Let the system of linear equations 4x + + 2z = 0 2x - y + z = 0 +...

For non-trivial solution

\left|\begin{cases}4 & \lambda & 2 \\ 2 & - 1 & 1 \\ \mu & 2 & 3\end{cases}\right| = 0

$⇒ - 20 - 6\lambda + \lambda\mu + 8 + 2\mu = 0$

$⇒ 2\mu - 6\lambda + \lambda\mu = 12$

when $\mu = 6, 12 - 6\lambda + 6\lambda = 12$

which is satisfied by all $\lambda$

When $\lambda = 2, 4\mu = 24$

When $\lambda = 3, 5\mu = 30$

When $\mu = - 6, - 12 - 6\lambda - 6\lambda = 12$