Evaluate the determinant: left| {begin{array} | vedclass

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(D) Let $\Delta = \left| {\begin{array}{*{20}{c}}{b + c}& a & a\b& {c + a}& b\c& c& {a + b}\end{array}} ight|$.Apply the operation $R_1 	o R_1 -

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$4abc$

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(D) Let $\Delta = \left| {\begin{array}{*{20}{c}}{b + c}& a & a\b& {c + a}& b\c& c& {a + b}\end{array}} ight|$.Apply the operation $R_1 	o R_1 - (R_2 + R_3)$:$\Delta = \left| {\begin{array}{*{20}{c}}{b + c - (b + c)}& {a - (c + a + c)}& {a - (b + a + b)}\b& {c + a}& b\c& c& {a + b}\end{array}} ight| = \left| {\begin{array}{*{20}{c}}0& {-2c}& {-2b}\b& {c + a}& b\c& c& {a + b}\end{array}} ight|$.Expanding along the first row:$\Delta = 0 - (-2c) \left| {\begin{array}{*{20}{c}}b& b\c& {a + b}\end{array}} ight| + (-2b) \left| {\begin{array}{*{20}{c}}b& {c + a}\c& c\end{array}} ight|$.$\Delta = 2c(b(a + b) - bc) - 2b(bc - c(c + a))$.$\Delta = 2c(ab + b^2 - bc) - 2b(bc - c^2 - ac)$.$\Delta = 2abc + 2b^2c - 2bc^2 - 2b^2c + 2bc^2 + 2abc$.$\Delta = 4abc$.

Evaluate the determinant: $\left| {\begin{array}{*{20}{c}}{b + c}& a& a\\b& {c + a}& b\\c& c& {a + b}\end{array}} \right|$

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