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

Prove that: $A^2=0$

Thanks in advance!

Answer: 
Hint: Try to reason on possible characteristic polynomial of A.
$A^n=O\_2=> det(A^n)=0=>det(A)=0=>A^2=Tr(A).A=>A^n=[Tr(A)]^{n-1}.A=>[Tr(A)]^{n-1}.A=O\_2.$

I. $Tr(A)=0=>A^2=Tr(A).A=O\_2;$

II. $A=O\_2=>A^2=O\_2.$