I'm trying to comprehend a calculation given in a paper. http://www.math.utah.edu/~bresslof/publications/15-6.pdf
The task is the following:
We want to find a relation between $\omega$ and $\tau$. \begin{equation} \label{eq:matrix} \begin{pmatrix} 1 & -\omega & \alpha_I cos(\omega \tau) & \alpha_I sin(\omega \tau) \\ \omega & 1 & -\alpha_I sin(\omega \tau) & \alpha_I cos(\omega \tau) \\ -\alpha_E cos(\omega \tau) & -\alpha_E sin(\omega \tau) & 1 & -\omega \\ \alpha_E sin(\omega \tau) & -\alpha_E cos(\omega \tau) & \omega & 1 \end{pmatrix} \begin{pmatrix} U_E \\ V_E \\ U_I \\ V_I \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \end{equation}
In the paper, the authors argue that the determinant of the matrix must vanish if a nontrivial solution exists. But this gives us merely one condition. So, they assume that $V_E = 0$ and $U_I = 0$. By doing so, we get four conditions, which indeed allow us to find a relation between $\omega$ and $\tau$.
But how can one argue that this is a legitimate assumption?