First slide

System of linear equations

Question

The system of linear equations $x + λ y - z = 0, λ x - y - z = 0, x + y - λ z = 0$ has a non-trivial solution for

Moderate

A

Infinity many values of $λ$
B

Exactly one value of $λ$
C

Exactly two values of $λ$
D

Exactly three values of $λ$

Solution

$T h e s y s t e m h a s n o n - t r i v i a l s o l u t i o n \Rightarrow Δ = 0$
$\Rightarrow |\begin{matrix} 1 & λ & - 1 \\ λ & - 1 & - 1 \\ 1 & 1 & - λ \end{matrix}| = 0 \Rightarrow λ + 1 - λ (- λ^{2} + 1) - (λ + 1) = 0$
$\Rightarrow λ (λ^{2} - 1) = 0$
$\Rightarrow λ = 0, \pm 1$
$∴ λ$ has three values

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