Prove that the following statements are equivalent.

Question

Let A be a nxn Matrix whose characteristic polynomial XA can be decomposed into a product of linear factors.

1. A^n = 0
2. A is nilpotent
3. 0 is the only Eigenvalue of A.

I did the steps 1) => 2) but I'm stuck on 2) => 3) and 3) => 1). I'm unsure how to use the fact that the characteristic Polynomial can be decomposed into a product of linear factors. Isn't the characteristic polynomial in 3) just equal to the determinant of A since the only eigenvalue is 0?

1. => 2) is trivial since k = n with A^k = 0 so A is nilpotent

2. => 3): $$A^k = 0$$ $$Av = \lambda v$$ $$A^kv = \lambda A^{k-1}v$$ $$\lambda A^{k-1}v = 0$$