The set of all values of $\lambda$for which the system of linear equations  $2x_1-2x_2+x_3=\lambda x_1$, $2x_1-3x_2+2x_3=\lambda x_2$ ,  $-x_1+2x_3=\lambda x_3$ has a non-trivial solution

$The condition for a homogeneous system of equations to have a non-trivial solution is \Delta =0$

$\Rightarrow$$\left|\begin{array}{l}2-\lambda -2 1\\ 2 -3-\lambda 2\\ -1 2-\lambda \end{array}\right|=0$

      $R_1\to R_1+R_3,R_2\to R_1+2R_3$

$$\begin{gathered} \Rightarrow \left|\begin{array}{l}1-\lambda 0 1-\lambda \\ 01-\lambda 2-2\lambda \\ -1 2-\lambda \end{array}\right|=0 \\ \Rightarrow {\left(1-\lambda \right)}^2\left|\begin{array}{l}1 0 1\\ 0 1 2\\ -1 2-\lambda \end{array}\right|=0 \end{gathered}$$

                    $C_3\to C_3+C_1$

$\Rightarrow {\left(1-\lambda \right)}^2\left|\begin{array}{l}1 0 0\\ 0 1 2\\ -1 2 -\lambda +1\end{array}\right|=0$

$$\begin{gathered} \Rightarrow {\left(1-\lambda \right)}^2\left[-\lambda +1-4\right]=0 \\ \Rightarrow \lambda =1,-3 \end{gathered}$$