Suppose we have a symmetric matrix A such that its diagonal elements are positive. What can we say about the diagonal elements of its inverse? Are also they positive?
My attempt:
We know the inverse of a symmetric matrix is symmetric.
Suppose our matrix is $2*2$ $$\begin{bmatrix}a & b\\ b & d\end{bmatrix}$$
and the inverse is $$\begin{bmatrix}a' & b'\\ b' & d'\end{bmatrix}$$
So in order to the product to be identity matrix :$$\begin{bmatrix}a & b\\ b & d\end{bmatrix}\begin{bmatrix}a' & b'\\ b' & d'\end{bmatrix}=\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}$$
we have:$$aa'+bb'=1$$$$bb'+dd'=1$$
so then:$$a'=\frac{1-bb'}{a} , d'=\frac{1-bb'}{d}$$ as a and d are positive, $1-bb'$ should be positive to fulfill the requirement, so:$$bb'<1$$.
So does this mean that the statement does not always hold? Or my approach is not complete yet?