Nilpotent Matrix Question Link
I was wondering if anyone could help with the latter parts of the question (b & c). I have concluded from part A that Matrix "A" is Nilpotent as det(A)=0 and tr(A)=0, and that Matrix "B" is not Nilpotent as det(B) does not equal 0 and tr(B)=0. However, I do not know where to go from here.
HINT:
$D^k=0$ but $D^{k-1}\neq 0$. Therefore there exists a vector $x$ such that $$ D^{k-1}x\neq 0$$
Show $x,D x,D^2x,\text{...},D^{k-1}x$ are linearly independent. To do this consider a linear combination and set it equal to zero then show the coefficients must all be zero as follows
$$a_0x+a_1D x+a_2D^2x+\text{...}+a_{k-1}D^{k-1}x=0$$
Apply $D$ to both sides of this equation $(k-1)$ times
you should get
$$a_0D^{k-1}x=0$$
this shows $a_0=0$
continue ...