Question:
Let \( G \) be the set of all matrices of the form:
\[ \begin{bmatrix} x & x \\ x & x \end{bmatrix}, \quad x \in \mathbb{R} – \{0\} \]
Find the identity element with respect to matrix multiplication.
Solution:
Step 1: Let identity matrix be
\[ E = \begin{bmatrix} y & y \\ y & y \end{bmatrix} \]
Step 2: Multiply with a general element
\[ \begin{bmatrix} x & x \\ x & x \end{bmatrix} \cdot \begin{bmatrix} y & y \\ y & y \end{bmatrix} = \begin{bmatrix} 2xy & 2xy \\ 2xy & 2xy \end{bmatrix} \]
Step 3: Compare with original matrix
For identity:
\[ \begin{bmatrix} 2xy & 2xy \\ 2xy & 2xy \end{bmatrix} = \begin{bmatrix} x & x \\ x & x \end{bmatrix} \]
So,
\[ 2xy = x \Rightarrow 2y = 1 \Rightarrow y = \frac{1}{2} \]
Step 4: Identity matrix
\[ E = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{bmatrix} \]
Final Answer:
\[ \boxed{ \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{bmatrix} } \]