Question

The set of all values of λ for which the system of linear equations x - 2y - 2z = λx,

x + 2y + z = λy and - x – y = λz has a non-trivial solution.

1. Contains exactly two elements.
2. Contains more than two elements.
3. Is a singleton.
4. Is an empty set

Answer 1

Correct Answer - Option 3 : Is a singleton.

The given system of linear equations is
x – 2y – 2z = λx

x + 2y + z = λy

-x – y – λz = 0

which can be rewritten as

(1 – λ)x – 2y – 2z = 0

⇒ x + (2 – λ) y + z = 0

x + y + λz = 0

Now, for non-trivial solution, we should have

\(\left| {\begin{array}{*{20}{c}} {1 - {\rm{\lambda }}}&{ - 2}&{ - 2}\\ 1&{2 - {\rm{\lambda }}}&1\\ 1&1&{\rm{\lambda }} \end{array}} \right|\) = 0

[∴ If (a₁ x + b₁y + c₁ z = 0, a₂ x + b₂ y + c₂z = 0, a₃ x + b₃ y + c₃ z = 0]

has a non-trivial solution, then

\(\left. {{\rm{\;}}\left| {\begin{array}{*{20}{c}} {{{\rm{a}}_1}}&{{{\rm{b}}_1}}&{{{\rm{c}}_1}}\\ {{{\rm{a}}_2}}&{{{\rm{b}}_2}}&{{{\rm{c}}_2}}\\ {{{\rm{a}}_3}}&{{{\rm{b}}_3}}&{{{\rm{c}}_3}} \end{array}} \right| = 0} \right]\)

⇒ (1 – λ)[(2 - λ)λ – 1] + 2[λ – 1] – 2[1 – 2 + λ] = 0

⇒ (λ – 1)[λ² - 2λ + 1 + 2 – 2] = 0

⇒ (λ – 1)³ = 0

⇒ λ = 1