Eigenvectors, Eigenvalues and Idempotent

[mpm]

I have a question that deals with all three of the terms in the title. I'm not really even sure where to begin on this. I was hoping someone could help.

Question:

An n x n matrix A is said to be idempotent if A^2 = A. Show that if λ is an eigenvalue of an independent matrix, then λ must either be 0 or 1.

If I could get some help on this I would really appreciate it.

Thanks,

mpm

 

[BerkMath]

Well you know that there must exist a nonzero vector v such that A*v=lamda*v. Now play with this statement by applying A again, and rearanging terms so that you end up with only expressions involving lambda. Then solve for lambda.