The eigenvector you found, , is not correct. Could you show us how you arrived at that answer?

If we call the matrix , we have , which, for gives us the system:

which gives us

.

You can see that is an eigenvector of because

But be sure to see that any nonzero multiple of is also an eigenvector of . So, if you had, for example, , you would be okay.

I'm not sure what you are asking here. Please clarify.