MEI FP1 Study Resources Further Pure Matrices 1-4

MEI Further Pure Mathematics 1 Matrices Section 1: Introduction to matrices Study Plan Background You have probably never met matrices before. Don’t be frightened by them. The matrices work in FP1 provides a fairly straightforward introduction. Matrices have many applications in higher mathematics and it is likely that if you pursue mathematics or science at a higher level, you will make use of them. In this section matrices are introduced using simple practical contexts, and you will look at some simple properties of matrices. You will then look at how matrices can be used to describe transformations, which is the main context for the matrices work in FP1.

Detailed work plan 1. The opening Discussion Point is worth looking at, but don’t worry if you can’t see how to represent the arrangements symbolically. There is some help in the Notes and Examples, but you may prefer to come back to this at the end of the section. 2. Read pages 1 – 3 carefully. Answer the “Discussion Point” questions to make sure that you have understood. Answers to these are in the back of the textbook. There is an example showing calculations with matrices in the Notes and Examples. 3. Exercise 1A Attempt all the questions. You should find that this is a fairly straightforward exercise. 4. For some extra practice in calculating with matrices, try the interactive questions Adding and subtracting 2 × 2 matrices. 5. Read pages 6 – 12. This is extremely important as it forms the basis for much of the work in this chapter. Activities 1.1 and 1.2 are worth doing as they will help you understand the way matrices are used to describe transformations. Example 1.3 is very important as it gives you the matrix for a rotation through any angle about the origin. Using rotation matrices involves the use of some of the Trigonometry in C2. If you haven’t yet covered this work, look at the Trigonometry notes which will give you the help you need. Activity 1.3 will also help. There are some more examples on rotation matrices in the Notes and Examples.

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MEI FP1 Matrices Section 1 Study plan 6. To investigate matrix transformations further, try the Flash resource Transformations (read the instruction sheets first) or the Excel spreadsheet Investigating matrix transformations (there are instructions for this in the Notes and Examples).

7. Exercise 1B Attempt Questions 1 – 6. If you have time, try the enrichment questions 7 and 8 as well. 8. You can also test yourself on basic matrix transformations using the Matrix matchings activity.

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MEI Further Pure Mathematics 1 Matrices Section 1: Introduction to matrices Notes and Examples These notes contain subsections on  Opening discussion point  Matrices  Transformations

Opening Discussion Point Look at the opening discussion point on page 1. It’s useful to explain what is happening in terms of the coordinates of the points within the pack of cards. From the first diagram to the second, the pack has been tilted in the y direction so that the x and z coordinates of each point within the pack remain unchanged but the y coordinates have altered. The y coordinate of each point has increased by an amount which depends on its height up the pack. The higher the point, the more the y coordinate has increased. In other words, the y coordinate has increased by an amount proportional to its coordinate. A point in the first diagram with coordinates (x, y, z) has been moved to a point with coordinates (x, y + tz, z) in the second diagram, where t is a positive constant. (This type of transformation is called a shear. You will meet it again in Example 1.1 in the textbook). From the second diagram to the third, the whole pack has been rotated through an angle of 90° anticlockwise about the x axis. The effect of this is that the x coordinate of each point remains unchanged, but the x coordinates of points in the second diagram become their y coordinates in the third diagram, and the y coordinates of point in the second diagram become their negative x coordinates in the third diagram. A point in the second diagram with coordinates (X, Y, Z) has been moved to a point with coordinates (-Y, X, Z) in the third diagram. The overall effect of both transformations could be written as (x, y, z)  (x, y + tz, z)  (-(y + tz), x, z) (Note: in the textbook Answers, t is taken to be 1).

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MEI FP1 Matrices Section 2 Notes and Examples Don’t worry if you find this difficult! You will be looking at how transformations are described in the second half of this section, but this example is harder than most of the ones that you will encounter. For the most part, you will be working in two dimensions only. If you have found this explanation hard to follow, come back to it when you have completed the section.

Matrices A matrix is simply a way of storing information. In the textbook and in Exercise 1A you will see some simple uses for matrices. In this section you learn to add and subtract matrices, and to multiply a matrix by a number.

Example 1

 2 3  3 4  A is the matrix   , B is the matrix  .  1 0   1 2  Find (i) A+B (ii) A–B (iii) 2A (iv) 3B – A Solution (i) (ii) (iii) (iv)

 2 3   3 4   1 7  A+B=  + =   1 0   1 2   0 2   2 3   3 4   5 1 A–B=  – =   1 0   1 2   2 2   2 3  4 6 2A = 2  =   1 0   2 0   3 4   2 3   9 12   2 3   11 9  3B – A = 3  – =      1 2   1 0   3 6   1 0   4 6 

To practice matrix arithmetic, use the interactive questions Adding and subtracting 2 by 2 matrices.

