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Question
\[\begin{vmatrix}0 & b^2 a & c^2 a \\ a^2 b & 0 & c^2 b \\ a^2 c & b^2 c & 0\end{vmatrix} = 2 a^3 b^3 c^3\]
Solution
\[∆ = \begin{vmatrix}0 & b^2 a & c^2 a \\ a^2 b & 0 & c^2 b \\ a^2 c & b^2 c & 0\end{vmatrix}\]
\[ = \frac{1}{abc}\begin{vmatrix}0 & b^3 a & c^3 a \\ a^3 b & 0 & c^3 b \\ a^3 c & b^3 c & 0\end{vmatrix} \left[\text{ Multiplying the three columns by a, b and c }\right]\]
\[\]
\[ = \frac{abc}{abc}\begin{vmatrix}0 & b^3 & c^3 \\ a^3 & 0 & c^3 \\ a^3 & b^3 & 0\end{vmatrix} \left[\text{ Taking out a, b and c common from the three rows }\right] \]
\[ = b^3 \begin{vmatrix}a^3 & c^3 \\ a^3 & 0\end{vmatrix} + c^3 \begin{vmatrix}a^3 & 0 \\ a^3 & b^3\end{vmatrix} = 2 a^3 b^3 c^3 \left[\text{ Expanding along }R_1 \right]\]
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