Let $A$ be a $3\times3$ matrix with $A^3+A+I_3=0$ with coefficients in $\mathbb{Q}$. Find the rational canonical form of $A$ and the characteristic polynomial of $A^2+I_3$.
Let $p(x)=x^3+x+1$. Since $p(A)=0$, the minimal polynomial of $A$, say $m_A(x)$, needs to divide $p(x)$. However, since $p(x)$ is irreducible over $\mathbb{Q}$, we must have that $m_A(x)=x^3+x+1$, and since $A$ is a $3\times3$ matrix, we have that $m_A(x)=\chi_A(x)$ - characteristic polynomial. Thus, $A$ has only one invariant factor, namely $m_A(x)$. Hence,
$$ R_A=\begin{pmatrix} 0 & 0 & -1\\ 1 & 0 & 0\\ 0 & 1 & -1 \end{pmatrix} $$
I think this part is alright, but I do not know how to find the characteristic polynomial of $A^2+I$. I tried using the fact that $A=-A^3+I_3$ and plug it in, but I just found that $A^2+I=A^6-2A$. I also know that if $A$ has an eigenvalue $\lambda$, then $A^2$ has an eigenvalue $\lambda^2$, but I don't know how either of these things can help me.
Any help will be appreciated.
Hint: $A^3+A+I=0$ implies that $(-A)(A^2+I)=I$, use the reference to find the characteristic polynomial of $A^2+I$.
Characteristic polynomial of an inverse