Linear Algebra, eigenvectors problem

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.

2 is also true because if Av = (lamda)v then (2A)v = (2lambda)v

Can someone help show 3? If Av = lambda v

then (A^-1)Av =A^-1(lambda)v

I am not sure how to show it