The set of the all values of λ for which the system of linear equations:
$2x_1-2x_2 + x_3 = λx_1$
$2x_1-3x_2+2x_3 = λx_2$
$-x_1+2x_2 = λx_3$ has a non-trivial solution,
Answer & explanation
Correct answer: option 1
The given system of equations is
$x_1 (2-λ)-2x_2+x_3=0$
$2x_1-(3+λ)x_2+2x_3=0$
$-x+2x_2-\lambda x_3=0$
Clearly, it is a homogenous system of equations and will have non-trivial solutions, if
$\begin{vmatrix}2-λ&-2&1\\2&-(3+λ)&2\\-1&2&-λ\end{vmatrix}=0$
$⇒λ^3+λ^2-5λ+3=0$
$⇒λ^2(λ-1)+2λ(λ-1)-3(λ-1)=0$
$⇒(λ-1) (λ^2+2λ-3)=0$
$⇒(λ-1) (λ-1) (λ+3)=0$
$⇒λ=1, 1, 3$.
Hence, there are two values of λ.