The value of $k$ for which the system of equations $x+ky+3z=0$, $4x+3y+kz=0$, $2x+y+2z=0$ has a non-trivial solution is:

For a homogeneous system of linear equations to have a non-trivial solution, the determinant of the coefficient matrix must be zero ($|A|=0$).
$$|A|=\left|\begin{array}{ccc}1& k& 3\\ 4& 3& k\\ 2& 1& 2\end{array}\right|=0$$
Expanding along the first row:
$$1(6-k)-k(8-2k)+3(4-6)=0$$
$$6-k-8k+2k^2-6=0$$
$$2k^2-9k=0$$
$$k(2k-9)=0$$
$$k=0\text{ or }k=\frac{9}{2}$$