$∣ ∣ ∣ \begin{matrix} a^{2} + a b + b^{2} & a^{2} - a b + b^{2} a + b & a - b \end{matrix} ∣ ∣ ∣ =$

2 a^{3}

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b^{3}

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- 2 a^{3}

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- 2 b^{3}

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Solution

The correct option is D $- 2 b^{3}$
$∣ ∣ ∣ \begin{matrix} a^{2} + a b + b^{2} & a^{2} - a b + b^{2} a + b & a - b \end{matrix} ∣ ∣ ∣$
By column trasformation ,
$C_{1} \to C_{1} - C_{2}$
$∣ ∣ ∣ \begin{matrix} a^{2} + a b + b^{2} - (a^{2} - a b + b^{2}) & a^{2} - a b + b^{2} a + b - (a - b) & a - b \end{matrix} ∣ ∣ ∣$
$∣ ∣ ∣ \begin{matrix} 2 a b & a^{2} - a b + b^{2} 2 b & a - b \end{matrix} ∣ ∣ ∣ = 2 a b (a - b) - (2 b) (a^{2} - a b + b^{2})$
$= 2 a^{2} b - 2 a b^{2} - 2 b a^{2} + 2 a b^{2} - 2 b^{3}$
$= - 2 b^{3}$

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