8.Many atoms per unit cell problem 3.5(Atomic chain with 3different_spring_constants)

Hi, can anyone help me work out the eigenvalues of omega in 3.5 of working class 8.
In the answers they state that they use the same eigenvectors as in 3.4(ka =0) however 2 of the eigenvectors don’t seem to be an eigenvector in the new situation k=pi/a. Is the answer using new eigenvectors? and if so how would we go about geussing those?

The solution tries the previous eigenvectors as an ansatz and it turns out that they are in fact eigenvectors, so you can use them.