of square matrices like:
$$
M = \begin{pmatrix}
A & B\\
C & D
\end{pmatrix}
$$
I want to calculate its determinant $|M|$. Consider that all components are square matrices of course.
Can this be related somehow to the components $A$, $B$, $C$ and $D$?

Note that $$\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} A & 0 \\ C & I \end{pmatrix} \begin{pmatrix} I & A^{-1}B \\ 0 & D - CA^{-1}B \end{pmatrix} = \begin{pmatrix} A - BD^{-1}C & BD^{-1} \\ 0 & I \end{pmatrix} \begin{pmatrix} I & 0 \\ C & D\end{pmatrix}$$
The middle factorization is valid assuming $A$ is invertible and the last factorization is valid assuming $D$ is invertible. Further, $$\det \left( \begin{pmatrix} X_{11} & X_{12}\\ 0 & X_{22} \end{pmatrix} \right) = \det(X_{11}) \det(X_{22})$$ and $$\det \left( \begin{pmatrix} X_{11} & 0\\ X_{21} & X_{22} \end{pmatrix} \right) = \det(X_{11}) \det(X_{22})$$
Hence, $$\det \left( \begin{pmatrix} A & B \\ C & D \end{pmatrix} \right) = \det(A) \det(D-CA^{-1}B) = \det(A-BD^{-1}C) \det(D)$$
The middle one assumes $A$ is invertible, and the last one assumes $D$ is invertible.
