First slide

System of linear equations

Question

The value of $λ$ for which the system of equations $x - 2 y - 2 z = λ x, x + 2 y + z = λ y, - x - y - λ z = 0$ has a non-trivial solution is

Moderate

A

1
B

-1
C

2
D

-2

Solution

The system has a non – trivial solution $\Rightarrow Δ = 0$
$\Rightarrow |\begin{matrix} 1 - λ & - 2 & - 2 \\ 1 & 2 - λ & 1 \\ - 1 & - 1 & - λ \end{matrix}| = 0 a p p l y R_{1} \to R_{1} + R_{3}, R_{2} \to R_{2} + R_{3} \Rightarrow |\begin{matrix} - λ & - 3 & - 2 - λ \\ 0 & 1 - λ & 1 - λ \\ - 1 & - 1 & - λ \end{matrix}| = 0 \Rightarrow |\begin{matrix} λ & - 3 & - 2 - λ \\ 0 & 1 - λ & 1 - λ \\ 1 & - 1 & - λ \end{matrix}| = 0 \Rightarrow λ [- λ (1 - λ) + 1 (1 - λ)] - 0 + 1 [- 3 (1 - λ) + (1 - λ) (2 + λ)] = 0 \Rightarrow λ {(λ - 1)}^{2} - {(λ - 1)}^{2} = 0 \Rightarrow {(λ - 1)}^{3} = 0 \Rightarrow λ = 1 \Rightarrow λ = 1$

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