A) \[{{a}^{3}}+{{b}^{3}}\]
B) \[-({{a}^{3}}+{{b}^{3}})\]
C) \[2\,({{a}^{3}}+{{b}^{3}})\]
D) \[-2\,({{a}^{3}}+{{b}^{3}})\]
Correct Answer: D
Solution :
Let \[A=\left| \begin{matrix} a & b & a+b \\ b & a+b & a \\ a+b & a & b \\ \end{matrix} \right|\] Applying \[({{R}_{1}}\to {{R}_{1}}+{{R}_{2}}+{{R}_{3}})\] \[=2(a+b)\left| \begin{matrix} 1 & 1 & 1 \\ b & a+b & a \\ a+b & a & b \\ \end{matrix} \right|\] Applying \[({{C}_{1}}\to {{C}_{1}}-{{C}_{2}},{{C}_{2}}\to {{C}_{2}}-{{C}_{3}})\] \[=2(a+b)\left| \begin{matrix} 0 & 0 & 1 \\ -a & b & a \\ b & a-b & b \\ \end{matrix} \right|\] \[=2(a+b)[1\{-a(a-b)-{{b}^{2}}\}]\] \[=2(a+b)(-{{a}^{2}}+ab-{{b}^{2}})\] \[=-2({{a}^{3}}+{{b}^{3}})\]
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