Question: > 
> Let $A$ and $B$ be two nilpotent matrices. Prove that $A+B$ is nilpotent.
> 
> 
> 

I'm new to the study of matrices, I already proved that $AB$ is nilpotent too (that was the previous question) but I have no clue about solving this one.
Plus, I was wondering if I have to prove that it exists an $n$ such that $(A+B)^n=0$ or such that $A^n+B^n=0$.

Any help?

EDIT:

I just know that:
$(A+B)^k=(A+B)(A+B)...(A+B)$ $k$ times, so I need to prove that $(A+B)=0$ 

Is this enough to prove what I need, right?

Answer: 
Take for example the nilpotent matrices

$$A=\begin{pmatrix}2&-1\\4&-2\end{pmatrix}\;,\;\;\;B=\begin{pmatrix}0&1\\0&0\end{pmatrix}\implies\text{ what is}\;\;A+B \;?$$

Why that sum can't be nilpotent ?
I think you're missing a hypothesis: the group of the matrices has to be commutative! Otherwise the thesis doesn't hold:

let 
$
A=
\begin{bmatrix} 
0 & 1 \\
0 & 0 
\end{bmatrix}$ and 
$
B=
\begin{bmatrix} 
0 & 0 \\
1 & 0 
\end{bmatrix}$

We have that $A^2=0$ and $B^2=0$, but $(A+B)^2=
\begin{bmatrix} 
1 & 0 \\
0 & 1 
\end{bmatrix}$
 which is definitely not the zero matrix!