Hey math88.

You should eliminate two rows (since they are all linearly dependent) and you will get x1 - 2x2 + 2x3 = 0.

Now you need to convert that to a vector with a free parameter (call it t) by letting one variable equal to t and solving for the other parameters (in terms of t) and your vector will be (x1(t),x2(t),x3(t)) and this is your eigenvector. You can then normalize this to have length 1 (this is typically done for eigenvectors).