# Determinant of block matrix with off-diagonal blocks conjugate of each other.

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!

• Was that Theorem found by You? – Ganesh Feb 11 '16 at 14:19
• – anon Feb 11 '16 at 14:21

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