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

Answer: (B)

$\frac {1} {2} \begin{bmatrix} 1 & 1 \\\ 1 & 1\\\ \end{bmatrix}$

Answer 2

$M=\begin{bmatrix}x & x \\\ x & x\end{bmatrix}, \forall x \in R$ 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 $PM=\begin{bmatrix}\frac{1}{2} & \frac{1}{2} \\\ \frac{1}{2} & \frac{1}{2}\end{bmatrix}\begin{bmatrix}x & x \\\ x & x\end{bmatrix}=M$
$\therefore\, M P=M=P M$