1beginvmatrix 1 - a - a- 2 - b c1 - b - b- 2 - c a1 - c - c- 2 - a bend vmatrix -1

The correct option is D 01/a amp; a^2 amp; bc 1/b amp; b^2 amp; ca 1/c amp; c^2 amp; ab =1/abc 1 amp; a^3 amp; abc 1 amp; b^3 ...

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Properties of Determinants

1beginvmatrix...

Question

$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 / a & a^{2} & b c 1 / b & b^{2} & c a 1 / c & c^{2} & a b \end{matrix} ∣ ∣ ∣ ∣ ∣ =$

abc

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\frac{1}{a b c}

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ab+bc+ca

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0.0

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∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} - b c 1 & b & b^{2} - a c 1 & c & c^{2} - a b \end{matrix} ∣ ∣ ∣ ∣ ∣ =

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} 1 & b & b^{2} 1 & c & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ =

$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} 1 & b & b^{2} 1 & c & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a)$

$∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a + b + c)$

A = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣, B = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a^{2} & b^{2} & c^{2} a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣, C = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a & b & c a^{2} & b^{2} & c^{2} a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣,

∣ ∣ ∣ ∣ \begin{matrix} 1 & a & b c 1 & b & c a 1 & c & a b \end{matrix} ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} 1 & b & b^{2} 1 & c & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣

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