Question

Find the real values of r which the following system of linear equations has a non-trivial solution. Also find the non-trivial solutions:
$2rx-2y+3z=0x+ry+2z=02x+rz=0$

Answer 1

$△=0\Rightarrow r^3-2r-4=0\Rightarrow (r-2)(r^2+2r+2)=0$
$\therefore r=2$ as other factor gives imaginary roots. Putting $r=2$, the equation reduces to $4x-2y+3z=0,x+2y+2z=0$ and $x+z=0$
Putting $x+z=0$, they reduce to $x-2y=0,2y+z=0$ and $x+z=0$.
Choosing $x=k$, we get $y=\frac{k}{2},z=-k$ which given the required solutions.