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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.

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I removed a tag "mat" since I assume it was created in error. Appologies if this should have some meaning that escapes me. –  quid Feb 22 '13 at 21:03
    
@quid: Thanks. The "mat" was, obviously, a miscarried "matrices". –  Felix Goldberg Feb 22 '13 at 21:57
    
Are you interested in the size of the largest entry of $A^{-1}$, or its location? 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
    
@GregMartin: Only the location (this time). –  Felix Goldberg Feb 22 '13 at 22:46

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