The characteristic polynomial for traceless $2\times 2, 3\times 3, 4\times 4$ matrices $A$ are
\begin{align} x^2+&\det A \\ x^3-\frac{1}{2}{\rm Tr}A^2 x - &\det A \\ x^4-\frac{1}{2}{\rm Tr}A^2x^2-\frac{1}{3}{\rm Tr}A^3x+&\det A \end{align}
does this pattern continue? That is, would we have
$$x^5-\frac{1}{2}{\rm Tr}A^2x^3-\frac{1}{3}{\rm Tr}A^3x^2-\frac{1}{\color{red}4}({\rm Tr}A^4+\color{red}{??} \ {\rm Tr}A^2{\rm Tr}A^2)x+\det A$$ for the characteristic polynomial for $5\times 5$ matrices. That is, do the coefficients of ${\rm Tr}A^k$ continue to be $-k^{-1}$?
Edit: it appears there is a determinant formula on the wikipedia page from which this follows. I would still appreciate if anyone knows of a more elementary reason for the pattern.