Matrices problem

Hi, I'm having trouble figuring this out:
A is an nxn matrix. It's given that A²=A, A=\=0, A=\=I (I is nxn)
prove or disprove: Ax=0 has only the trivial solution
I tried to disprove that with the determinant:
det(A²)=det(A)
2det(A)=det(A)
det(A)=0
Hence A is singular
BUT im not sure that this det(A²)=2det(A) is right, and I can't fathom why A=\=0, A=\=I (I is nxn) is given, so I suspect I'm mistaken.
help is very welcomed

Your solution is incorrect, because det(A^2)=det(A)^2. But this is the right track. Can you follow from here?

consider A =

[1 0]
[0 0].

Thanks to the both of you!