Let a symmetric matrix $A$ be: $$A = \begin{bmatrix} a & -b & -c\\ -b & d & -e\\ -c & -e & f\end{bmatrix}$$ All the values $a, b, \cdots , f \in \mathbb{R}^+$ and infact all the components are $\geq 1$. I have an optimization problem to minimize the trace of inverse of this matrix. Also, all the values in the matrix are somehow related (i.e increasing $a$ increases $b, c, \cdots f$). So there are two options,
$\textbf{Case 1:}$ Maximize $a$ which increases the value diagonal components but decreases the off diagonal components (or makes them more negative), or
$\textbf{Case 2:}$ Minimize $a$ which increases the off diagonal components (makes them less negative) but minimizes the diagonal components.
Is there a theorem of way to prove that either of the two cases is the option to go when looking to minimize trace($A^{-1}$)?