I'm teaching a course in physics, and I need a simple and intuitive proof that a matrix ($3\times3$, but it doesn't matter) has exactly 1 invariant which is linear in its entries, 2 that are quadratic, etc. When I say "invariance" I mean under orthonormal transformation of the axes $A\to Q A Q^{-1}$ for orthonormal $Q$.

For example, that every quadratic invariant scalar is a combination of $\operatorname{tr}(A)^2$ and $\operatorname{tr}(A^2)$.

The proof for the linear case is trivial (and intuitive) but I can't find a generalization for the quadratic case.