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$\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = $
$3abc + {a^3} + {b^3} + {c^3}$
$3abc - {a^3} - {b^3} - {c^3}$
$abc - {a^3} + {b^3} + {c^3}$
$abc + {a^3} - {b^3} - {c^3}$
Solution
(b) $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = \left| {\,\begin{array}{*{20}{c}}{a + b + c}&{a + b + c}&{a + b + c}\\b&c&a\\c&a&b\end{array}\,} \right|$,
$({R_1} \to {R_1} + {R_2} + {R_3})$
=$(a + b + c)$ $\left| {\,\begin{array}{*{20}{c}}{1}&{1}&{1}\\b&c&a\\c&a&b\end{array}\,} \right|$ =$(a + b + c)$ $\left| {\,\begin{array}{*{20}{c}} 1&1&1 \\ b&c&a \\ c&a&b \end{array}\,} \right|$ =$(a + b + c)$ $\left| {\,\begin{array}{*{20}{c}} 1&0&0 \\ b&{b – c}&{c – a} \\ c&{c – a}&{a – b} \end{array}\,} \right|$
= $3abc – {a^3} – {b^3} – {c^3}$, (After simplification).
