I have an equation of the form $C=B^{T}AB$ where $A$ is an $n \times n$ block diagonal matrix, $B$ is an $n \times p$ matrix, and so $C$ is a symmetric matrix. I would like to establish conditions around when $\det(C)=0$ based on my entries of $A$ and $B$ which are pretty long expressions.
I've managed to do this for the special case when $A$ is exactly a diagonal matrix, but I'd like to generalize my results if possible to when A is block diagonal (or even just symmetric). Are there any known properties for the determinant of the matrix product form of $C=B^T AB$ when B is not square and A is block diagonal/symmetric?
If $p\ne n$ then either $B$ or $B^T$ will fail to be injective. Therefore the product of them, as a map, will fail to be injective -- therefore the determinant will be $0$.
If $p=n$ then $|B^TAB|=|B^T||A|B| = |B|^2 |A|=0$ if and only if one of $A,B$ has a zero determinant.