Relation between diagonal entries of $A^{-1}$ and inverse values of $a_{ii}$ for positive definite $A$. I'd like to expand upon this question. Namely, it says that if $A$, $A=A^T$, is a positive definite matrix, then it holds that \begin{equation}\tag{*}(A^{-1})_{ii}\ge \frac1{A_{ii}}.\end{equation}
Can we prove the converse, i.e.,
if (*) holds for all $1\le i\le n$, then $A$ is positive definite?
OK, as suggested by @Klaus and @Jan I accept this answer and continue here.

No, that's even wrong for numbers, e.g. $A = -1$.
