A) \[9{{a}^{2}}(a+b)\]
B) \[9{{b}^{2}}(a+b)\]
C) \[{{a}^{2}}(a+b)\]
D) \[{{b}^{2}}(a+b)\]
E) \[9{{b}^{2}}(a-b)\]
Correct Answer: B
Solution :
The determinant value of matrix \[3AB={{3}^{3}}(-1)(3)=-81\] \[\left| \begin{matrix} a & a+b & a+2b \\ a+2b & a & a+b \\ a+b & a+2b & a \\ \end{matrix} \right|\] \[=\left| \begin{matrix} 3(a+b) & 3(a+b) & 3(a+b) \\ a+2b & a & a+b \\ a+b & a+2b & a \\ \end{matrix} \right|\] \[{{R}_{1}}\to {{R}_{1}}+{{R}_{2}}+{{R}_{3}}\] \[=3(a+b)\left| \begin{matrix} 1 & 0 & 0 \\ a+2b & -2b & -b \\ a+2b & b & -b \\ \end{matrix} \right|\] \[{{C}_{2}}\to {{C}_{2}}-{{C}_{1}}\] \[{{C}_{3}}\to {{C}_{3}}-{{C}_{1}}\] \[=3(a+b)\left| \begin{matrix} -2b & -b \\ b & -b \\ \end{matrix} \right|\] \[=3{{b}^{2}}(a+b)\left| \begin{matrix} 2 & 1 \\ -1 & 1 \\ \end{matrix} \right|\] \[=3{{b}^{2}}(a+b)(2+1)\] \[=9{{b}^{2}}(a+b)\]
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