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(B) Let $\Delta = \left| {\begin{array}{ccc} a - b - c & 2a & 2a \ 2b & b - c - a & 2b \ 2c & 2c & c - a - b \end{array}} ight|$.Apply the operati

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$(a + b + c)^3$

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(B) Let $\Delta = \left| {\begin{array}{ccc} a - b - c & 2a & 2a \ 2b & b - c - a & 2b \ 2c & 2c & c - a - b \end{array}} ight|$.Apply the operation $R_1 	o R_1 + R_2 + R_3$:$\Delta = \left| {\begin{array}{ccc} a + b + c & a + b + c & a + b + c \ 2b & b - c - a & 2b \ 2c & 2c & c - a - b \end{array}} ight|$.Take $(a + b + c)$ common from $R_1$:$\Delta = (a + b + c) \left| {\begin{array}{ccc} 1 & 1 & 1 \ 2b & b - c - a & 2b \ 2c & 2c & c - a - b \end{array}} ight|$.Apply $C_2 	o C_2 - C_1$ and $C_3 	o C_3 - C_1$:$\Delta = (a + b + c) \left| {\begin{array}{ccc} 1 & 0 & 0 \ 2b & -(a + b + c) & 0 \ 2c & 0 & -(a + b + c) \end{array}} ight|$.Expanding along $R_1$:$\Delta = (a + b + c) [1 \cdot (-(a + b + c)) \cdot (-(a + b + c)) - 0] = (a + b + c)^3$.

$\left| {\begin{array}{ccc} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{array}} \right| = $

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