I have the following symmetrical functions $$f_i = \sum_{j=1}^3 \tau_{ij}\ \frac{1}{c_j}$$ $$c_i = \sum_{j=1}^3 \tau_{ij}\ \frac{1}{f_j}$$ $$\tau_{ij} = \tau_{ji}$$
where $\tau_{ij}$ is given number, $f_i$ and $c_i$ are unknown number.
I write them as matrix form $$ \left[ \begin{matrix} f_{1}\\ f_{2}\\ f_{3}\\ \end{matrix} \right] = \left[ \begin{matrix} \tau_{11} \ \ \tau_{12} \ \ \tau_{13}\\ \tau_{21} \ \ \tau_{22} \ \ \tau_{23}\\ \tau_{31} \ \ \tau_{32} \ \ \tau_{33}\\ \end{matrix} \right]\left[ \begin{matrix} 1/c_{1}\\ 1/c_{2}\\ 1/c_{3}\\ \end{matrix} \right]$$
$$ \left[ \begin{matrix} c_{1}\\ c_{2}\\ c_{3}\\ \end{matrix} \right] = \left[ \begin{matrix} \tau_{11} \ \ \tau_{12} \ \ \tau_{13}\\ \tau_{21} \ \ \tau_{22} \ \ \tau_{23}\\ \tau_{31} \ \ \tau_{32} \ \ \tau_{33}\\ \end{matrix} \right]\left[ \begin{matrix} 1/f_{1}\\ 1/f_{2}\\ 1/f_{3}\\ \end{matrix} \right]$$
Can I get any relation between $f_i$ and $c_i$,say $f_i = \rho c_i$?
or can I solve for $f_i$ and $c_i$?