The set of all values of λ for which the system of linear equations: 2 x_1 - 2 x_2 + x_3 = λ x_1 2 x_1 - 3 x_2 + 2 x _3 = λ x_2 - x_1 + 2 x _2 = λ…

$$( 2 - \lambda ) {x}_{1} - 2 {x}_{2} + {x}_{3} = 0$$(2−λ)x1​−2x2​+x3​=0

$$2 {x}_{1} - ( \lambda + 3 ) {x}_{2} + 2 {x}_{3} = 0$$2x1​−(λ+3)x2​+2x3​=0

$$- {x}_{1} + 2 {x}_{2} - \lambda {x}_{3} = 0$$−x1​+2x2​−λx3​=0

The systems of linear equations will have a non-trivial solution

$$\Rightarrow | \begin{matrix} 2 - \lambda & - 2 & 1 \\ 2 & - \lambda - 3 & 2 \\ - 1 & 2 & - \lambda \end{matrix} | = 0$$⇒∣2−λ2−1​−2−λ−32​12−λ​∣=0

$$\Rightarrow ( 2 - \lambda ) [ {\lambda }^{2} + 3 \lambda - 4 ] + 2 [ - 2 \lambda + 2 ] + 1 [ 4 - \lambda - 3 ] = 0$$⇒(2−λ)[λ2+3λ−4]+2[−2λ+2]+1[4−λ−3]=0

$$2 {\lambda }^{2} + 6 \lambda - 8 - {\lambda }^{3} - 3 {\lambda }^{2} + 4 \lambda - 4 \lambda + 4 + 1 - \lambda = 0$$2λ2+6λ−8−λ3−3λ2+4λ−4λ+4+1−λ=0

$$- {\lambda }^{3} - {\lambda }^{2} + 5 \lambda - 3 = 0$$−λ3−λ2+5λ−3=0

$${\lambda }^{3} + {\lambda }^{2} - 5 \lambda + 3 = 0$$λ3+λ2−5λ+3=0

$$( \lambda - 1 ) ( {\lambda }^{2} + 2 \lambda - 3 ) = 0$$(λ−1)(λ2+2λ−3)=0

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

$$\Rightarrow \lambda = -3 , 1 , 1$$⇒λ=−3,1,1