left| {begin{array}{*{20}{c}}a&b&cb&c&ac&a&be | vedclass

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(B) Let $\Delta = \left| {\begin{array}{*{20}{c}}a&b&c\b&c&a\c&a&b\end{array}} ight|$.Applying the operation ${R_1} 	o {R_1} + {R_2} + {R_3}$,we

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$3abc - {a^3} - {b^3} - {c^3}$

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(B) Let $\Delta = \left| {\begin{array}{*{20}{c}}a&b&c\b&c&a\c&a&b\end{array}} ight|$.Applying the operation ${R_1} 	o {R_1} + {R_2} + {R_3}$,we get:$\Delta = \left| {\begin{array}{*{20}{c}}{a + b + c}&{a + b + c}&{a + b + c}\b&c&a\c&a&b\end{array}} ight|$$= (a + b + c) \left| {\begin{array}{*{20}{c}}1&1&1\b&c&a\c&a&b\end{array}} ight|$Applying ${C_2} 	o {C_2} - {C_1}$ and ${C_3} 	o {C_3} - {C_1}$:$= (a + b + c) \left| {\begin{array}{*{20}{c}}1&0&0\b&c-b&a-b\c&a-c&b-c\end{array}} ight|$$= (a + b + c) [ (c - b)(b - c) - (a - b)(a - c) ]$$= (a + b + c) [ -(c - b)^2 - (a^2 - ac - ab + bc) ]$$= (a + b + c) [ -(c^2 - 2bc + b^2) - a^2 + ac + ab - bc ]$$= (a + b + c) [ -c^2 + 2bc - b^2 - a^2 + ac + ab - bc ]$$= (a + b + c) [ -a^2 - b^2 - c^2 + ab + bc + ca ]$$= -(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$$= -(a^3 + b^3 + c^3 - 3abc)$$= 3abc - a^3 - b^3 - c^3$.

$\left| {\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}} \right| = $

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