The set of all values of λ for which the system of linear equations...

For non-trivial solution, $∆ = 0$.

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

$⇒ \left(1 - \lambda\right)\left(- 2\lambda + {\lambda}^{2} + 1\right) + 1\left(- \lambda + 1\right) - 1\left(- 1 + 2 - \lambda\right) = 0$

$⇒ {\lambda}^{2} - 3{\lambda}^{2} + 3\lambda - 1 = 0$

$⇒ {\left(\lambda - 1\right)}^{3} = 0$

$⇒ \lambda = 1$

Singleton set