Let $A$ be a symmetric positive definite matrix. I am interested in finding a general expression for the $A^{-1}$. Using the Cayley-Hamilton Theorem (https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem), we have that $A^{-1}$ can be expressed as \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}
I am bit puzzled finding the coefficients $c_i$, when $A$ is symmetric and positive definite. Also, would it be possible to avoid evaluating the $det{A}$, that is finding a expression equivalent to it?