in order to find the eigenvector for the eigenvalue r , you first subtract r from the diagonal of the matrix and then find the kernel of that matrix. this is because the eigenvector needs to solve the equation Av=rv, or 0 = Av-rv=Av-rIv=(A-rI)v. the best way to do it is to find its reduced row echelon form and then solve the system.
for example with the 0 eigenvalue you get
(almost reduced form...)
now, a vector is in the kernel of that matrix only if it has the form (0,-x,3x), so the base for the kernel is the vector (0,-1,3) which is the eigenvector. if the base is of higher dimension then you have more than 1 eigenvector for that specified eigenvalue