If a+b+c= S,then the value of left|begin{arra | vedclass

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(B) Let $\Delta = \left|\begin{array}{ccc} S+c & a & b \ c & S+a & b \ c & a & S+b \end{array}ight|$.Applying the operation $C_1 	o C_1 + C_2 + C

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$2S^3$

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(B) Let $\Delta = \left|\begin{array}{ccc} S+c & a & b \ c & S+a & b \ c & a & S+b \end{array}ight|$.Applying the operation $C_1 	o C_1 + C_2 + C_3$:$\Delta = \left|\begin{array}{ccc} S+c+a+b & a & b \ c+S+a+b & S+a & b \ c+a+S+b & a & S+b \end{array}ight|$.Since $a+b+c = S$,we have $S+c+a+b = S+S = 2S$.$\Delta = \left|\begin{array}{ccc} 2S & a & b \ 2S & S+a & b \ 2S & a & S+b \end{array}ight| = 2S \left|\begin{array}{ccc} 1 & a & b \ 1 & S+a & b \ 1 & a & S+b \end{array}ight|$.Applying $R_2 	o R_2 - R_1$ and $R_3 	o R_3 - R_1$:$\Delta = 2S \left|\begin{array}{ccc} 1 & a & b \ 0 & S & 0 \ 0 & 0 & S \end{array}ight|$.Expanding along the first column:$\Delta = 2S 	imes [1(S 	imes S - 0 	imes 0)] = 2S 	imes S^2 = 2S^3$.

If $a+b+c= S$,then the value of $\left|\begin{array}{ccc} S+c & a & b \\ c & S+a & b \\ c & a & S+b \end{array}\right|$ is . . . . . . .

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