Question

Find the sum of all real values of $r$ for which the following system of linear equations has a non-trivial solution.
$2rx-2y+3z=0$
$x+ry+2z=0$
$2x++rz=0$

Answer 1

Given a system of equations has a non-trivial solution.

$\Delta =\left|\begin{array}{ccc}2r& -2& 3\\ 1& r& 2\\ 2& 0& r\end{array}\right|=0$

$\Rightarrow 2r^3+2(r-4)-6r=0$

$\Rightarrow r^3-2r-4=0$

$\Rightarrow (r-2)(r^2+2r+2)=0$

one root is real, and the remaining two are imaginary.

$\therefore r=2$

$\therefore$ Sum of real values of $r$ $=2$