Let $G$ be the set of all matrices of the form

\begin{equation}

\begin{pmatrix}

1&a\\

0&1\\

\end{pmatrix}

\end{equation}

with $a\in \mathbf{Z}$ and matrix multiplication as the binary operation.Prove that $G$ is an abelian group isomorphic to $\mathbf{Z}$.

 

Proof:

\begin{equation}

\begin{pmatrix}

1&a_1\\

0&1\\

\end{pmatrix}\begin{pmatrix}

1&a_2\\

0&1\\

\end{pmatrix}=\begin{pmatrix}

1&a_2+a_1\\

0&1\\

\end{pmatrix}

\end{equation}

Done.