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Evaluation of a Determinant

If α, β, γ ar...

Question

If $α, β, γ$ are the roots of $p x^{3} + q x^{2} + r = 0,$ then the value of the determinant
$∣ ∣ ∣ ∣ ∣ \begin{matrix} α β & β γ & γ α β γ & γ α & α β γ α & α β & β γ \end{matrix} ∣ ∣ ∣ ∣ ∣$ is

p

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q

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0

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r

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Solution

The correct option is C $0$
Operation $C_{1} \to C_{1} + C_{2} + C_{3}$ gives $(α β + β γ + γ α)$ $∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & β γ & γ α 1 & γ α & α β 1 & α β & β γ \end{matrix} ∣ ∣ ∣ ∣ ∣$
From the given equation, $α β + β γ + γ α = 0.$
So, the value of determinant is $0$

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