A) \[\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]\]
B) \[\left[ \begin{matrix} 2 & 0 \\ 0 & 2 \\ \end{matrix} \right]\]
C) \[\left[ \begin{matrix} 1/2 & 0 \\ 0 & 1/2 \\ \end{matrix} \right]\]
D) \[\left[ \begin{matrix} 0 & 2 \\ 2 & 0 \\ \end{matrix} \right]\]
Correct Answer: B
Solution :
Consider a \[2\times 2\] matrix whose \[|A|=2\] \[A=\left[ \frac{4}{2}\,\,\frac{1}{1} \right]\] Now, \[adj\,\,A=\left[ \begin{matrix} 1 & -2 \\ -1 & 4 \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & -1 \\ -2 & 4 \\ \end{matrix} \right]\] \[\Rightarrow \] \[A\,\,(adj\,\,A)=\left[ \begin{matrix} 4 & 1 \\ 2 & 1 \\ \end{matrix} \right]\,\,\left[ \begin{matrix} 1 & -1 \\ -2 & 4 \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} 4-2 & -4+4 \\ 2-2 & -2+4 \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} 2 & 0 \\ 0 & 2 \\ \end{matrix} \right]\]
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