Prove the following - - b 2 - c 2 ab ac ab c 2 - a 2 bc ca cb a 2 - b 2 - - b 2 - c 2 a 2 a 2 b 2 c 2 - a 2 b 2 c 2 c 2 a 2 - b 2 - -4 a 2 b 2 c 2

Take out a , b , c from C_1 , C_2 and C_3 respectively. =abc a amp; c amp; a+c a+b amp; b amp; a b ...

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Definition of a Determinant

Prove the fol...

Question

Prove the following :
$∣ ∣ ∣ ∣ ∣ \begin{matrix} b^{2} + c^{2} & a b & a c a b & c^{2} + a^{2} & b c c a & c b & a^{2} + b^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} b^{2} + c^{2} & a^{2} & a^{2} b^{2} & c^{2} + a^{2} & b^{2} c^{2} & c^{2} & a^{2} + b^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 4 a^{2} b^{2} c^{2}$

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∣ ∣ ∣ ∣ ∣ \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} ∣ ∣ ∣ ∣ ∣ = k a^{2} b^{2} c^{2}

k

∣ ∣ ∣ ∣ ∣ \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^{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} ∣ ∣ ∣ ∣ ∣ = k a^{2} b^{2} c^{2},

k

(a^{2} + b^{2}) (a^{2} + b^{2}) - (a^{2} - b^{2}) (a^{2} - b^{2}) = 4 a^{2} b^{2}

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