= Solution
A square homogeneous <linear system> has a nonzero solution exactly when its <determinant> vanishes. Expanding along the second row gives
$$
\begin{aligned}
\det A&=3\left[(18-\lambda)+2\lambda(1-\lambda)\right]
-3\left[(9-\lambda)+\lambda(2-\lambda)\right]\\
&=3(9-\lambda^2).
\end{aligned}
$$
Consequently
$$
\boxed{\lambda=3\quad\text{or}\quad\lambda=-3.}
$$
For an explicit check, the <null space> contains $(3,-1,1)^{\mathsf T}$ at $\lambda=3$ and $(3,1,-1)^{\mathsf T}$ at $\lambda=-3$.
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