The set of all values of  λ for which the system of linear equations 2x1−2x2+x3=λx1,2x1−3x2+2x3=λx2  and  −x1+2x2=λx3 has a non-trivial solution

Given system of linear equations ⇒(2−λ)x1−2x2+x3=0----1 2x1−3x2+2x3=λx2----2 ⇒2x1−(3+λ)x2+2x3=0----3 −x1+2x2=λx3 ⇒−x1+2x2−λx3=0Since , the system has non-trivial solution∴2−λ−212−(3+λ)2−12λ=0⇒(2−λ)3λ+λ2−4+2(−2λ+2)+1(4−3)−λ=0⇒(2−λ)λ2+3λ−4+4(1−λ)+(1−λ)=0⇒ (2−λ)(λ+4)(λ−1)+5(1−λ)=0⇒(λ−1)[(2−λ)(λ+4)−5]=0⇒(λ−1)λ2+2λ−3=0⇒(λ−1)[(λ−1)(λ+8)]=0⇒(λ−1)2(λ+3)=0⇒ λ=1,1,−3Hence , λ contains two elements.