4.03e Successive transformations: matrix products

1. The triangle \(T\) has vertices \(A ( 2,1 ) , B ( 2,3 )\) and \(C ( 0,1 )\).

The triangle \(T ^ { \prime }\) is the image of \(T\) under the transformation represented by the matrix $$\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$

1. Find the coordinates of the vertices of \(T ^ { \prime }\)
2. Describe fully the transformation represented by \(\mathbf { P }\) The \(2 \times 2\) matrix \(\mathbf { Q }\) represents a reflection in the \(x\)-axis and the \(2 \times 2\) matrix \(\mathbf { R }\) represents a rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
3. Write down the matrix \(\mathbf { Q }\) and the matrix \(\mathbf { R }\)
4. Find the matrix \(\mathbf { R Q }\)
5. Give a full geometrical description of the single transformation represented by the answer to part (d).

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

1. Draw a diagram showing the image of the unit square under the transformation represented by \(\mathbf { C }\). The transformation represented by \(\mathbf { C }\) is equivalent to a transformation S followed by another transformation T.
2. Given that S is a shear with the \(y\)-axis invariant in which the image of the point ( 1,1 ) is ( 1,2 ), write down the matrix that represents \(S\).
3. Find the matrix that represents transformation T and describe fully the transformation T .

8 The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { l l } 0 & 3 \\ 3 & 0 \end{array} \right)\).

1. The diagram in the printed answer book shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { X }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
2. The transformation represented by \(\mathbf { X }\) 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.

6 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } 0 & 2 \\ - 1 & 0 \end{array} \right)\).

1. The diagram in the Printed Answer Book shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { M }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\), indicating clearly the coordinates of \(A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\).
2. The transformation represented by \(\mathbf { M }\) is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a possible pair of transformations P and Q and state the matrices that represent them.

3 Fig. 3 shows the unit square, OABC , and its image, \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\), after undergoing a transformation. \begin{figure}[h] \end{figure}

1. Write down the matrix \(\mathbf { P }\) representing this transformation.
2. The parallelogram \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is transformed by the matrix \(\mathbf { Q } = \left( \begin{array} { r r } 2 & - 1 \\ 0 & 3 \end{array} \right)\). Find the coordinates of the vertices of its image, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\), following this transformation.
3. Describe fully the transformation represented by \(\mathbf { Q P }\).

2 Fig. 2 shows the unit square, OABC , and its image, \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\), after undergoing a transformation. \begin{figure}[h] \end{figure}

1. Write down the matrix \(\mathbf { T }\) representing this transformation. The quadrilateral \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is reflected in the \(x\)-axis to give a new quadrilateral, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).
2. Write down the matrix representing reflection in the \(x\)-axis.
3. Find the single matrix that will transform OABC onto \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).

4 The transformations \(T _ { A }\) and \(T _ { B }\) are represented by the matrices \(\mathbf { A }\) and \(\mathbf { B }\) respectively, where \(\mathbf { A } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\)

1. Describe geometrically the single transformation consisting of \(T _ { A }\) followed by \(T _ { B }\).
2. By considering the transformation \(\mathrm { T } _ { \mathrm { A } }\), determine the matrix \(\mathrm { A } ^ { 423 }\). The transformation \(\mathrm { T } _ { \mathrm { C } }\) is represented by the matrix \(\mathbf { C }\), where \(\mathbf { C } = \left( \begin{array} { l l } \frac { 1 } { 2 } & 0 \\ 0 & \frac { 1 } { 3 } \end{array} \right)\). The region \(R\) is defined by the set of points \(( x , y )\) satisfying the inequality \(x ^ { 2 } + y ^ { 2 } \leqslant 36\). The region \(R ^ { \prime }\) is defined as the image of \(R\) under \(\mathrm { T } _ { \mathrm { C } }\).
3. 1. Find the exact area of the region \(R ^ { \prime }\).
   2. Sketch the region \(R ^ { \prime }\), specifying all the points where the boundary of \(R ^ { \prime }\) intersects the coordinate axes.

8 The diagram below shows a rectangle \(R _ { 1 }\) which has vertices \(( 0,0 ) , ( 3,0 ) , ( 3,2 )\) and \(( 0,2 )\).

1. On the diagram, draw:
   1. the image \(R _ { 2 }\) of \(R _ { 1 }\) under a rotation through \(90 ^ { \circ }\) clockwise about the origin;
   2. the image \(R _ { 3 }\) of \(R _ { 2 }\) under the transformation which has matrix $$\left[ \begin{array} { l l } 4 & 0 \\ 0 & 2 \end{array} \right]$$
2. Find the matrix of:
   1. the rotation which maps \(R _ { 1 }\) onto \(R _ { 2 }\);
   2. the combined transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-5_913_910_1228_598}

