Please see the attached image.
The first line just finds the eigenvalues of that matrix.
The second line finds the eigenvectors.
The third line just takes row 1 and row 3 of that matrix and find the determinant.
The fourth line just takes row 2 and row 4 of that matrix and find the determinant.
Because the two sets of equations are identitical, the eigenval
ues are double degenerate in the later case. Thus the evectors are not fixed.
But in the former case, the eigenvalues/eigenvecotrs are different.
THe solution is the later but I don't understand why the former part gives different answers.