The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel to the x-axis, scale factor 14, followed by a rotation anticlockwise about the origin through angle \(\frac{1}{3} \pi\).

(a) Show that \(2\mathbf{M} = \begin{pmatrix} 14 & -\sqrt{3} \\ 14\sqrt{3} & 1 \end{pmatrix}\).

(b) Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by M.

The unit square S in the x-y plane is transformed by M onto the rectangle P.

(c) Find the matrix which transforms P onto S.