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

Easy

A

one value of $λ$
B

two values of $λ$
C

three values of $λ$
D

Infinitely many values of $λ$

Solution

System of equations have non-trivial solution $\Rightarrow Δ = 0$
$\begin{array}{l} \Rightarrow |\begin{matrix} 1 & λ & - 1 \\ λ & - 1 & - 1 \\ 1 & 1 & - λ \end{matrix}| = 0 \\ \Rightarrow 1 (λ + 1) - λ (- λ^{2} + 1) - 1 (λ + 1) = 0 \\ \Rightarrow λ^{3} - λ = 0 \\ \Rightarrow λ (λ^{2} - 1) = 0 \\ \Rightarrow λ = 0, 1, - 1 . \end{array}$

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