A) \[\left[ \begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{matrix} \right]\]
B) \[\left[ \begin{matrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{2} \\ \end{matrix} \right]\]
C) \[\left[ \begin{matrix} 0 & 0 & 2 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \\ \end{matrix} \right]\]
D) \[\left[ \begin{matrix} 0 & 0 & \frac{1}{2} \\ 0 & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & 0 \\ \end{matrix} \right]\]
Correct Answer: D
Solution :
Given, \[A=\left[ \begin{matrix} 0 & 0 & 2 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \\ \end{matrix} \right]\] \[\therefore \] \[|A|=-8\] \[{{C}_{11}}=0,\,{{C}_{12}}=0,\,{{C}_{13}}=-4,\,{{C}_{21}}=0,\,{{C}_{22}}=-4,\] \[{{C}_{23}}=0,\,{{C}_{31}}=-4,\,{{C}_{32}}=0,\,{{C}_{33}}=0\] \[adj\,(A)=C'=\left[ \begin{matrix} 0 & 0 & -4 \\ 0 & -4 & 0 \\ -4 & 0 & 0 \\ \end{matrix} \right]\] \[{{A}^{-1}}=\frac{adj\,(A)}{|A|}=\frac{1}{-8}\left[ \begin{matrix} 0 & 0 & -4 \\ 0 & -4 & 0 \\ -4 & 0 & 0 \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} 0 & 0 & 1/2 \\ 0 & 1/2 & 0 \\ 1/2 & 0 & 0 \\ \end{matrix} \right]\]
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