First slide

System of linear equations

Question

The system of equations $λ x + y + z = 0, - x + λ y + z = 0, - x - y + λ z = 0$ will have a non- trivial solution for real values of $λ$ are

Moderate

A

0,1
B

0,-1
C

0,2
D

0

Solution

$T h e c o n d i t i o n f o r h o m o g e n e o u s s y s t e m t o h a v e a n o n - t r i v i a l s o l u t i o n i s Δ = 0$
$\begin{array}{l} \Rightarrow |\begin{matrix} λ & 1 & 1 \\ - 1 & λ & 1 \\ - 1 & - 1 & λ \end{matrix}| = 0 \\ \Rightarrow λ [λ^{2} + 1] - [- λ + 1] + [1 + λ] = 0 \\ \Rightarrow λ^{3} + λ + λ - 1 + 1 + λ = 0 \Rightarrow λ^{3} + 3 λ = 0 \\ \Rightarrow λ (λ^{2} + 3) = 0 \\ \Rightarrow λ = 0 \end{array}$

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