Let $\mathbb{M}^{3\times3}$ denote the set of real, symmetric, and positive definite $3\times3$ matrices.
Given $A \in \mathbb{M}^{3\times3}$ with the property $\det A=1$.
In my engineering application, I observed that $\mathrm{tr} A$ and $\mathrm{tr} A^{-1}$ are somehow related; $\mathrm{tr}$ denotes the trace. In particular, if I plot $\mathrm{tr} A$ and $\mathrm{tr} A^{-1}$ in a 2d scatter plot, the tuples $\{\mathrm{tr} A, \mathrm{tr} A^{-1} \}$ are located close to the diagonal line. I know that $\mathrm{tr} A \geq 3$ and $\mathrm{tr} A^{-1} \geq 3$.
I was wondering if there is maybe a mathematical relation between these two quantities, that could explains the relation. Admittedly, I do not have enough mathematical background to derive a possible relation. I also do not know if the information that I provided are enough to address this problem -- if not, please let me know.
Any help or ideas are greatly appreciated!