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Question
If G is the set of all matrices of the form
\[\begin{bmatrix}x & x \\ x & x\end{bmatrix}, \text{where x } \in R - \left\{ 0 \right\}\] then the identity element with respect to the multiplication of matrices as binary operation, is ______________ .
Options
\[\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\]
\[\begin{bmatrix}- 1/2 & - 1/2 \\ - 1/2 & - 1/2\end{bmatrix}\]
\[\begin{bmatrix}1/2 & 1/1 \\ 1/2 & 1/2\end{bmatrix}\]
\[\begin{bmatrix}- 1 & - 1 \\ - 1 & - 1\end{bmatrix}\]
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Solution
\[\text{ Let }\begin{bmatrix}x & x \\ x & x\end{bmatrix}\in G \text{ and }\begin{bmatrix}e & e \\ e & e\end{bmatrix}\in G \text{ such that }\]
\[\begin{bmatrix}x & x \\ x & x\end{bmatrix} \begin{bmatrix}e & e \\ e & e\end{bmatrix} = = \begin{bmatrix}x & x \\ x & x\end{bmatrix} = \begin{bmatrix}e & e \\ e & e\end{bmatrix}\begin{bmatrix}x & x \\ x & x\end{bmatrix}\]
\[\begin{bmatrix}x & x \\ x & x\end{bmatrix} \begin{bmatrix}e & e \\ e & e\end{bmatrix} = \begin{bmatrix}x & x \\ x & x\end{bmatrix}\]
\[\begin{bmatrix}2ex & 2ex \\ 2ex & 2ex\end{bmatrix} = \begin{bmatrix}x & x \\ x & x\end{bmatrix}\]
\[2ex = x\]
\[e = \frac{1}{2} \in R - \left\{ 0 \right\}\]
\[\text{ Thus },\begin{bmatrix}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{bmatrix}\in G,\text{ is the identity element in G} .\]
Notes
The question in the book has some error, so, none of the options are matching with the solution. The solution is created according to the question given in the book.