6

1. The matrix \(\mathbf { X }\) is defined by \(\left[ \begin{array} { l l } 1 & 2 \\ 3 & 0 \end{array} \right]\).
   1. Given that \(\mathbf { X } ^ { 2 } = \left[ \begin{array} { c c } m & 2 \\ 3 & 6 \end{array} \right]\), find the value of \(m\).
   2. Show that \(\mathbf { X } ^ { 3 } - 7 \mathbf { X } = n \mathbf { I }\), where \(n\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
2. It is given that \(\mathbf { A } = \left[ \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right]\).
   1. Describe the geometrical transformation represented by \(\mathbf { A }\).
   2. The matrix \(\mathbf { B }\) represents an anticlockwise rotation through \(45 ^ { \circ }\) about the origin. Show that \(\mathbf { B } = k \left[ \begin{array} { r r } 1 & - 1 \\ 1 & 1 \end{array} \right]\), where \(k\) is a surd.
   3. Find the image of the point \(P ( - 1,2 )\) under an anticlockwise rotation through \(45 ^ { \circ }\) about the origin, followed by the transformation represented by \(\mathbf { A }\). \(7 \quad\) The variables \(y\) and \(x\) are related by an equation of the form $$y = a x ^ { n }$$ where \(a\) and \(n\) are constants. Let \(Y = \log _ { 10 } y\) and \(X = \log _ { 10 } x\).

8 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows two triangles, \(T _ { 1 }\) and \(T _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{504b79bf-1bcc-4fa7-a7a0-689c21a8b03a-04_866_883_1318_550}

1. Find the matrix of the stretch which maps \(T _ { 1 }\) to \(T _ { 2 }\).
2. The triangle \(T _ { 2 }\) is reflected in the line \(y = x\) to give a third triangle, \(T _ { 3 }\). On Figure 3, draw the triangle \(T _ { 3 }\).
3. Find the matrix of the transformation which maps \(T _ { 1 }\) to \(T _ { 3 }\).

6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$ Describe fully the geometrical transformation represented by each of the following matrices:

1. A ;
2. B ;
3. \(\quad \mathbf { A } ^ { 2 }\);
4. \(\quad \mathbf { B } ^ { 2 }\);
5. AB.
   □

7 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left[ \begin{array} { c c } - 1 & - \sqrt { 3 } \\ \sqrt { 3 } & - 1 \end{array} \right]$$

1. 1. Calculate the matrix \(\mathbf { A } ^ { 2 }\).
   2. Show that \(\mathbf { A } ^ { 3 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
2. Describe the single geometrical transformation, or combination of two geometrical transformations, corresponding to each of the matrices:
   1. \(\mathrm { A } ^ { 3 }\);
   2. A.

8 The diagram shows two triangles, \(T _ { 1 }\) and \(T _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{d74d6295-d5b8-46da-8812-c5bf7c7a35f1-09_972_967_358_589}

1. Find the matrix which represents the stretch that maps triangle \(T _ { 1 }\) onto triangle \(T _ { 2 }\).
2. The triangle \(T _ { 2 }\) is reflected in the line \(y = \sqrt { 3 } x\) to give a third triangle, \(T _ { 3 }\). Find, using surd forms where appropriate:
   1. the matrix which represents the reflection that maps triangle \(T _ { 2 }\) onto triangle \(T _ { 3 }\);
   2. the matrix which represents the combined transformation that maps triangle \(T _ { 1 }\) onto triangle \(T _ { 3 }\).
      (2 marks)

5

1. The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { c c } - 2 & c \\ d & 3 \end{array} \right]\).
   Given that the image of the point \(( 5,2 )\) under the transformation represented by \(\mathbf { A }\) is \(( - 2,1 )\), find the value of \(c\) and the value of \(d\).
   [0pt] [4 marks]
2. The matrix \(\mathbf { B }\) is defined by \(\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 2 } & \sqrt { 2 } \\ - \sqrt { 2 } & \sqrt { 2 } \end{array} \right]\).
   1. Show that \(\mathbf { B } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   2. Describe the transformation represented by the matrix \(\mathbf { B }\) as a combination of two geometrical transformations.
   3. Find the matrix \(\mathbf { B } ^ { 17 }\). \(6 \quad \mathrm {~A}\) curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$$

11

1. Specify fully the transformations represented by the following matrices.

9 The transformation Too the plane has associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } - 1 & 0 \\ - 2 & 1 \end{array} \right)\).

1. On the grid in the Printed Answer Booklet, plot the image \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) of the unit square OABC under the transformation T.
2. 1. Calculate the value of \(\operatorname { det } \mathbf { M }\).
   2. Explain the significance of the value of \(\operatorname { det } \mathbf { M }\) in relation to the image \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\).
3. T is equivalent to a sequence of two transformations of the plane.
   1. Specify fully two transformations equivalent to T .
   2. Use matrices to verify your answer.

