Verify invariant line property

6 Given that \(y = m x\) is an invariant line of the transformation with matrix \(\left( \begin{array} { r r } 1 & 2 \\ 2 & - 2 \end{array} \right)\), determine the possible values of \(m\). Section B (113 marks)
Answer all the questions.

The matrix \(\mathbf{C}\) is defined by $$\mathbf{C} = \begin{bmatrix} 3 & 2 \\ -4 & 5 \end{bmatrix}$$ Prove that the transformation represented by \(\mathbf{C}\) has no invariant lines of the form \(y = kx\) [4 marks]

You are given that the matrix \(\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\) represents a transformation T.

1. You are given that the line with equation \(y = kx\) is invariant under T. Determine the value of \(k\). [4]
2. Determine whether the line with equation \(y = kx\) in part (a) is a line of invariant points under T. [1]

Transformation M is represented by matrix \(\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\).

1. On the diagram in the Printed Answer Booklet draw the image of the unit square under M. [2]
2. 1. Show that there is a constant \(k\) such that \(\mathbf{M} \begin{pmatrix} x \\ kx \end{pmatrix} = 5 \begin{pmatrix} x \\ kx \end{pmatrix}\) for all \(x\). [2]
   2. Hence find the equation of an invariant line under M. [1]
   3. Draw the invariant line from part (ii) (B) on your diagram for part (i). [1]

$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix},$$ where \(k\) is constant. A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\).

1. Find the value of \(k\) for which the line \(y = 2x\) is mapped onto itself under \(T\). [3]
2. Show that \(\mathbf{A}\) is non-singular for all values of \(k\). [3]
3. Find \(\mathbf{A}^{-1}\) in terms of \(k\). [2]