A) - 2
B) 0
C) 81
D) None of these
Correct Answer: B
Solution :
\[\left| \,\begin{matrix} 31 & 37 & 92 \\ 31 & 58 & 71 \\ 31 & 105 & 24 \\ \end{matrix}\, \right|\,=\,\left| \,\begin{matrix} 31 & 37 & 92 \\ 0 & 21 & -21 \\ 0 & 47 & -47 \\ \end{matrix}\, \right|\];by \[\begin{align} & {{R}_{3}}\to {{R}_{3}}-{{R}_{2}} \\ & {{R}_{2}}\to {{R}_{2}}-{{R}_{1}} \\ \end{align}\] = \[\left| \,\begin{matrix} 31 & 129 & 92 \\ 0 & 0 & -21 \\ 0 & 0 & -47 \\ \end{matrix}\, \right|\] = 0; (by \[{{C}_{2}}\to {{C}_{2}}+{{C}_{3}})\].
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