4. The transformation \(T\) in the plane consists of a translation in which the point \(( x , y )\) is transformed to the point ( \(x + 2 , y - 2\) ), followed by a reflection in the line \(y = x\).

1. Determine the \(3 \times 3\) matrix which represents \(T\).
2. Determine how many invariant points exist under the transformation \(T\).

1. A point \(P\) is reflected in the line \(y = k x\), where \(k\) is a constant. It is then rotated anticlockwise about \(O\) through an acute angle \(\theta\), where \(\cos \theta = 0 \cdot 8\). The resulting transformation matrix is given by \(T\), where

$$T = \frac { 1 } { 85 } \left[ \begin{array} { r r } - 84 & - 13 \\ - 13 & 84 \end{array} \right]$$

1. Determine the value of \(k\).
   Find the invariant points of \(T\).

5. $$\mathbf { A } = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\ \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { A }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { B }\), is a reflection in the line \(y = - x\)
2. Write down the matrix \(\mathbf { B }\). Given that \(U\) followed by \(V\) is the transformation \(T\), which is represented by the matrix \(\mathbf { C }\), (c) find the matrix \(\mathbf { C }\).
   (d) Show that there is a real number \(k\) for which the point \(( 1 , k )\) is invariant under \(T\).

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   VJYV SIHIL NI JIIYM ION OC

1. $$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$

1. 1. Describe fully the single geometrical transformation \(P\) represented by the matrix \(\mathbf { P }\).
   2. Describe fully the single geometrical transformation \(Q\) represented by the matrix \(\mathbf { Q }\). The transformation \(P\) followed by the transformation \(Q\) is the transformation \(R\), which is represented by the matrix \(\mathbf { R }\).
2. Determine \(\mathbf { R }\).
3. 1. Evaluate the determinant of \(\mathbf { R }\).
   2. Explain how the value obtained in (c)(i) relates to the transformation \(R\).

5

1. The transformation \(T _ { 1 }\) is defined by the matrix $$\left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right]$$ Describe this transformation geometrically.
2. The transformation \(T _ { 2 }\) is an anticlockwise rotation about the origin through an angle of \(60 ^ { \circ }\). Find the matrix of the transformation \(T _ { 2 }\). Use surds in your answer where appropriate.
   (3 marks)
3. Find the matrix of the transformation obtained by carrying out \(T _ { 1 }\) followed by \(T _ { 2 }\).
   (3 marks)

6 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \sqrt { 3 } & 3 \\ 3 & - \sqrt { 3 } \end{array} \right]$$

1. 1. Show that $$\mathbf { M } ^ { 2 } = p \mathbf { I }$$ where \(p\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   2. Show that the matrix \(\mathbf { M }\) can be written in the form $$q \left[ \begin{array} { c c } \cos 60 ^ { \circ } & \sin 60 ^ { \circ } \\ \sin 60 ^ { \circ } & - \cos 60 ^ { \circ } \end{array} \right]$$ where \(q\) is a real number. Give the value of \(q\) in surd form.
2. The matrix \(\mathbf { M }\) represents a combination of an enlargement and a reflection. Find:
   1. the scale factor of the enlargement;
   2. the equation of the mirror line of the reflection.
3. Describe fully the geometrical transformation represented by \(\mathbf { M } ^ { 4 }\).

6 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a rectangle \(R _ { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-4_652_1136_470_429}

1. The rectangle \(R _ { 1 }\) is mapped onto a second rectangle, \(R _ { 2 }\), by a transformation with matrix \(\left[ \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right]\).
   1. Calculate the coordinates of the vertices of the rectangle \(R _ { 2 }\).
   2. On Figure 1, draw the rectangle \(R _ { 2 }\).
2. The rectangle \(R _ { 2 }\) is rotated through \(90 ^ { \circ }\) clockwise about the origin to give a third rectangle, \(R _ { 3 }\).
   1. On Figure 1, draw the rectangle \(R _ { 3 }\).
   2. Write down the matrix of the rotation which maps \(R _ { 2 }\) onto \(R _ { 3 }\).
3. Find the matrix of the transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\).

7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a triangle with vertices \(A ( 1,1 ) , B ( 3,1 )\) and \(C ( 3,2 )\). \includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-04_1114_1141_552_360}

1. The triangle \(D E F\) is obtained by applying to triangle \(A B C\) the transformation T represented by the matrix $$\left[ \begin{array} { r r } 2 & 2 \\ - 2 & 2 \end{array} \right]$$
   1. Calculate the coordinates of \(D , E\) and \(F\).
   2. Draw the triangle \(D E F\) on Figure 1.
2. Given that T is a combination of an enlargement and a rotation, find the exact value of:
   1. the scale factor of the enlargement;
   2. the magnitude of the angle of the rotation.