First slide

System of linear equations

Question

The system of equations $\begin{array}{r} λ x + y + z = 0 \\ - x + λ y + z = 0 \\ - x - y + λ z = 0 \end{array}$ will have a non-trivial solution if real values of $λ$ are

Moderate

A

0, 1
B

0, – 1
C

0, 2
D

0

Solution

The system of equation will have a non-trivial solution if and only if
$∆ = 0$ where
$Δ = |\begin{matrix} λ & 1 & 1 \\ - 1 & λ & 1 \\ - 1 & - 1 & λ \end{matrix}| = λ^{3} + 3 λ = λ (λ^{2} + 1)$
Thus , $Δ = 0, λ \in R \Rightarrow λ = 0$

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