Matrix analysis , inequalities , diagonal entries

Question

Suppose we have an invertible positive definite matrix and we know its diagonal elements when seen as a vector are in increasing order , what can be said about diagonal vector of its inverse ? , if it's in increasing order or decreasing order etc

Answer 1

No:

for example

$\begin{pmatrix} 3 & 1 & 1 \\ 1 & 4 & 3 \\ 1 & 3 & 5 \end{pmatrix}$

has an inverse where the diagonal elements are not in order

$\begin{pmatrix} \;\;\frac{11}{30} & -\frac1{15} & -\frac{1}{30}\\ -\frac1{15} & \;\;\frac{14}{30} & -\frac4{15} \\ -\frac{1}{30} & -\frac4{15} & \;\;\frac{11}{30} \end{pmatrix}$