Question

The number of values of $k$ for which the linear equations
$4x+ky+2z=0$
$kx+4y+z=0$
$2x+2y+z=0$
posses a non-zero solution is

Answer 1

For the system to have non-zero solution :
$\Delta =0$
$\Delta =\left|\begin{array}{ccc}4& k& 2\\ k& 4& 1\\ 2& 2& 1\end{array}\right|=0$
Expanding along row, we get
$\Rightarrow 4(4-2)-k(k-2)+2(2k-8)=0$
$\Rightarrow k^2-6k+8=0$
$\Rightarrow k=2\text{or}4$
$\therefore$ Two values of $k$ exist.