Characteristic polynomial of real matrix with complex eigenvalues

Question

I need some clarification regarding the characteristic polynomial for a real matrix with complex eigenvalues. I am given the matrix

$$A=\begin{pmatrix} -1 & -5 & 4 \\ 1 & 1 & -1 \\ 0 & 0 & 3 \end{pmatrix}$$

which has the eigenvalues $3, 2i, -2i$, i.e. the characteristic polynomial is $(x-3)(x-2i)(x+2i)$. I am wondering how the characteristic polynomial would look like, if it is explicitly stated that $A$ is from $\mathbb{R}^{3\times 3}$, i.e. the eigenvalues should also be from $\mathbb{R}$ (at least if I understand the definition of eigenvalues correctly). In this case, only the eigenvalue $3$ should be considered...

Answer 1

Note: There are infinite many square matrices over $\mathbb{R}$ that have complex eigenvalues.

For example $$A=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ has $\iota $ and $-\iota$ as two complex conjugate eigenvalues. We can only say if $a+\iota b $ is some root of the characteristic polynomial of the such matrix then $a-\iota b $ must satisfy the characteristic polynomial. I hope this is helpful if I understand your doubt correctly.