Practicing Success
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, |
contains two elements contains more than two elements is an empty set is a singleton set |
contains two elements |
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. |