I am trying to find a nice way of computing the determinant of the matrix
\begin{equation} M= \begin{bmatrix} A & B \\ C & D \end{bmatrix} \in \mathbb{R}^{T\times T} \end{equation}
where $A \in \mathbb{R}^{M\times N}$, $B \in \mathbb{R}^{M\times (T-N)}$, $C \in \mathbb{R}^{(T-M)\times N}$ and $D \in \mathbb{R}^{(T-M)\times (T-N)}$. Furthermore, $(A)_{i,j} = f_i(x_j)$, $(C)_{i,j} = g_i(x_j)$ where $f$ and $g$ are differentiable functions.
I know there are nice ways to compute it when either $A$ or $D$ are invertible but is there a way to do it in the more general case above?
When $A$ is invertible, $$|M|=|A||D-CA^{-1}B|$$ A similar formula holds when $D$ is invertible. The question is specifically if such formulas can be extended to give $|M|$ in the case where neither $A$ nor $D$ is invertible (indeed, both could be non-square).