For a symmetric, invertible matrix $A=(a_{ij})\in \mathbb{R}^{n\times n}$ with (at least two) nonzero off-diagonal elements, is it possible to bound in absolute value the smallest entry of its inverse in terms of the $a_{ij}$? And if so how?

If this is difficult to answer in such a general setting, would assuming that $A$ is either positive definite or sparse help simplifying the problem? Are there other assumptions that could help simplifying the problem?