Find all values of $k$ for which the following homogeneous systems of linear equations have a non-trivial solution:
(a)
$$
\begin{aligned}
x+k y & =0, \\
k x+4 y & =0,
\end{aligned}
$$
$$
x_1+k x_2+4 x_3=0,
$$
(b)
$$
\begin{array}{r}
k x_1+x_2+2 x_3=0 \\
2 x_1+k x_2+8 x_3=0 .
\end{array}
$$
(c)
$$
\begin{aligned}
x+k y+2 z & =0, \\
3 x-k y-2 z & =0,
\end{aligned}
$$
$$
\begin{aligned}
(k+1) x-2 y-4 z & =0, \\
k x+3 y+6 z & =0 .
\end{aligned}
$$