Let S be the set of all λ ∈ R for which the system of linear...

For no solution $\Delta = 0$ and at least one of ${\Delta}_{1}, {\Delta}_{2}, {\Delta}_{3}$ is non-zero.
\Delta = \left|\begin{cases}2 & - 1 & 2 \\ 1 & - 2 & \lambda \\ 1 & \lambda & 1\end{cases}\right| = - \left(\lambda - 1\right)\left(2\lambda + 1\right)
{\Delta}_{1} = \left|\begin{cases}2 & - 1 & 2 \\ - 4 & - 2 & \lambda \\ 4 & \lambda & 1\end{cases}\right| = - 2\left({\lambda}^{2} + 6\lambda - 4\right)
$\Delta = 0 ⇒ \lambda = 1, - \dfrac{1}{2}$

And for $\lambda = 1, - \dfrac{1}{2}, {∆}_{1} \ne 0$
Hence, $S = \left\{1, - \dfrac{1}{2}\right\}$