4.03e Successive transformations: matrix products

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } \cos 2 \theta & - \sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta \end{array} \right) \left( \begin{array} { l l } 1 & k \\ 0 & 1 \end{array} \right)\), where \(0 < \theta < \pi\) and \(k\) is a non-zero constant. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations, one of which is a shear.

1. Describe fully the other transformation and state the order in which the transformations are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Find, in terms of \(k\), the value of \(\tan \theta\) for which \(\mathbf { M - I }\) is singular.
4. Given that \(k = 2 \sqrt { 3 }\) and \(\theta = \frac { 1 } { 3 } \pi\), show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(3 y + \sqrt { 3 } x = 0\).

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 } \\ \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right) \left( \begin{array} { c c } 14 & 0 \\ 0 & 1 \end{array} \right)\).

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations in the \(x - y\) plane. Give full details of each transformation, and make clear the order in which they are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf { M }\).
4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Given that the area of triangle \(D E F\) is \(28 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(A B C\).

3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } 7 & 0 \\ 0 & 1 \end{array} \right)\).

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations in the \(x - y\) plane. Give full details of each transformation, and make clear the order in which they are applied. [4]
2. Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf { M }\).
   The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(P Q R\) .
3. Given that the area of triangle \(P Q R\) is \(35 \mathrm {~cm} ^ { 2 }\) ,find the area of triangle \(D E F\) .

1 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 1 & b \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } a & 0 \\ 0 & 1 \end{array} \right)\), where \(a\) and \(b\) are positive constants.

1. 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.
   The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto parallelogram \(O P Q R\).
2. Find, in terms of \(a\) and \(b\), the matrix which transforms parallelogram \(O P Q R\) onto the unit square.
   It is given that the area of \(O P Q R\) is \(2 \mathrm {~cm} ^ { 2 }\) and that the line \(\mathrm { x } + 3 \mathrm { y } = 0\) is invariant under the transformation represented by \(\mathbf { M }\).
3. Find the values of \(a\) and \(b\).

4 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 } \\ \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right)$$

1. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
2. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { B }\).
   The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A B }\) onto triangle \(P Q R\).
3. Show that the triangles \(D E F\) and \(P Q R\) have the same area.
4. Find the matrix which transforms triangle \(P Q R\) onto triangle \(D E F\).
5. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { A B }\).

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right) \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\).

1. 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. Find the values of \(\theta\), for \(0 \leqslant \theta \leqslant \pi\), for which the transformation represented by \(\mathbf { M }\) has exactly one invariant line through the origin, giving your answers in terms of \(\pi\).

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

1. 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.
   The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto parallelogram \(O P Q R\).
2. Find, in terms of \(k\), the area of parallelogram \(O P Q R\) and the matrix which transforms \(O P Q R\) onto the unit square.
3. 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 \(\mathbf { M }\).

7. (i) $$\mathbf { A } = \left( \begin{array} { r r } 6 & k \\ - 3 & - 4 \end{array} \right) , \text { where } k \text { is a real constant, } k \neq 8$$ Find, in terms of \(k\),

1. \(\mathbf { A } ^ { - 1 }\)
2. \(\mathbf { A } ^ { 2 }\) Given that \(\mathbf { A } ^ { 2 } + 3 \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 5 & 9 \\ - 3 & - 5 \end{array} \right)\)
3. find the value of \(k\).
   (ii) $$\mathbf { M } = \left( \begin{array} { c c } - \frac { 1 } { 2 } & - \sqrt { 3 } \\ \frac { \sqrt { 3 } } { 2 } & - 1 \end{array} \right)$$ The matrix \(\mathbf { M }\) represents a one way stretch, parallel to the \(y\)-axis, scale factor \(p\), where \(p > 0\), followed by a rotation anticlockwise through an angle \(\theta\) about \(( 0,0 )\).
   1. Find the value of \(p\).
   2. Find the value of \(\theta\).

$$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$ The matrix \(\mathbf { P }\) represents a geometrical transformation \(U\)

1. Describe \(U\) fully as a single geometrical transformation. The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through \(240 ^ { \circ }\) anticlockwise about the origin followed by an enlargement about ( 0,0 ) with scale factor 6
2. Determine the matrix \(\mathbf { Q }\), giving each entry in exact numerical form. Given that \(U\) followed by \(V\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\)
3. determine the matrix \(\mathbf { R }\) (ii) The transformation \(W\) is represented by the matrix $$\left( \begin{array} { c c } - 2 & 2 \sqrt { 3 } \\ 2 \sqrt { 3 } & 2 \end{array} \right)$$ Show that there is a real number \(\lambda\) for which \(W\) maps the point \(( \lambda , 1 )\) onto the point ( \(4 \lambda , 4\) ), giving the exact value of \(\lambda\) \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
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4. $$\mathbf { A } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$

