(a) The matrix \(\mathbf{M} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 7 & 0 \\ 0 & 1 \end{pmatrix}\) represents a stretch followed by a shear. The first matrix \(\begin{pmatrix} 7 & 0 \\ 0 & 1 \end{pmatrix}\) is a stretch parallel to the \(x\)-axis with a scale factor of 7. The second matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\) is a shear with the \(x\)-axis fixed, mapping \((0,1)\) to \((2,1)\).

(b) To find the invariant lines, consider \(\mathbf{M} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7x + 2y \\ y \end{pmatrix}\). For a line \(y = mx\) to be invariant, \(mx = m(7x + 2mx)\). This simplifies to \(2m^2 + 6m = 0\), giving solutions \(m = 0\) and \(m = -3\). Thus, the invariant lines are \(y = 0\) and \(y = -3x\).

(c) The area of \(PQR\) is \(35 \text{ cm}^2\). The area of \(DEF\) is related by the determinant of the transformation matrix \(\mathbf{M}\), which is \(|7| = 7\). Therefore, \(\text{Area of } DEF = \frac{35}{7} = 5 \text{ cm}^2\).