Question

If G is the set of all matrices of the form $\left[\begin{array}{cc}x& x\\ x& x\end{array}\right],\mathrm{where}x\in R-\left\{0\right\}$, then the identity element with respect to the multiplication of matrices as binary operation, is

(a) $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

(b) $\left[\begin{array}{cc}-1/2& -1/2\\ -1/2& -1/2\end{array}\right]$

(c) $\left[\begin{array}{cc}1/2& 1/1\\ 1/2& 1/2\end{array}\right]$

(d) $\left[\begin{array}{cc}-1& -1\\ -1& -1\end{array}\right]$

Answer 1

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


Disclaimer: The question in the book has some error, so, none of the options are matching with the solution. The solution is created according to the question given in the book.