Question

The identity element in the group $M = \left\{ \begin{bmatrix} x & x \\ x & x\\ \end{bmatrix} | x \ \in \ R, x \neq 0 \right\}$ with respect to matrix multiplication is

• A. $\begin{bmatrix}1 & 1 \\1 & 1\\\end{bmatrix} $
• B. $\frac {1} {2} \begin{bmatrix} 1 & 1 \\ 1 & 1\\ \end{bmatrix} $
• C. $\begin{bmatrix}1 & 0 \\0 & 1\\ \end{bmatrix} $
• D. $\begin{bmatrix}0 & 1 \\1 & 0\\\end{bmatrix} $

Answer 1

The Correct Option is B

$M=\begin{bmatrix}x & x \\ x & x\end{bmatrix}, \forall x \in R$ and $x \neq 0$
Let $P$ be the identity element in the group
i.e. $P=\begin{bmatrix}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{bmatrix}=\frac{1}{2}\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}$
$P$ is obtained by putting $x=\frac{1}{2}$
$\therefore \, M P=\begin{bmatrix}x & x \\ x & x\end{bmatrix}\begin{bmatrix}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{bmatrix}=M$
and $PM=\begin{bmatrix}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{bmatrix}\begin{bmatrix}x & x \\ x & x\end{bmatrix}=M$
$\therefore\, M P=M=P M$