$A\in M_2(\mathbb C)$ 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.

$A$ can be put into upper triangular form, and since $A$ is nilpotent the its eigenvalues are both zero and the upper triangular form will have zeroes on its diagonal. Squaring this matrix gives zero, and so $A^2 = 0$ also.
Note that this same argument gives that any $n$ by $n$ nilpotent matrix over ${\mathbb C}$, when raised to the $n$th power, gives the zero matrix. 
