I wonder if there is a "qualitative way" of predicting from the structure ix of the matrix $A$ which entry of $A^{-1}$ will be the largest. I am specially interested in the case that $A$ is a symmetric $M$-matrix (and so $A^{-1}$ entrywise nonnegative).

There are many nice results like this for the zero pattern so I have some hope something might be possible.

sizeof the largest entry of $A^{-1}$, or itslocation? If the latter, then I don't know that I have any useful comments, unless $A$ is nearly diagonal or something strong. – Greg Martin Feb 22 '13 at 22:28