System of linear equations

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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 , then sum of cubes of $λ$ is

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Solution

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

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System of linear equations

Remember concepts with our Masterclasses.

The system of linear equations x+λy−z=0,λx−y−z=0;x+y−λz=0has a non-trivial solution , then sum of cubes of λis

System has non-trivial solution⇒Δ=0 ⇒1λ−1λ−1−111−λ=0 ⇒1λ+1-λ-λ2+1-1λ+1=0 ⇒λλ2-1=0 ⇒λ=-1,0,1 ∴sum of cubes of λ=-1+0+1=0

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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 , then sum of cubes of $λ$ is