$A\in M_2(\mathbb C)$ and $A $ is nilpotent then $A^2=0$. How to prove this? $A\in M_2(\mathbb C)$ and $A $ is nilpotent then $A^2=0$. How to prove this?
I am not getting enough hints to start.

Since $0$ is the only eigenvalue of a nilpotent matrix, its trace (sum of eigenvalues) and determinant (product of eigenvalues) are both $0$. These two relations suggest that $A$ is of the form 
\begin{bmatrix}
a & \frac{-a^2}{c} \\ 
c & -a
\end{bmatrix}
Square it to find that it is the zero matrix.
