The matrix \(\mathbf{M}\) is given by \(\mathbf{M} = \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\), where \(k\) is a constant and \(k \neq 0\) and \(k \neq 1\).

(a) The matrix \(\mathbf{M}\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied. [2]

The unit square in the \(x-y\) plane is transformed by \(\mathbf{M}\) onto parallelogram \(OPQR\).

(b) Find, in terms of \(k\), the area of parallelogram \(OPQR\) and the matrix which transforms \(OPQR\) onto the unit square. [3]

(c) Show that the line through the origin with gradient \(\frac{1}{k-1}\) is invariant under the transformation in the \(x-y\) plane represented by \(\mathbf{M}\). [3]