That is awesome!!!! Then I have to find other method to derive the convergence... Basically I want to see in a Gaussian process, how the covariance matrix changes as all the points go to one point by using scaling the sampling space. I will try other methods to see whether any structure of $B$ would be helpful. Thank you so much for your awesome answer!

Thank you for your reply! But I do not think things could be that easy since there maybe interactions of $det(A)$ and $\sum A_{ij}$, and $A_{ij}$ have different signs which make the numerator term more complicated. But that is a good point!