Coefficients of characteristic polynomial and eigenvalues

Question

Let $A$ be a $3x3$ symmetric (thus, orthogonally diagonalizable) real matrix. We know that its characteristic polynomial is (in the variable $λ$) $$-λ^3 + tr(A)λ^2-cλ+det(A)$$ where $c\in \mathbb{R}$ is the sum of the principal minors of order 2 of A. If $α$, $β$, $γ$ are the eigenvalues of $A$ (not necessarily different), we know that we can write $tr(A)$ and $det(A)$ as sum and product (respectively) of the eigenvalues, and this is sometimes useful to calculate the eigenvalues: $$ \left\{ \begin{array}{c} tr(A) =α+β+γ \\ det(A) = αβγ \\ \end{array} \right. $$

QUESTION

Is there a way to express $c$ as equal to an expression with eigenvalues in it, so that we can use a system of the form $$ \left\{ \begin{array}{c} tr(A) =α+β+γ \\ det(A) = αβγ \\ c = ? \end{array} \right. $$

and we can calculate the 3 eigenvalues?

APPENDIX

My purpose is to use this to diagonalize the $3x3$ matrix of the quadratic terms of a quadric surface (the “complete” matrix is a $4x4$). See this question.

Answer 1

The answer is yes :

$$ c=\alpha\beta+\beta\gamma+\gamma\alpha $$

In general those polynoms (for diagonalized matrix) will be Newton's polynom for multi-nominal expression (number of indeteminate will be the size of the matrix).