A) \[9{{a}^{2}}(a+b)\]
B) \[9{{b}^{2}}(a+b)\]
C) \[{{a}^{2}}(a+b)\]
D) \[{{b}^{2}}(a+b)\]
Correct Answer: B
Solution :
Operating\[{{C}_{1}}\to {{C}_{1}}+{{C}_{2}}+{{C}_{3}}\]. We get the value of given determinant as \[\left| \,\begin{matrix} 3a+3b & a+b & a+2b \\ 3a+3b & a & a+b \\ 3a+3b & a+2b & a \\ \end{matrix}\, \right|\] = \[3\,(a+b)\,\left| \,\begin{matrix} 1 & a+b & a+2b \\ 1 & a & a+b \\ 1 & a+2b & a \\ \end{matrix}\, \right|\] Operate \[{{R}_{3}}\to {{R}_{3}}-{{R}_{1}}\], \[{{R}_{2}}\to {{R}_{2}}-{{R}_{1}}\] = \[3\,(a+b)\,\left| \,\begin{matrix} 1 & a+b & a+2b \\ 0 & -b & -b \\ 0 & b & -2b \\ \end{matrix}\, \right|\] \[=3(a+b)\,(2{{b}^{2}}+{{b}^{2}})=9{{b}^{2}}(a+b)\].
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