What can we conclude from the equality $m_A(x) = m_B(x)$?

Question

Suppose $A,B \in M_{n\times n}(\mathbb C)$ and $m_A(x) = m_B(x)$.

Is one of the following propositions is true?

(1) $f_A(x) = f_B(x)$

(2) A is invertible if and only if B is invertible.

I think that both of the propositions above are true but I didn't manage to prove them.

In case the propositions are correct can anyone give me an hint on how to prove them?

Otherwise, please provide a counter-example.

Note : $f_A(x)$ is the characteristic polynomial of $A$ and $m_A(x)$ is the minimal polynomial of $A$.

Answer 1

Hints:

1) Take $A=\text{diag}(1,1,2), B = \text{diag}(1,2,2).$

2) Eigenvalues of a matrix are roots of its characteristic polynomial as well as its minimal polynomial i.e. minimal and characteristic polynomial only differ by multiplicities of their roots.