The first method works fine

Howdo "we know v1= 0"? Why can't v1 be any number at all?

No, it doesn't. It means an eigenvector is of the form (a, 0) for any number a.but that means the eigenvector is (0,0) !! how is this possible?

Yes, it is. It is your statement "and we know v1 = 0" that is incorrect. We have no reason to say v1= 0.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....ifyou use it correctly!