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 ______________ .

विकल्प

• \[\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\]

• \[\begin{bmatrix}- 1/2 & - 1/2 \\ - 1/2 & - 1/2\end{bmatrix}\]

• \[\begin{bmatrix}1/2 & 1/1 \\ 1/2 & 1/2\end{bmatrix}\]

• \[\begin{bmatrix}- 1 & - 1 \\ - 1 & - 1\end{bmatrix}\]

Answer 1

\[\text{ Let }\begin{bmatrix}x & x \\ x & x\end{bmatrix}\in G \text{ and }\begin{bmatrix}e & e \\ e & e\end{bmatrix}\in G \text{ such that }\]
\[\begin{bmatrix}x & x \\ x & x\end{bmatrix} \begin{bmatrix}e & e \\ e & e\end{bmatrix} = = \begin{bmatrix}x & x \\ x & x\end{bmatrix} = \begin{bmatrix}e & e \\ e & e\end{bmatrix}\begin{bmatrix}x & x \\ x & x\end{bmatrix}\]
\[\begin{bmatrix}x & x \\ x & x\end{bmatrix} \begin{bmatrix}e & e \\ e & e\end{bmatrix} = \begin{bmatrix}x & x \\ x & x\end{bmatrix}\]
\[\begin{bmatrix}2ex & 2ex \\ 2ex & 2ex\end{bmatrix} = \begin{bmatrix}x & x \\ x & x\end{bmatrix}\]
\[2ex = x\]
\[e = \frac{1}{2} \in R - \left\{ 0 \right\}\]
\[\text{ Thus },\begin{bmatrix}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{bmatrix}\in G,\text{  is the identity element in G} .\]