First slide

Introduction to Determinants

Question

If $|\begin{matrix} (β + γ - α - δ)^{4} & (β + γ - α - δ)^{2} & 1 \\ (γ + α - β - δ)^{4} & (γ + α - β - δ)^{2} & 1 \\ (α + β - γ - δ)^{4} & (α + β - γ - δ)^{2} & 1 \end{matrix}|$ $= - k (α - β) (α - γ) (α - δ) (β - γ) (β - δ) (γ - δ)$ , then the value of k is _________.

Moderate

Solution

Let $D = |\begin{matrix} (β + γ - α - δ)^{4} & (β + γ - α - δ)^{2} & 1 \\ (γ + α - β - δ)^{4} & (γ + α - β - δ)^{2} & 1 \\ (α + β - γ - δ)^{4} & (α + β - γ - δ)^{2} & 1 \end{matrix}|$
Applying $R_{1} \to R_{1} - R_{3}, R_{2} \to R_{2} - R_{3}$
$= |\begin{matrix} (β + γ - α - δ)^{4} - (α + β - γ - δ)^{4} & (β + γ - α - δ)^{2} - (α + β - γ - δ)^{2} & 0 \\ (γ + δ - β - δ)^{4} - (α + β - γ - δ)^{4} & (γ + α - β - δ)^{2} - (α + β - γ - δ)^{2} & 0 \\ (α + β - γ - δ)^{4} & (α + β - γ - δ)^{2} & 1 \end{matrix}|$
$= 4 (β - δ) (γ - α) . 4 (α - δ) (γ - β) \times |\begin{array}{ccc} (β + γ - α - δ)^{2} + (α + β - γ - δ)^{2} & 1 & 0 \\ (γ + α - β - δ)^{2} + (α + β - γ - δ)^{2} & 1 & 0 \\ (α + β - γ - δ)^{4} & (α + β - γ - δ)^{2} & 1 \end{array}|$
Apply $R_{1} \to R_{1} - R_{2}$
$= 16 (β - δ) (γ - α) (α - δ) \cdot 4 (γ - δ) (β - α) \times |\begin{array}{ccc} 1 & 0 & 0 \\ (γ + α - β - δ)^{2} + (α + β - γ - δ)^{2} & 1 & 0 \\ (α + β - γ - δ)^{4} & (α + β - γ - δ)^{2} & 1 \end{array}|$
$= - 64 (α - β) (α - γ) (α - δ) (β - γ) (β - δ) (γ - δ)$

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