1 The matrix \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k ( k \neq 0 )\), followed by a shear, \(x\)-axis fixed, with \(( 0,1 )\) mapped to \(( k , 1 )\).

1. Show that \(\mathbf { M } = \left( \begin{array} { c c } k & k
   0 & 1 \end{array} \right)\).
2. The transformation represented by \(\mathbf { M }\) has a line of invariant points. Find, in terms of \(k\), the equation of this line.
   \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-02_2722_43_107_2005} The unit square \(S\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto the parallelogram \(P\).
3. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\).
4. Given that the area of \(P\) is \(3 k ^ { 2 }\) units \({ } ^ { 2 }\), find the possible values of \(k\).