8 The set \(X\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { r r } x & - y
y & x \end{array} \right)\), where \(x\) and \(y\) are real numbers which are not both zero.

1. (a) The matrices \(\left( \begin{array} { c c } a & - b
   b & a \end{array} \right)\) and \(\left( \begin{array} { c c } c & - d
   d & c \end{array} \right)\) are both elements of \(X\). Show that \(\left( \begin{array} { c c } a & - b
   b & a \end{array} \right) \left( \begin{array} { c c } c & - d
   d & c \end{array} \right) = \left( \begin{array} { c c } p & - q
   q & p \end{array} \right)\) for some real numbers \(p\) and \(q\) to be found in terms of \(a , b , c\) and \(d\).
   (b) Prove by contradiction that \(p\) and \(q\) are not both zero.
2. Prove that \(X\), under matrix multiplication, forms a group \(G\).
   [0pt] [You may use the result that matrix multiplication is associative.]
3. Determine a subgroup of \(G\) of order 17.