Let $ A \in M_n(\mathbb{R}) $ be a matrix which satisfies $ A^2 + A + 5I_n = 0 $. Find the characteristic polynomial $ p_A $

Question

Problem: Let $ A \in M_n(\mathbb{R}) $ be a matrix which satisfies $ A^2 + A + 5I_n = 0 $. Find the characteristic polynomial $ p_A $

I don't really know how to find $ p_A $.We can write $ p_A = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \cdots a_1 x + a_0 $. I know that by the Cayley-Hamilton theorem $ A $ is a root of $ p_A $. Also the polynomial $ f(x) = x^2 + x +5 $ satisfies $ f(A) = 0 $. I don't know how to proceed, I know $ p_A $ can be written as $ p_A(x) = f(x) \cdot g(x) $ where $ g(x) $ is some polynomial s.t. $ \deg g \leq n-2 $, but what next?

Thanks in advance for help!

Answer 1

The only possible complex eigenvalues of $A$ are $\lambda = \frac{-1 - i\sqrt{19}}{2}$ and $\overline\lambda$. It follows that \begin{equation} \det(x I_n - A) = p_A(x) = (x-\lambda)^\alpha (x - \overline\lambda)^{n-\alpha} \end{equation} As $p_A$ is real, it follows that $\alpha = n -\alpha$, hence $n$ is even and finally \begin{equation} p_A(x) = (x^2 + x + 5)^{n/2} \end{equation}