Question

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

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

Answer 1

Answer: (C)

$A_{x}=\begin{bmatrix}x & x \\ x & x\end{bmatrix}\, \&\, A_{e}=\begin{bmatrix}e & e \\ e & e\end{bmatrix}$
$A_{x} A_{e}=A_{x}$
$\begin{bmatrix}2 x e & 2 x e \\ 2 x e & 2 x e\end{bmatrix}=\begin{bmatrix}x & x \\ x & x\end{bmatrix}$
$\therefore 2 x e=x$
$\therefore e=1 / 2 $
$\therefore A_{e}=\begin{bmatrix}1 / 2 & 1 / 2 \\ 1 / 2 & 1 / 2\end{bmatrix}=1 / 2\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}$