6 The set \(S\) consists of all non-singular \(2 \times 2\) real matrices \(\mathbf { A }\) such that \(\mathbf { A Q } = \mathbf { Q A }\), where
$$\mathbf { Q } = \left( \begin{array} { l l }
1 & 1
0 & 1
\end{array} \right)$$
- Prove that each matrix \(\mathbf { A }\) must be of the form \(\left( \begin{array} { l l } a & b
0 & a \end{array} \right)\). - State clearly the restriction on the value of \(a\) such that \(\left( \begin{array} { l l } a & b
0 & a \end{array} \right)\) is in \(S\). - Prove that \(S\) is a group under the operation of matrix multiplication. (You may assume that matrix multiplication is associative.)