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(B) Let $A = \begin{bmatrix} x & x \ x & x \end{bmatrix}$. The identity element $E = \begin{bmatrix} e & e \ e & e \end{bmatrix}$ must satisfy $AE =

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$\begin{bmatrix} 3/4 & 3/4 \ 3/4 & 3/4 \end{bmatrix}$

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(B) Let $A = \begin{bmatrix} x & x \ x & x \end{bmatrix}$. The identity element $E = \begin{bmatrix} e & e \ e & e \end{bmatrix}$ must satisfy $AE = A$.$\begin{bmatrix} x & x \ x & x \end{bmatrix} \begin{bmatrix} e & e \ e & e \end{bmatrix} = \begin{bmatrix} 2xe & 2xe \ 2xe & 2xe \end{bmatrix} = \begin{bmatrix} x & x \ x & x \end{bmatrix}$.Thus,$2xe = x \implies e = 1/2$. The identity element is $E = \begin{bmatrix} 1/2 & 1/2 \ 1/2 & 1/2 \end{bmatrix}$.Let the inverse of $A = \begin{bmatrix} 1/3 & 1/3 \ 1/3 & 1/3 \end{bmatrix}$ be $A^{-1} = \begin{bmatrix} y & y \ y & y \end{bmatrix}$.Then $AA^{-1} = E \implies \begin{bmatrix} 1/3 & 1/3 \ 1/3 & 1/3 \end{bmatrix} \begin{bmatrix} y & y \ y & y \end{bmatrix} = \begin{bmatrix} 1/2 & 1/2 \ 1/2 & 1/2 \end{bmatrix}$.Multiplying the matrices: $\begin{bmatrix} 2y/3 & 2y/3 \ 2y/3 & 2y/3 \end{bmatrix} = \begin{bmatrix} 1/2 & 1/2 \ 1/2 & 1/2 \end{bmatrix}$.Equating the elements: $2y/3 = 1/2 \implies y = 3/4$.Therefore,$A^{-1} = \begin{bmatrix} 3/4 & 3/4 \ 3/4 & 3/4 \end{bmatrix}$.

$G = \left\{ \begin{bmatrix} x & x \\ x & x \end{bmatrix} : x \in \mathbb{R} \setminus \{0\} \right\}$ is a group with respect to matrix multiplication. In this group,the inverse of $\begin{bmatrix} 1/3 & 1/3 \\ 1/3 & 1/3 \end{bmatrix}$ is

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