polinomial squared problem

4)A)

there is $\displaystyle A=\left(\begin{array}{ccc}i & 0 & 1\\0 & 3 & 0\\0 & 0 & i\end{array}\right)$ find polinomial $\displaystyle 0\neq Q(t)=at^{2}+bt+c\in C_{3}[t]$

so $\displaystyle [Q(A)]^{2}=0$

B)there is $\displaystyle A\in M_{nxn}^{F}$ matrices which is not diagonisable.

prove that if F=C (the field is C) then there is a polinomial $\displaystyle 0\neq Q(t)\in C_{n}[t] so [Q(A)]^{2}=0$

c)

does the previos part id true if F=R

?

dont knot from where to start,any starting guidence?

Re: polinomial squared problem

i solved part A by taking the caracteristic polinomial of A$\displaystyle p(t)=(t-i)^{2}(t-3)$

then we take d(t)=(t-i)(t-3)

$\displaystyle d^{2}(t)=(t-i)^{2}(t-3)^{2}=(t-i)^{2}(t-3)(t-3)$

by cayly hemilton

$\displaystyle p(A)=(A-iI)^{2}(A-3I)=0$

so

$\displaystyle d^{2}(A)=(A-iI)^{2}(A-3I)^{2}=(A-iI)^{2}(A-3I)(A-3I)=0(A-3I)=0$

what is the difference between part A and part B

why cant i solve it the same way?