The set of all values of λ for which the system of linear equations x – 2y – 2z = λx x + 2y + z = λy – x

Engineering

Mathematics

Solving System of Linear Equation Cramers Rule

Question

The set of all values of λ for which the system of linear equations

x – 2y – 2z = λx

x + 2y + z = λy

– x – y = λz

has a non-trivial solution

is a singletion

contains exactly two elements

is an empty set

contains more than two elements

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

$Δ = | \begin{matrix} 1 - λ & - 2 & - 2 \\ 1 & 2 - λ & 1 \\ - 1 & - 1 & - λ \end{matrix} | = 0$

$\Rightarrow | \begin{matrix} 3 - λ & 0 & - 2 \\ λ - 1 & 1 - λ & 1 \\ 0 & λ - 1 & - λ \end{matrix} | = 0$

$\Rightarrow (λ - 1) | \begin{matrix} 3 - λ & 0 & - 2 \\ λ - 1 & - 1 & 1 \\ 0 & 1 & - λ \end{matrix} | = 0$

⇒ (λ – 1) ((3 – λ) (λ – 1) –2 (λ –1)) = 0

⇒ (λ – 1)2 (3 – λ – 2) = 0 ⇒ (λ – 1)2 (1 – λ) = 0 ⇒ λ = 1.