Cayley-Hamilton theorem for symmetric positive definite matrix

Question

Let $A$ be a symmetric positive definite matrix. We know that by Cayley-Hamilton theorem \begin{aligned}A^{-1}={\frac {(-1)^{n-1}}{\det A}}(A^{n-1}+c_{n-1}A^{n-2}+\cdots +c_{1}I_{n}).\end{aligned}

How can we simplify this expression, by knowing that $A$ is symmetric and positive definite?

Specific questions: Can we say the $c_j$ ($j=1,...,n-1$) are positive?

Answer 1

Let $A=\begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}>0$, then $A^{-1}=\begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}=\frac{-1}{1}(\begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}+(-3)I)$.

$c_1$ is negative here.