Absolutely valid, and an approach I've used, but any changes made to constraints or the order of the input matrix requires extremely tedious manual calculations. I was hoping for a direction that relies on matrix mathematics and could be implemented programmatically. 3x3, not so bad - 9x9 will...
Gonna preface by saying I never thought linear algebra would be a class I would regret not taking so much... but in short the goal is to reduce an arbitrary symmetric NxN system using a set of auxiliary constraint relationships, e.g. for a 3x3
\begin{bmatrix}
V_1\\
V_2\\
V_3\\
\end{bmatrix}
=...