A quick question $$X = |( \operatorname{adj} (A^{-1}(B^{-1}){2B^{T}})|
= |(\operatorname{adj}(2B^T) \operatorname{adj}(B^{-1})\operatorname{adj}(A^{-1}))| $$
$$ = |2B^T|^2 |B^{-1}|^2 |A^{-1}|^2 = 2^6 |B|^2 \cfrac{1}{(|A||B|)^2} = \cfrac{2^6}{|A|^2}$$ Have I done anything wrong?