Partitioned matrix of partitioned matrices Please, help solve this question:
Given the partitioned matrix
\begin{equation} 
P=\left( \begin{array} {c,c} A \quad B \\ C \quad 0 \end{array} \right)
\end{equation}
where A is a 2x2 block matrix, B is a 2x1 matrix, C is a 1x2 matrix and 0 is a 1x1 zero matrix, such that
\begin{equation} 
A=\left( \begin{array} {c,c,c} A_{11} \quad A_{12}\quad B_{11} \\ A_{21} \quad A_{22} \quad B_{21}\\ C_{11} \quad \, C_{12} \quad \, 0 \end{array} \right)
\end{equation}
Additionally, $A_{11}, A_{12}, A_{21}$ and $A_{22}$ themselves are 2x2, 2x3, 3x2 and 3x3 matrices, respectively. So, we are dealing with matrices of matrices of matrices.
Now, I know that $Det(P) = Det(A)Det(-C A^{-1} B)$ but I have difficulty in manipulating the block matrix multiplication, $C A^{-1} B$. Please, how do I solve it?
Thank you.

Simply take determinants on both sides of the identity
$$
\left( \begin{array} {c,c} A \quad 0 \\ C \quad I 
\end{array} \right)
\left( \begin{array} {cc} I & A^{-1}B \\  0 & -CA^{-1}B 
\end{array} \right)=
\left( \begin{array} {cc} A \quad B \\ C \quad 0 
\end{array} \right)
$$
