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Standard XII

Mathematics

Definition of a Determinant

If α, β, γ ...

Question

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

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Solution

Since, $α, β, γ$ are the roots of the equation $x^{3} + p x + q = 0$
$\Rightarrow α + β + γ = 0$
Consider, $∣ ∣ ∣ ∣ ∣ \begin{matrix} α & β & γ β & γ & α γ & α & β \end{matrix} ∣ ∣ ∣ ∣ ∣$
$C_{1} \to C_{1} + C_{2} + C_{3}$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} α + β + γ & β & γ α + β + γ & γ & α α + β + γ & α & β \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & β & γ 0 & γ & α 0 & α & β \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= 0$

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If α,β,γ are the roots of the equation x3+px+q=0, then the value of the determinant ∣∣ ∣ ∣∣αβγβγαγαβ∣∣ ∣ ∣∣ is ..............

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