For small oscillations, my textbook equation for amplitude says:
$(V-\omega^2T) \cdot a=0$ where $a$ is a column vector in which each component $a_i$ is related to $q_i$ as $q_i=a_i\cos(\omega t-\gamma)$. Now it says for a solution to exist $\det(V-\omega^2T)=0$ must be satisfied.
My objection is: if it's non-zero it has an inverse and simple matrix algebra would do the trick. Although the problem we might face here is that the RHS is 0, which would give $a=0$. But if we make $\det(V-\omega^2T)=0$, doesn't it imply that the system has infinitely many solutions, or no solution- again as useless as previous case. I somehow get a feeling that the author is trying to use the argument that $x \cdot y=0$ and we want some value of $x$, hence $y=0$ but I am not sure can we extend this line of reasoning to matrices.