I am not sure about this question. I have two claims and I need to know if anyone of them is correct.
Claims:
a) There exists a real matrix for which the characteristic polynomial is: $t(t-1)(t^2+t+1)$ and the minimal polynomial is $t(t-1).$
b) Let $A$ and $B$ be $n\times n$ matrices. If there exists a polynomial $q$ such that $q(A)=0$ but $q(B) \neq 0$, then $A$ and $B$ are not similar.
What I think:
a) Basically, the minimal polynomial is the monic polynomial of smallest degree that "resets" a matrix, so the minimal polynomial divides a polynomial say $q$(some matrix)$=0$, and it also divides the characteristic polynomial. Because of the last sentence, I think that the claim is true.
b) Similar matrices are matrices that follow $A=x^{-1}Bx$, and because of that if $q(A)=0$ and $q(B) \neq 0$ then they are not similar. so I think that this claim is correct a well.
Are both claims true?