Describe enlargement or stretch from matrix

6.

1. $$\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r } - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$
   1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\).
   2. Describe fully the single transformation represented by the matrix \(\mathbf { B }\). The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\).
   3. Find \(\mathbf { C }\).
   4. \(\mathbf { M } = \left( \begin{array} { c c } 2 k + 5 & - 4 \\ 1 & k \end{array} \right)\), where \(k\) is a real number. Show that \(\operatorname { det } \mathbf { M } \neq 0\) for all values of \(k\).

7 Describe fully the geometrical transformation represented by each of the following matrices:

1. \(\left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\),
2. \(\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\),
3. \(\left( \begin{array} { l l } 1 & 0 \\ 0 & 6 \end{array} \right)\),
4. \(\left( \begin{array} { r r } 0.8 & 0.6 \\ - 0.6 & 0.8 \end{array} \right)\).

3 The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 3 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively under the transformation represented by the matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 0 \\ 0 & \frac { 1 } { 2 } \end{array} \right)\). \begin{figure}[h] \end{figure}

1. Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly.
2. Describe fully the transformation represented by the matrix \(\mathbf { M }\).

1 Transformation A is represented by matrix \(\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and transformation B is represented by matrix \(\mathbf { B } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 3 \end{array} \right)\).

1. Describe transformations A and B .
2. Find the matrix for the composite transformation A followed by B .

$$\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$

1. Describe fully the transformations described by each of the matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\). [4]

It is given that the matrix \(\mathbf{D} = \mathbf{CA}\), and that the matrix \(\mathbf{E} = \mathbf{DB}\).

1. Find \(\mathbf{D}\). [2]
2. Show that \(\mathbf{E} = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}\). [1]

The triangle \(ORS\) has vertices at the points with coordinates \((0, 0)\), \((-15, 15)\) and \((4, 21)\). This triangle is transformed onto the triangle \(OR'S'\) by the transformation described by \(\mathbf{E}\).

1. Find the coordinates of the vertices of triangle \(OR'S'\). [4]
2. Find the area of triangle \(OR'S'\) and deduce the area of triangle \(ORS\). [3]