A) \[\left( \begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix} \right)\]
B) \[\frac{1}{2}\left( \begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix} \right)\]
C) \[\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right)\]
D) \[\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix} \right)\]
Correct Answer: B
Solution :
Let \[\left[ \begin{matrix} a & a \\ a & a \\ \end{matrix} \right]\]be the identity element then \[\left[ \begin{matrix} x & x \\ x & x \\ \end{matrix} \right]\left[ \begin{matrix} a & a \\ a & a \\ \end{matrix} \right]=\left[ \begin{matrix} x & x \\ x & x \\ \end{matrix} \right]\] i.e., \[2ax=x\Rightarrow a=\frac{1}{2}\], \[(\because x\ne 0)\] Identity element =\[\frac{1}{2}\left[ \begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix} \right]\].
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