Linear algebra, eigenvectors and eigenvalues

[ann.r221]

If v is an eigenvector of an invertible matrix A, which of the following is/are necessarily true?

(1) v is also an eigenvector of 2A
(2) v is also an eigenvector of A^2
(3) v is also an eigenvector of A^-1

A) 1 only
B) 2 only
C) 3 only
D) 1 and 3 only
E) 1,2 and 3

I am pretty sure 2 is true because if we look at Av = (lamba)v.
A^2v = A(lambda)v = (Av)(lambda) = (lambda)v(lambda) = (lambda)^2 v

So A^2v = (lambda)^2v. So that should be that v is an eigenvector for A^2 as well. I am not sure about the others can someone help?

 

[Dick]

You did so well on b), isn't a) just as easy? (2A)v=? For c) use A^(-1)A=I.