4 The matrix \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by a rotation of \(60 ^ { \circ }\) anticlockwise about the origin followed by a one-way stretch in the \(x\)-direction, scale factor \(d ( d \neq 0 )\).

1. Find \(\mathbf { M }\) in terms of \(d\).
2. The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto a parallelogram of area \(\frac { 1 } { 2 } d ^ { 2 }\) units \({ } ^ { 2 }\). Show that \(d = 2\).
   The matrix \(\mathbf { N }\) is such that \(\mathbf { M N } = \left( \begin{array} { l l } 1 & 1
   \frac { 1 } { 2 } & \frac { 1 } { 2 } \end{array} \right)\).
3. Find \(\mathbf { N }\).
4. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { M N }\).