Combined transformation matrix product

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 \(\mathbf{M} = \begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}\). [4]
2. The transformation represented by \(\mathbf{M}\) has a line of invariant points. Find, in terms of \(k\), the equation of this line. [3]

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

1. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\). [1]
2. Given that the area of \(P\) is \(3k^2\) units\(^2\), find the possible values of \(k\). [2]

$$\mathbf{A} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \quad \mathbf{B} = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$$ The transformation represented by \(\mathbf{B}\) followed by the transformation represented by \(\mathbf{A}\) is equivalent to the transformation represented by \(\mathbf{P}\).

1. Find the matrix \(\mathbf{P}\). [2]

Triangle \(T\) is transformed to the triangle \(T'\) by the transformation represented by \(\mathbf{P}\). Given that the area of triangle \(T'\) is 24 square units,

1. find the area of triangle \(T\). [3]

Triangle \(T'\) is transformed to the original triangle \(T\) by the matrix represented by \(\mathbf{Q}\).

1. Find the matrix \(\mathbf{Q}\). [2]