Question: Let the matrix $A=(a\_{ij})=Mat(2, R)$, satisfies $A^{2011}=0$

Prove that: $A^2=0$

Thanks in advance!

Answer: 
Consider the minimal polynomial $p(x)$ of $A$. By hypothesis, $p(x) \mid x^{2011}$, whence $p(x) = x^k$ for some $k\in \mathbb{N}$. Also $deg(p(x)) \leq 2$, and so $p(x) = x$ or $p(x) = x^2$.

In either case, $A^2 = 0$.
The purpose of this exercise, of course, is to illustrate that if $A$ is an $n\times n$ nilpotent matrix, then $A^n=0$. However, since $n=2$ here, we have a more elementary proof.

As $A^{2011}=0$, $A$ is singular. So, its two rows are linearly dependent and hence $A=uv^\top$ for some vectors $u$ and $v$. It follows that for $m\ge2$, $A^m=(uv^\top)^m=uv^\top uv^\top\cdots uv^\top=(v^\top u)^{m-1}A$. So, ...