Inverse of Positive definite matrix

Question

Assume $P$ is a non-negative positive definite matrix. It is well known what $P^{-1}$ is also positive definite and thus all its diagonal entries are positive. Can we say something about the off diagonal entries of $P^{-1}$? In particular can we say that $$P_{ij}>0 \implies P^{-1}_{ij}<0,\space \forall i\neq j$$.

Answer 1

Here is a random counterexample to your conjecture: $$ C=\pmatrix{4&4&4\\ 4&8&1\\ 4&1&8},\ \det(C)=28, \ C^{-1}=\frac{1}{28}\pmatrix{63&-28&-28\\ -28&16&12\\ -28&12&16}. $$ One can easily check that $C\succ0$ using Sylvester's criterion.

For those entrywise positive $P$s that satisfy your conjecture, $P^{-1}$ belongs to the class of $M$-matrices.