The real value of $k$ for which the system of equations $2kx-2y+3z=0$, $x+ky+2z=0$, $2x+kz=0$, has a non-trivial solution is:

For a homogeneous system to have a non-trivial solution, the determinant of the coefficient matrix must be zero:
$$\left|\begin{array}{ccc}2k& -2& 3\\ 1& k& 2\\ 2& 0& k\end{array}\right|=0$$
Expanding along the 3rd row:
$$2[(-2)(2)-(k)(3)]-0+k[(2k)(k)-(1)(-2)]=0$$
$$2[-4-3k]+k[2k^2+2]=0$$
$$-8-6k+2k^3+2k=0$$
$$2k^3-4k-8=0$$
Divide by 2:
$$k^3-2k-4=0$$
By inspection, if $k=2$: $2^3-2(2)-4=8-4-4=0$.Factoring: $(k-2)(k^2+2k+2)=0$.The quadratic $k^2+2k+2$ has discriminant $D=2^2-4(1)(2)=-4$, so it has no real roots.The only real value is $k=2$.