A) \[{{(a+b+c)}^{2}}\]
B) \[{{(a+b+c)}^{3}}\]
C) \[(a+b+c)(ab+bc+ca)\]
D) None of these
Correct Answer: B
Solution :
\[\left| \,\begin{matrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \\ \end{matrix}\, \right|\] = \[\left| \,\begin{matrix} -\Sigma a & 0 & 2a \\ \Sigma a & -\Sigma a & 2b \\ 0 & \Sigma a & c-a-b \\ \end{matrix}\, \right|\] , \[\left( \begin{align} & {{C}_{1}}\to {{C}_{1}}-{{C}_{2}} \\ & {{C}_{2}}\to {{C}_{2}}-{{C}_{3}} \\ \end{align} \right)\] = \[{{(\Sigma a)}^{2}}\,\left| \,\begin{matrix} -1 & 0 & 2a \\ 1 & -1 & 2b \\ 1 & 1 & c-a-b \\ \end{matrix}\, \right|={{(\Sigma a)}^{3}}\], (on expansion) = \[{{(a+b+c)}^{3}}\].
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