(Solved):   \( .3 \) Consider the set \( \mathcal{G} \) of \( 3 \times 3 \) matrices defined as follow ...

 

\( .3 \) Consider the set \( \mathcal{G} \) of \( 3 \times 3 \) matrices defined as follows: \[ \mathcal{G}=\left\{\left[\begin{array}{lll} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array}\right] \in \mathbb{R}^{3 \times 3} \mid x, y, z \in \mathbb{R}\right\} \] We define - as the standard matrix multiplication. Is \( (\mathcal{G}, \cdot) \) a group? If yes, is it Abelian? Justify your answer.