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

a^{2} b^{2} c^{2}

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4 a b c

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4 a^{2} b^{2} c^{2}

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2 a^{2} b^{2} c^{2}

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Solution

The correct option is D $4 a^{2} b^{2} c^{2}$
Performing $R_{1} \to R_{1} - R_{2} - R_{3}$
$∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & - 2 c^{2} & - 2 b^{2} b^{2} & c^{2} + a^{2} & b^{2} c^{2} & c^{2} & a^{2} + b^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$(- 2) ∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & c^{2} & b^{2} b^{2} & c^{2} + a^{2} & b^{2} c^{2} & c^{2} & a^{2} + b^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
Performing $R_{3} \to R_{3} - R_{1}$
$(- 2) ∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & c^{2} & b^{2} b^{2} & c^{2} + a^{2} & b^{2} c^{2} & 0 & a^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
Performing $C_{3} \to C_{3} - C_{1}$
$(- 2) ∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & c^{2} & b^{2} b^{2} & c^{2} + a^{2} & 0 c^{2} & 0 & a^{2} - c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$(- 2) [(- c^{2}) (b^{2} a^{2} - b^{2} c^{2}) + b^{2} (- c^{4} - c^{2} a^{2})]$
$= 4 a^{2} b^{2} c^{2}$

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Similar questions

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

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

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

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

(a + b)^{2} - 2 a b

(a + b)^{2} - (a - b)^{2}

4 a b

- 4 a b

2 a^{2} + 2 b^{2}

2 a^{2} - 2 b^{2}

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