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, $A^2 = 0$.
Hint: Try to reason on possible characteristic polynomial of A.