After you found the eigenvalue you must substitute the eignevalue into the formula:

(eigenvalueI-A)X=0

since the eigenvalue is 3 your matrix will look like:

2,2

-2,-2

carry this matrix to reduced row echelon form:

1,1

0,0

X is your eigenvector so the above matrix multiplied by X must equal 0. Say X is the column matrix = [a b] transposed. you get a=-b and b is arbitrary. Your eigenvector will then be:

-1

1