My book discusses the oscillations of systems for small pertubations around an equilibrium point. By considering taylors expansions of the potential and kinetic energy they are led to a set of equations for the amplitude factors. I have attached the crucial lines as a picture. My question is to how they solve the system of equations. They say that we have n solutions so any nontrivial solution requires the determinant of the system of equations to be zero. I think I am misunderstanding something here, because how does that solve the system of equations? Everyone knows that a system has no solutions if the determinant is zero. So what information does requiring the determinant to be zero give?