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I am working on finding the determinant of the following block matrix $$ \begin{pmatrix} C & D \\ D^* & C \\ \end{pmatrix}, $$ where $C$ and $D$ are $4 \times 4$ matrices with complex entries that do not commute. I have looked up a theorem that states $$ \det\begin{pmatrix} A & B \\ C & D \\ \end{pmatrix}=\det(A-B)\det(A+B), $$ when $A=D$ and $B=C$, but does there exist a similar simplification for my situation?

Any and all help is much appreciated!

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Hint: Suppose $C$ is invertible [otherwise use the matrix $C - \lambda I$ in place of $C$ for that will certainly be invertible for infinitely many $\lambda \in \mathbb{C}$].

Write $\begin{pmatrix} C & D \\ D^{*} & C \end{pmatrix} = \begin{pmatrix} I & 0 \\ D^{*}C^{-1} & I \end{pmatrix} \begin{pmatrix} C & D \\ 0 & C - D^{*}C^{-1}D \end{pmatrix}$

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