8 The operation \(*\) is defined on the elements \(( x , y )\), where \(x , y \in \mathbb { R }\), by
$$( a , b ) * ( c , d ) = ( a c , a d + b ) .$$
It is given that the identity element is \(( 1,0 )\).
- Prove that \(*\) is associative.
- Find all the elements which commute with \(( 1,1 )\).
- It is given that the particular element \(( m , n )\) has an inverse denoted by \(( p , q )\), where
$$( m , n ) * ( p , q ) = ( p , q ) * ( m , n ) = ( 1,0 ) .$$
Find \(( p , q )\) in terms of \(m\) and \(n\).
- Find all self-inverse elements.
- Give a reason why the elements \(( x , y )\), under the operation \(*\), do not form a group.