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Let $Δ_{1} = |\begin{matrix} y^{5} z^{6} (z^{3} - y^{3}) & x^{4} z^{6} (x^{3} - z^{3}) & x^{4} y^{5} (y^{3} - x^{3}) \\ y^{2} z^{3} (y^{6} - z^{6}) & {xz}^{3} (z^{6} - x^{6}) & {xy}^{2} (x^{6} - y^{6}) \\ y^{2} z^{3} (z^{3} - y^{3}) & {xz}^{3} (x^{3} - z^{3}) & {xy}^{2} (y^{3} - x^{3}) \end{matrix}|$ and $Δ_{2} = |\begin{array}{ccc} x & y^{2} & z^{3} \\ x^{4} & y^{5} & z^{6} \\ x^{7} & y^{8} & z^{9} \end{array}|$ . Then $Δ_{1} Δ_{2}$ is equal to

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$Δ_{2}^{6}$

$Δ_{2}^{4}$

$Δ_{2}^{2}$

$Δ_{2}^{3}$

answer is C.

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Detailed Solution

The given determinant $Δ_{1}$ is obtained by corresponding cofactors of determinant $Δ_{2}$ and hence, $Δ_{1} = Δ_{2}^{2}$ .

Now, $Δ_{1} Δ_{2} = Δ_{2}^{2} Δ_{2} = Δ_{2}^{3}$ .

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