Finding all characteristic/minimal polynomials of a matrix

Question

I've been given this problem which I've never encountered before, neither in class or in tutorials.

It goes like so:

Given $A\in M_4(\mathbb{R})$ such that for the polynomial $p(x)=(x-2)(x^2+9)$ :

$p(A)=0$

Find all possible minimal polynomials and all possible characteristic polynomials.

What are the real eigenvalues of $A$ and what are their algebraic/geometric multiplicities in relations to the minimal/characteristic polynomial?

In my solution, or the one I've tried to come up with, I've got that:

$m_A(x) \in \{x-2,x+3i,x-3i,x^2+9, (x-2)(x+3i),(x-2)(x-3i),(x-2)(x^2+9)\}$

Now I'm pretty stuck, I was taught that the characteristic polynomial is unique to $A$, yet the question asks to find all possible characteristic polynomial.

Plus the only eigenvalue $\lambda$ I'm seeing such that $\lambda \in \mathbb{R}$ is $\lambda=2$, the question hints there are more than a single eigenvalue within $\mathbb{R}$.

Any advice/direction of thought would be helpful,

Thanks in advance!

Answer 1

The minimal polynomial of $A$ must be monic, divide $p(x)$ and have real coefficients and be non-constant. So, it must be $x-2$, $x^2+9$ or $p(x)$ itself.

In order to find the possible characteristic polynomials, you have to keep in mind that the minimal polynomial divides the characteristic polynomial and that both polynomials have the same roots. Besides, the characteristic polynomial has degree $4$.

• If the minimal polynomial is $x-2$, then the characteristic polynomial is $(x-2)^4$.
• If the minimal polynomial is $x^2+9$, then the characteristic polynomial is $(x^2+9)^2$.
• And if the minimal polynomial is $p(x)$, then the characteristic polynomial is $(x-2)^2(x^2+9)$.