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The value of the determinant $\left| {\,\begin{array}{*{20}{c}}0&{{b^3} - {a^3}}&{{c^3} - {a^3}}\\{{a^3} - {b^3}}&0&{{c^3} - {b^3}}\\{{a^3} - {c^3}}&{{b^3} - {c^3}}&0\end{array}\,} \right|$ is equal to
${a^3} + {b^3} + {c^3}$
${a^3} - {b^3} - {c^3}$
$0$
$ - {a^3} + {b^3} + {c^3}$
Solution
(c) $\left| {\,\begin{array}{*{20}{c}}0&{{b^3} – {a^3}}&{{c^3} – {a^3}}\\{{a^3} – {b^3}}&0&{{c^3} – {b^3}}\\{{a^3} – {c^3}}&{{b^3} – {c^3}}&0\end{array}\,} \right|$
$({b^3} – {a^3})({c^3} – {a^3})\left| {\,\begin{array}{*{20}{c}}0&1&1\\{{a^3} – {b^3}}&1&1\\{{a^3} – {c^3}}&1&1\end{array}\,} \right| = 0$
$[{C_2} \to {C_2} – {C_1}$ and ${C_3} \to {C_3} – {C_1}]$ and then taking out common
$({b^2} – {a^3})$ from $II^{nd}$ column and ( ${c^3} – {a^3}$) from $III^{rd}$ column].
