How to compute the determinant of this block matrix?

Question

$$M = \left[\begin{matrix} -C & -A \\ A^\top & 0 \end{matrix} \right]$$ I found a paper using $\det(M) = \det(A^\top C^{-1}A)$ but don't know how to prove this.

Answer 1

Performing Gaussian elimination,

$$\begin{bmatrix} \mathrm I & \mathrm O\\ \mathrm A^\top \mathrm C^{-1} & \mathrm I \end{bmatrix} \begin{bmatrix} - \mathrm C & -\mathrm A \\ \mathrm A^\top & \mathrm O \end{bmatrix} = \begin{bmatrix} - \mathrm C & -\mathrm A \\ \mathrm O & -\mathrm A^\top \mathrm C^{-1} \mathrm A \end{bmatrix}$$

Note that the determinant of a block triangular matrix is the product of the determinants of the diagonal blocks.