$2 x_{1}-2 x_{2}+x_{3}=\lambda x_{1}$

$2 x_{1}-3 x_{2}+2 x_{3}=\lambda x_{2}$

$-x_{1}+2 x_{2}=\lambda x_{3}$

$(2-\lambda) x_{1}-2 x_{2}+x_{3}=0$

$2 x_{1}-(3+\lambda) x_{2}+2 x_{3}=0$

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

$\angle=0$

$\left|\begin{array}{ccc}{2-\lambda} & {-2} & {1} \\ {2} & {-(3+\lambda)} & {2} \\ {-1} & {2} & {-\lambda}\end{array}\right|=0$

${R_1} \to {R_1} + {R_3}$

$\left|\begin{array}{ccc}{1-\lambda} & {0} & {1-\lambda} \\ {2} & {-(3+\lambda)} & {2} \\ {-1} & {2} & {-\lambda}\end{array}\right|=0$

$=(1-\lambda)[(3+\lambda)(\lambda)-4]+(1-\lambda)[4-(3+\lambda)]=0$

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

$=(1-\lambda)\left[\lambda^{2}+2 \lambda-3\right]=0$

$=(1-\lambda)[\lambda(\lambda+3)-1(\lambda+3)]=0$

$=(1-\lambda)(\lambda-1)(\lambda+3)=0$

$\lambda=1,1,3$