Relation between square matrix A and inverse of partial Gram matrix

Question

Given a $N \times N$ matrix $A$ (it has structure, specifically it is a real-valued Vandermonde matrix if that helps) from which the first $L \ll N$ columns are taken to form $\tilde{A}$.

Now the inverse Gram matrix of $\tilde{A}$ is formed:

$$ G = \left(\tilde{A}^T \tilde{A}\right)^{-1} $$

Is there any meaningful relation or expression to exploit between either:

1. entries of $A$
2. entries of $A^{-1}$
3. Determinant of $A$

and/or

1. G
2. $\operatorname{Tr}(G)$

?