Let $ n$ be a positive integer. For a ring $ A$ and matrix $ M \in \mathrm{Mat}_{n \times n}(A)$ , let $ \chi_{M}(t) = \det(M-t \operatorname{id}_{n}) = (-1)^{n}(t^{n} – \sigma_{M,1}t^{n-1} + \dotsb + (-1)^{n}\sigma_{M,n}) \in A[t]$ be the characteristic polynomial. Here $ \sigma_{M,1}$ is the trace of $ M$ and $ \sigma_{M,n}$ is the determinant of $ M$ . Recall that $ \sigma_{M,1}$ is additive and $ \sigma_{M,n}$ is multiplicative, i.e. $ \sigma_{M_{1}+M_{2},1} = \sigma_{M_{1},1} + \sigma_{M_{2},1}$ and $ \sigma_{M_{1}M_{2},1} = \sigma_{M_{1},1}\sigma_{M_{2},1}$ .

Are there any matrix operations that preserve the middle coefficients $ \sigma_{M,2},\dotsc,\sigma_{M,n-1}$ in any reasonable sense?

This is related to the question Geometric interpretation of characteristic polynomial but I don’t yet understand whether it answers my question.