1. Describe the single geometrical transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a rotation of \(210 ^ { \circ }\) anticlockwise about centre \(( 0,0 )\).
2. Write down the matrix \(\mathbf { B }\), giving each element in exact form. The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\).
3. Find \(\mathbf { C }\). The hexagon \(H\) is transformed onto the hexagon \(H ^ { \prime }\) by the matrix \(\mathbf { C }\).
4. Given that the area of hexagon \(H\) is 5 square units, determine the area of hexagon \(H ^ { \prime }\)

7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the \(x\)-axis.
2. Write down the matrix \(\mathbf { Q }\). Given that \(V\) followed by \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
3. find the matrix \(\mathbf { R }\).
4. Show that there is a real number \(k\) for which the transformation \(T\) maps the point \(( 1 , k )\) onto itself. Give the exact value of \(k\) in its simplest form.

10. In your answers to this question, the elements of each matrix should be expressed in exact form in surds where necessary. The transformation \(U\), represented by the \(2 \times 2\) matrix \(\mathbf { P }\), is a rotation through \(45 ^ { \circ }\) anticlockwise about the origin.

1. Write down the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through \(60 ^ { \circ }\) anticlockwise about the origin.
2. Write down the matrix \(\mathbf { Q }\). The transformation \(U\) followed by the transformation \(V\) is the transformation \(T\). The transformation \(T\) is represented by the matrix \(\mathbf { R }\).
3. Use your matrices from parts (a) and (b) to find the matrix \(\mathbf { R }\).
4. Give a full geometric description of \(T\) as a single transformation.
5. Deduce from your answers to parts (c) and (d) that \(\sin 75 ^ { \circ } = \frac { 1 + \sqrt { 3 } } { 2 \sqrt { 2 } }\) and find the
   exact value of \(\cos 75 ^ { \circ }\), explaining your answers fully.

6. (i) $$\mathbf { A } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$

1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a rotation of \(45 ^ { \circ }\) clockwise about the origin.
2. Write down the matrix \(\mathbf { B }\), giving each element of the matrix in exact form. The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\).
3. Determine \(\mathbf { C }\).
   (ii) The trapezium \(T\) has vertices at the points \(( - 2,0 ) , ( - 2 , k ) , ( 5,8 )\) and \(( 5,0 )\), where \(k\) is a positive constant. Trapezium \(T\) is transformed onto the trapezium \(T ^ { \prime }\) by the matrix $$\left( \begin{array} { r r } 5 & 1 \\ - 2 & 3 \end{array} \right)$$ Given that the area of trapezium \(T ^ { \prime }\) is 510 square units, calculate the exact value of \(k\).

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8. (i) The transformation \(U\) is represented by the matrix \(\mathbf { P }\) where, $$P = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\ \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$

1. Describe fully the transformation \(U\). The transformation \(V\), represented by the matrix \(\mathbf { Q }\), is a stretch scale factor 3 parallel to the \(x\)-axis.
2. Write down the matrix \(\mathbf { Q }\). Transformation \(U\) followed by transformation \(V\) is a transformation which is represented by matrix \(\mathbf { R }\).
3. Find the matrix \(\mathbf { R }\).
   (ii) $$S = \left( \begin{array} { r r } 1 & - 3 \\ 3 & 1 \end{array} \right)$$ Given that the matrix \(\mathbf { S }\) represents an enlargement, with a positive scale factor and centre \(( 0,0 )\), followed by a rotation with centre \(( 0,0 )\),
   1. find the scale factor of the enlargement,
   2. find the angle and direction of rotation, giving your answer in degrees to 1 decimal place.

10. $$\mathbf { A } = \left( \begin{array} { c c } 3 \sqrt { } 2 & 0 \\ 0 & 3 \sqrt { } 2 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 0 & 1 \\ 1 & 0 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 } \\ \frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$

1. Describe fully the transformations described by each of the matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\). It is given that the matrix \(\mathbf { D } = \mathbf { C A }\), and that the matrix \(\mathbf { E } = \mathbf { D B }\).
2. Find \(\mathbf { D }\).
3. Show that \(\mathbf { E } = \left( \begin{array} { c c } - 3 & 3 \\ 3 & 3 \end{array} \right)\). The triangle \(O R S\) has vertices at the points with coordinates \(( 0,0 ) , ( - 15,15 )\) and \(( 4,21 )\). This triangle is transformed onto the triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation described by \(\mathbf { E }\).
4. Find the coordinates of the vertices of triangle \(O R ^ { \prime } S ^ { \prime }\).
5. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\) and deduce the area of triangle \(O R S\).

4. The transformation \(U\), represented by the \(2 \times 2\) matrix \(\mathbf { P }\), is a rotation through \(90 ^ { \circ }\) anticlockwise about the origin.

