Relation between eigenvalues and the characteristic polynomial of a matrix

Question

If $\lambda =2$ is an eigenvalue of $A \in M_{3 \times 3}(\mathbb{R})$ and $P(\lambda) =a_{0} + a_{1} \lambda+\dotsb+a_{k}\lambda^k$ such that $p(A) =0_{3×3}$, then

Which of the following is true

1. $p(-2)$ must be zero
2. $p(2)$ must be zero
3. $p(2) +p(-2)$ must be zero

I think only option 2 is correct by fundamental theorem of algebra. Is my answer is correct ???

Answer 1

The Fundamental Theorem of Algebra has nothing to do with it. What happens is that, being an eigenvalue, $2$ is a root of the minimal polynomial of $A$. On the other hand, $p(A)=0$, implies that the minimal polynomial of $A$ divides $p$. So $2$ is a root, of $p$, i.e. $p(2)=0$.

The two others will be true in some examples, but not in general.

Answer 2

According to Cayley Hamilton theorem, the matrix A satisfies its characteristic polynomial $$P(λ) =a_{o} +a_{1} λ+...+a_{k}λ^k$$

The eigenvalues are the zeros of $P(λ).$

Thus if $ λ=2$ is an eigenvalue of $A$, we have to have $$P(2)=0$$