Let $A=[A_{0}\ E;E^{T} \ B]$ be a real positive definite matrix and let $B$ be a principal submatrix. I am interested in tightly bounding $\frac{|B|}{|A|}$ from below in some "explicit" way that will involve the entries of $A$. ($A$ is actually a strictly diagonally dominant $M$-matrix, if that helps).

An obvious first approach is to try the Fischer-Hadamard inequality which gives:

$$ \frac{|B|}{|A|} \geq \frac{1}{|A_{0}|}. $$

This is not bad at all, but insufficient for my purposes (it gets the order of magnitude right in my examples but I need more than that).

Another approach is to let $C=A/B$ be the Schur complement and to observe that $|A|=|B||C|$ and so $\frac{|B|}{|A|} \geq \frac{1}{|C|}$; however, I can't find appropriate bounds on $|C|$ to use as the next step.

Do you know of such bounds? Or another way to tackle this problem?