1. Write down the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line \(y = - x\).
2. Write down the matrix \(\mathbf { Q }\). Given that \(U\) followed by \(V\) is transformation \(T\), which is represented by the matrix \(\mathbf { R }\), (c) express \(\mathbf { R }\) in terms of \(\mathbf { P }\) and \(\mathbf { Q }\),
   (d) find the matrix \(\mathbf { R }\),
   (e) give a full geometrical description of \(T\) as a single transformation.

6. Write down the \(2 \times 2\) matrix that represents

1. an enlargement with centre \(( 0,0 )\) and scale factor 8 ,
2. a reflection in the \(x\)-axis. Hence, or otherwise,
3. find the matrix \(\mathbf { T }\) that represents an enlargement with centre ( 0,0 ) and scale factor 8, followed by a reflection in the \(x\)-axis. $$\mathbf { A } = \left( \begin{array} { l l } 6 & 1 \\ 4 & 2 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } k & 1 \\ c & - 6 \end{array} \right) , \text { where } k \text { and } c \text { are constants. }$$
4. Find \(\mathbf { A B }\). Given that \(\mathbf { A B }\) represents the same transformation as \(\mathbf { T }\),
5. find the value of \(k\) and the value of \(c\).

6. $$\mathbf { P } = \left( \begin{array} { c c } - \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 } \\ \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(U\) maps the point \(A\), with coordinates \(( p , q )\), onto the point \(B\), with coordinates \(( 6 \sqrt { } 2,3 \sqrt { } 2 )\).
2. Find the value of \(p\) and the value of \(q\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\).
3. Write down the matrix \(\mathbf { Q }\). The transformation \(U\) followed by the transformation \(V\) is the transformation \(T\). The transformation \(T\) is represented by the matrix \(\mathbf { R }\).
4. Find the matrix \(\mathbf { R }\).
5. Deduce that the transformation \(T\) is self-inverse.

3. (i) Given that $$\mathbf { A } = \left( \begin{array} { r r } - 2 & 3 \\ 1 & 1 \end{array} \right) , \quad \mathbf { A } \mathbf { B } = \left( \begin{array} { r r r } - 1 & 5 & 12 \\ 3 & - 5 & - 1 \end{array} \right)$$

1. find \(\mathbf { A } ^ { - 1 }\)
2. Hence, or otherwise, find the matrix \(\mathbf { B }\), giving your answer in its simplest form.
   (ii) Given that $$\mathbf { C } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$
   1. describe fully the single geometrical transformation represented by the matrix \(\mathbf { C }\).
   2. Hence find the matrix \(\mathbf { C } ^ { 39 }\)

8 The matrix \(\mathbf { T }\) is given by \(\mathbf { T } = \left( \begin{array} { r r } 2 & 0 \\ 0 & - 2 \end{array} \right)\).

1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { T }\). [3]
2. The transformation represented by matrix \(\mathbf { T }\) is equivalent to a transformation \(A\), followed by a transformation B. Give geometrical descriptions of possible transformations A and B, and state the matrices that represent them.

9 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r } 0 & 3 \\ - 1 & 0 \end{array} \right)\).

1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\). The transformation represented by \(\mathbf { C }\) is equivalent to a rotation, R , followed by another transformation, S.
2. Describe fully the rotation R and write down the matrix that represents R .
3. Describe fully the transformation S and write down the matrix that represents S .

9

1. Write down the matrix, \(\mathbf { A }\), that represents an enlargement, centre ( 0,0 ), with scale factor \(\sqrt { 2 }\).
2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \\ - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { B }\).
3. Given that \(\mathbf { C } = \mathbf { A B }\), show that \(\mathbf { C } = \left( \begin{array} { r r } 1 & 1 \\ - 1 & 1 \end{array} \right)\).
4. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\).
5. Write down the determinant of \(\mathbf { C }\) and explain briefly how this value relates to the transformation represented by \(\mathbf { C }\).

7

1. Find the matrix that represents a rotation through \(90 ^ { \circ }\) clockwise about the origin.
2. Find the matrix that represents a reflection in the \(x\)-axis.
3. Hence find the matrix that represents a rotation through \(90 ^ { \circ }\) clockwise about the origin, followed by a reflection in the \(x\)-axis.
4. Describe a single transformation that is represented by your answer to part (iii).

9 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = x\) and has the same \(x\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h] \end{figure}

1. Write down the image of the point ( \(- 3,7\) ) under transformation T .
2. Write down the image of the point \(( x , y )\) under transformation T .
3. Find the \(2 \times 2\) matrix which represents the transformation.
4. Describe the transformation M represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).
5. Find the matrix representing the composite transformation of T followed by M .
6. Find the image of the point \(( x , y )\) under this composite transformation. State the equation of the line on which all of these images lie.