My question follows from <https://math.stackexchange.com/questions/3857976/inverse-inequality-of-symmetric-matrix>. Suppose we assume that $A$ and $B$ are two positive definite matrices with positive entries and $A\geq B $ entry wise.  

Can we say that $A^{-1}\leq B^{-1}$ entry wise?   

I tried with numeric examples in Matlab, but I am not getting any counter-example.
Any help would be really great.



$A\geq B\Rightarrow A^{-1}\leq B^{-1}$ entrywise for pos.def. symmetric matrices?
$$A=\left(
\begin{array}{cc}
 1 & \frac{1}{10} \\
 \frac{1}{10} & 1 \\
\end{array}
\right),\;\;A^{-1}=\left(
\begin{array}{cc}
 \frac{100}{99} & -\frac{10}{99} \\
 -\frac{10}{99} & \frac{100}{99} \\
\end{array}
\right),\;\;B=\left(
\begin{array}{cc}
 1 & 0 \\
 0 & 1 \\
\end{array}
\right)=B^{-1}$$
is a counter example.


