how many solutions does det(A - xI) have? is 6 a solution?

the eigenspace is not a null space, unless the eigenvalue is 0. it is the space spanned by the eigenvector(s) corresponding to a single eigenvalue.

having found the eigenvalues, say they are λ1 and λ2, solve the equation (A-λ1I)(x,y) = 0 and then (A - λ2I)(x,y) = 0.

these should be simple enough to solve without row-reduction.