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Change of Variables

Prove that:a ...

Question

Prove that :

$|\begin{matrix} a^{2} & b c & a c + c^{2} \\ a^{2} + a b & b^{2} & a c \\ a b & b^{2} + b c & c^{2} \end{matrix}| = 4 a^{2} b^{2} c^{2}$

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Solution

$Let LHS = Δ = |a^{2} b c a c + c^{2} a^{2} + a b b^{2} a c a b b^{2} + b c c^{2}| Δ = a b c |a c a + c a + b b a b b + c c| [Taking out a, b and c common from C_{1}, C_{2} and C_{3}]$

$= a b c |a c 0 a + b b - 2 b b b + c - 2 b| [Applying C_{3} \to C_{3} - C_{2} - C_{1}] = (a b c) (- 2 b) |a c 0 a + b b 1 b b + c 1| [Taking (- 2 b) common from C_{3}] = (a b c) (- 2 b) |a c 0 a - c 0 b b + c 1| [Applying R_{2} \to R_{2} - R_{3}] = (a b c) (- 2 b) \times 1 |a c a - c| [Exp anding along C_{3}] = (a b c) (- 2 b) (- 2 ac) = 4 a^{2} b^{2} c^{2} = RHS$

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∣ ∣ ∣ ∣ ∣ \begin{matrix} a^{2} & b c & a c + c^{2} a^{2} + a b & b^{2} & a c a b & b^{2} + b c & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 4 a^{2} b^{2} c^{2}

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