Solving a simple system of equations

Question

Given the simultaneous equations $$A\cos{(\sqrt{\lambda}\pi)} + B\sin{(\sqrt{\lambda}\pi)} = 0$$ $$A\cos{(2\sqrt{\lambda}\pi)}+B\sin{(2\sqrt{\lambda}\pi)} = 0$$ We want to show this has not trivial solutions (ie. solutions when $A\not=0$ and $B\not= 0$). In my notes I have that this gives non-trivial solutions when $$\sin{(2\sqrt{\lambda}\pi)}\cos{(\sqrt{\lambda}\pi)} - \cos{(2\sqrt{\lambda}\pi)}\sin{(\sqrt{\lambda}\pi)} = 0$$ but can't quite see why. Can someone explain, thanks.

Answer 1

If the system $$Au+Bv=0$$ $$Aw+Bz=0$$ has a non-trivial solution, then $$u=-\frac{Bv}A=\frac{Avw}{Az}=\frac{vw}z$$ Therefore, $$uz-vw=0$$

Remark: The case $z=0$ must be considered in a different but easy way.

Answer 2

A system $$ Ax=0 $$ has a notrivial solution if and only if $\det A=0$.

Answer 3

If you set up the corresponding augmented coefficient matrix, you'll see that the matrix has determinant $$\left(\sin{(2\sqrt{\lambda}\pi)}\cos{(\sqrt{\lambda}\pi)} - \cos{(2\sqrt{\lambda}\pi)}\sin{(\sqrt{\lambda}\pi)}\right)$$

We want the determinant to be zero in order to ensure there is a non-trivial solution.