Group of nonsingular 2x2 matrices Let $GL(2, \mathbb R)$ denote the group of all nonsingular $2 \times 2$ matrices over $\mathbb R$. Show that each of the following sets is a subgroup of $GL(2, \mathbb R).$
$(a)\quad S = \left\{\begin{bmatrix} a & b \\ c & d\end{bmatrix}\mid a, b, c, d \in \mathbb R \;\text{and}\;ad-bc = 1\right\}.$
$(b)\quad S = \left\{\begin{bmatrix} a & 0\\ 0 & a\end{bmatrix}\mid a \in \mathbb R \text{ and } a\neq 0\right\}.$
A group is a set with an operation attached to it. What exactly does $GL(2, \mathbb{R})$ mean? What is the operation here?
I suppose the question states: Let $GL(2, \mathbb{R})$ denote a group with ALL $2 \times 2$ matrices EXCEPT the identity matrix $\begin{smallmatrix} 1 & 0 \\\ 0 & 1 \end{smallmatrix}$.
Show that the following is a subgroup of that group...? Right? But I need an operation to complete this problem. What am I misunderstanding?
 A: ${\bf GL}(\color{blue}{2}, \color{red}{\mathbb R})$ is called the $\bf{G}$eneral $\bf L$inear group  consisting of $\color{blue}{2} \times \color{blue}2$ invertible matrices with $\color{red}{\text{real}}$ valued entries.  
The group operation is matrix multiplication.
$\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$ IS included in $GL(2\mathbb R)$. Indeed, it is the identity element in $GL(2 \mathbb R)$.
In (a), you are being asked to show that the set of all $2\times 2$ matrices whose determinant is $1$ is a subgroup of $GL(2, \mathbb R)$.
In (b), you are being asked to show that all $2\times 2$ diagonal matrices (whose entries on the diagonal are non-zero) is a subgroup of $GL(2, \mathbb R)$.
A: The operation you need is the standard multiplication of matrices:
$$\left(\begin{matrix} a & b \\\ c & d \end{matrix}\right)\left(\begin{matrix} a' & b' \\\ c' & d' \end{matrix}\right)=\left(\begin{matrix} aa'+bc' & ab'+bd' \\\ ca'+dc' & cb'+dd' \end{matrix}\right)$$
$GL(2,\mathbb{R})$ is the set of invertible matrices, meaning that $A$ is invertible if there exist a $B$ such that $AB=\operatorname{I}$, where $\operatorname{I}=\left(\begin{matrix} 1 & 0 \\\ 0 & 1 \end{matrix}\right)$. You need to check first that $GL(2,\mathbb{R})$ with this operation is a group, and the you have to show that those two are subgroups, i.e. they are closed under multiplication and inverse.
