This is a sequel to this question in which I sought to expand on this question. Let me put it straight. Given a non-singular symmetric real matrix $A\in\mathbb{R}^{n\times n}$ such that $A_{ii}>0$.
Can we conclude that $A$ is positive definite if $$(A^{-1})_{ii}\ge \frac1{A_{ii}}$$ holds for all $1\le i\le n$?
No. Random counterexample: $$ A=\pmatrix{ 2& 3&-3\\ 3& 2&-3\\-3&-3& 4}, \ A^{-1}=\frac12\pmatrix{1&3&3\\ 3&1&3\\ 3&3&5}. $$ $A$ is nonsingular as its determinant is $-2$, but $A$ isn't positive definite as it has an eigenvector $(1,-1,0)^T$ corresponding to the eigenvalue $-1$.