Nice form for determinant of a special matrix

Question

I have a positive definite symmetric matrix that looks like

where matrices $A,B,C,D$ are positive definite symmetric matrices. Is there a nice way to calculate the determinant? For example, the upper left block is block diagonal and its determinant is just $\det(A)\cdot\det(B)$.

I want to find the determinant of the whole matrix though.

I have provided an example of such matrix using dput in R. You just need to copy and paste to R for a reproducible example

Answer 1

Using the Schur complement, your determinant is $$ \det(A)\det(B)\det(C)\det\big(D - E'A^{-1}E - F'B^{-1}F - G'C^{-1}G\big). $$ Does this help?