The system of linear equations x + λy – z = 0 λx – y – z = 0 x + y

Engineering

Mathematics

Solving System of Linear Equation Cramers Rule

Question

The system of linear equations

x + λy – z = 0

λx – y – z = 0

x + y – λz = 0

has a non‑trivial solution for

exactly one value of λ.

exactly two values of λ.

infinitely many values of λ.

exactly three values of λ.

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For non‑trivial solution

$| \begin{matrix} 1 & λ & - 1 \\ λ & - 1 & - 1 \\ 1 & 1 & - λ \end{matrix} | = 0$

1 (λ + 1) – λ (– λ² + 1) – 1 (λ + 1) = 0

λ + 1 + λ³ – λ – λ – 1 = 0

λ³ – λ = 0 ⇒ λ = 0, ± 1.