TITLE: Groups involving matrices
QUESTION [5 upvotes]: Let $$H=
  \left\{\left( {\begin{array}{ccc}
   1 & b \\
    0 & 1 \\
  \end{array} } \right)\Bigg\vert b\in \mathbb{R}\right\}$$
How do I show that H is a group and under which operation? I have not done so much with matrices in algebra.
I believe there is a pattern/guide of which I have to follow like when showing subgroups.

REPLY [3 votes]: I assume the candidate operations are matrix addition and multiplication.  We can rule out matrix addition since
$\begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & b  + c \\ 0 & 2 \end{pmatrix} \notin H; \tag{1}$
$H$ is not closed under matrix addition.  However, we have
$\begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & b + c \\ 0 & 1 \end{pmatrix} \in H, \tag{2}$
which not only shows $H$ is closed under matrix multiplication, which is in fact associative (as pointed out by dragon), but that
$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \tag{3}$
is the identity element for $H$; also, we clearly have
$\begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix}^{-1} = \begin{pmatrix} 1 & -c \\ 0 & 1 \end{pmatrix} \in H; \tag{4}$
$H$ contains the multiplicative inverses of all its elements.  So it is indeed an (abelian) group.  Interestingly enough, all these group properties may be read off from (2), without referring them to the space of all matrices, e.g. it is clear from (2) that multiplication must be associative on $H$ etc. etc. etc.