The matrix \(\mathbf{M}\) is given by \(\mathbf{M} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 7 & 0 \\ 0 & 1 \end{pmatrix}\).

(a) The matrix \(\mathbf{M}\) represents a sequence of two geometrical transformations in the \(x-y\) plane. Give full details of each transformation, and make clear the order in which they are applied.

(b) Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf{M}\).

The triangle \(DEF\) in the \(x-y\) plane is transformed by \(\mathbf{M}\) onto triangle \(PQR\).

(c) Given that the area of triangle \(PQR\) is \(35 \text{ cm}^2\), find the area of triangle \(DEF\).