In this problem I am given a matrix $$B = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ Also, I am supposing I have a matrix $A$ that is a complex $4 \times 4$ matrix with $AB = 0$ Now, I need to describe the possibilities for the Jordan normal form of A.
EDIT: I should mention, all I tried was, considering an arbitrary complex matrix $A$ then we know that the fact that $AB = 0$ tells us that matrix $A$ has values only in the first and fourth row, so it has rank 2. Thus, I know $$A = \begin{bmatrix} 0 & a & b & 0 \\ 0 & e & f & 0 \\ 0 & i & j & 0 \\ 0 & m & n & 0 \end{bmatrix}$$. Thus, from here I was trying to pplay around with the characteristic polynomioal of A but didn't get anwhere helpful.
I have no idea how I to continue, can anyone help me?