The system of equations
$k x + (k + 1) y + (k - 1) z = 0$
$(k + 1) x + k y + (k + 2) z = 0$
$(k - 1) x + (k + 2) y + k z = 0$ has a non-trivial solution for

Exactly three values of k

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Exactly two real values of k

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Exactly one real value of k

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Infinite real values of k

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Solution

The correct option is C Exactly one real value of k
For non-trivial solution

$∣ ∣ ∣ ∣ \begin{matrix} k & k + 1 & k - 1 k + 1 & k & k + 2 k - 1 & k + 2 & k \end{matrix} ∣ ∣ ∣ ∣ = 0$
$R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} - R_{1}$
$∣ ∣ ∣ ∣ \begin{matrix} k & k + 1 & k - 1 1 & - 1 & 3 - 1 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0 k = - \frac{1}{2}$

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Similar questions

k x + (k + 1) y + (k - 1) z = 0

(k + 1) x + k y + (k + 2) z = 0

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