4 The set \(G\) is given by \(G = \{ \mathbf { M } : \mathbf { M }\) is a real \(2 \times 2\) matrix and det \(\mathbf { M } = 1 \}\).
- Show that \(G\) forms a group under matrix multiplication, × . You may assume that matrix multiplication is associative.
- The matrix \(\mathbf { A } _ { n }\) is defined by \(\mathbf { A } _ { n } = \left( \begin{array} { l l } 1 & 0
n & 1 \end{array} \right)\) for any integer \(n\). The set \(S\) is defined by \(\mathrm { S } = \left\{ \mathrm { A } _ { \mathrm { n } } : \mathrm { n } \in \mathbb { Z } , \mathrm { n } \geqslant 0 \right\}\).
- Determine whether \(S\) is closed under × .
- Determine whether \(S\) is a subgroup of ( \(G , \times\) ).
- Find a subgroup of ( \(G , \times\) ) of order 2 .
- By considering the inverse of the non-identity element in any such subgroup, or otherwise, show that this is the only subgroup of ( \(G , \times\) ) of order 2.
The set of all real \(2 \times 2\) matrices is denoted by \(H\).
- With the help of an example, explain why ( \(H , \times\) ) is not a group.