Finding eigenvectors of a 2x2 matrix

say you have the matrix:

A = (k, 1

0,2)

now i found the eigenvalues to be k and 2.

im trying to find the eigenvector for the eigenvalue k.

i thought its supposed to work when i do the following...

so if i use (A-lambda(I))v = 0, the matrix reduces to:

(k-k , 1

0, 2-k)

which is

(0, 1

0, 2-k)

so v2 = 0 and (2-k)v2 = 0

and we know v1 = 0...

but that means the eigenvector is (0,0) !! how is this possible?

because when i try working out the other way...

so back to our matrix A..

I do Av = lambda(v)

kv1 + v2 = kv1

2v2 = kv2

and i get eigenvector (1,0) which is correct..

so why does the first method not work?? i thought we can use any of those methods....(Doh)