Proving the Identity Matrix Property: A^2=A for n-Rowed Matrices | 20 Marks

[Hala91]

please help me prove this...

Homework Statement

Show that If "A" is an n-rowed matrix that satisfies A^2=A Then:
Row(A)+Row(I-A)=n

Homework Equations

The Attempt at a Solution

well since A is n-rowed that means that its an n*n matrix so Ax=I
as i guess so :
Row(A)=Rank(A)
Rank(I-A)+nullity(I-A)=Rank(A)+nullity(A)=n
please help if i find its solution I will be given 20 mark for it and i have been trying to solve it for over two day :S

 

[hunt_mat]

Examining the eigenvales might help, note that:
$$Ax=\lambda x\Rightarrow A^{2}x=\lambda Ax\Rightarrow Ax=\lambda^{2}x$$
I am not too sure what you mean by Row(A)

 

[earlh]

Multiply
$$A (I-A)$$ and solve it. what does that tell you?

 

[Hala91]

Thanks A lot guys I have proved it with your help :)