$A$ is matrix under $R$ which I know the following information about it:
$f_A(x)=(x+2)^4x^4$- Characteristic polynomial
$m_A(x)=(x+2)^2x^2$- Minimal polynomial.
I'm trying to find out
(i) $A$'s rank
(ii) $\dim$ $\ker(A+2I)^2$
(iii) $\dim$ $\ker (A+2I)^4$
(iv) the characteristic polynomial of $B=A^2-4A+3I$.
I believe that I don't have enough information to determine none of the above.
By the power of $x$ in the minimal polynomial I know that the biggest Jordan block of eigenvalue 0 is of size 2, so there can be two options of Jordan form for this eigenvalue: $(J_2(0),J_2(0))$ or $(J_2(0),J_1(0),J_1(0))$, therefore $A$'s rank can be $2$ or $3$. I'm wrong, please correct me.
How can I compute the rest?
Thanks for the answers.