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.