Transformations Although matrices have many applications, their use to describe transformations is one of the main topics in this chapter. By the time you have worked through this section and Exercise 1B, you should be familiar with the matrices for simple transformations such as reflections in the x axis, the y axis and the lines y = x and y = -x, rotations through 90° and 180°, enlargements and two-way stretches. However, if you

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MEI FP1 Matrices Section 2 Notes and Examples can’t remember the matrix, it is easy enough to work it out. All you need to do is to think about what happens to the x and y coordinates of a point after the transformation. This method is shown in the book, in the example of a flag being rotated through 90° anticlockwise, and you can try it out for yourself in Activity 1.1. Solutions to this Activity are given in the back of the textbook. Example 1.3 gives a very important result: the matrix which represents a  cos   sin   anticlockwise rotation through any angle  is given by  .  sin  cos   To deal with rotations through angles greater than 90°, you need to know a little about the sines and cosines of such angles. If you have already covered the trigonometry work in C2, this should be no problem. If you haven’t, look at the trigonometry notes which cover everything you will need to know, or refer to the AS Pure Mathematics textbook, chapter 10.

You can use the Flash resource Transformations to investigate matrix transformations. Read the instruction sheet first.

You can also use the spreadsheet Investigating matrix transformations. Change the transformation matrix on the left-hand side to look at different transformations of a square. You can zoom in or out using the buttons to the right of the graph. You can also change the initial shape by altering the OBJECT coordinates. Ignore “Inverse matrix” and “Determinant” for now.

Example 2 Find the matrix which describes (i) Anticlockwise rotation through 140° about the origin (ii) Clockwise rotation through 62° about the origin Solution (i) The matrix for anticlockwise rotation through angle  about the origin is  cos  sin    .  sin  cos 

(ii)

 cos140  sin140  In this case  = 140°, so the matrix is   cos140   sin140  0.766 0.642     0.642 0.766  A clockwise rotation involves a negative angle, so in this case  = -62°.  cos(62 )  sin(62 )   0.469 0.883  The matrix is     sin(62 ) cos(62 )   0.883 0.469  Alternatively you can use the matrix for clockwise rotation through  cos sin   angle  about the origin,   , with  = 62°. This will give the   sin  cos 

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MEI FP1 Matrices Section 2 Notes and Examples same result.

Example 3 Each of the following matrices represents a rotation about the origin. Find the angle and direction of rotation in each case.  12 3  12   0.6 0.8  (i) (ii)      1 1 3   0.8 0.6   2 2  0.966 0.259   0.766 0.643  (iii) (iv)      0.259 0.966   0.643 0.766  Solution In each case, compare the given matrix with the matrix for anticlockwise rotation  cos  sin   through angle ,  .  sin  cos  (i)

In this case cos  12 2,sin  12 . Both are positive, so the angle is in the first quadrant. arcsin  12  = 30°. The matrix represents an anticlockwise rotation of 30° about the origin.

(ii)

In this case cos  = -0.6, sin  = -0.8. Both are negative, so the angle is in the third quadrant. Using a calculator, arccos 0.6 = 53.1°. The corresponding angle in the third quadrant is 180° + 53.1° = 233.1°. The matrix represents an anticlockwise rotation of 233.1° about the origin. (OR a clockwise rotation of 126.9°).

(iii)

In this case cos  = -0.966, sin  = 0.259. cos  is negative and sin  is positive, so the angle is in the second quadrant. Using a calculator, arcsin 0.259 = 15°. The corresponding angle in the second quadrant is 165°. The matrix represents an anticlockwise rotation of 165° about the origin.

(iv)

In this case cos  = 0.766, sin  = -0.463. cos  is positive and sin  is negative, so the angle is in the fourth quadrant. Using a calculator, arcos 0.766 = 40°. The corresponding angle in the fourth quadrant is 320° or –40°. The matrix represents a clockwise rotation of 40° about the origin. (OR an anticlockwise rotation of 320°).

You can test yourself on matrix transformations using the Matrix matchings activity. The enrichment material on transformations in three dimensions, although beyond the syllabus, is well worth looking at. The ability to visualise mathematical problems in three dimensions is a useful one, and there is work

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MEI FP1 Matrices Section 2 Notes and Examples on vectors in three dimensions in C4. The textbook shows how the example in the opening Discussion Point can be expressed as a matrix. The transformation matrices you have met in this section are all 2  2 matrices. They represent mappings from two-dimensional space to twodimensional space. In the enrichment material, you meet 3  3 transformation matrices which represent mappings from three-dimensional space to threedimensional space. It is also possible to have transformation matrices which are not square. For example, when you draw the plan of a building to a scale of 1:100, you start with an object in three dimensional space and end up with 0 0  1 a two-dimensional drawing. You could write this as the matrix  100 1 .  0 100 0  So a matrix with m rows and n columns is used to describe a transformation from n dimensional space to m dimensional space.

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Further Pure Mathematics 1 Trigonometry needed for work on Matrices In your work on matrices, you will need to be able to deal with the sines and cosines of angles greater than 90°. If you haven’t yet done the work on Trigonometry in C2, this may be new to you. However, the notes and practice questions below should give you all the help you need.

Angles greater than 90° The diagram shows a point P with coordinates (x, y) joined to the origin. The line OP is 1 unit long and makes an angle  with the x axis. In this case  is between 0° and 90°. We say that P is in the first quadrant.

P (x, y) 1

y

You can see from the diagram that  y x O sin    y 1 x cos    x 1 y tan   x x and y are both positive, so sin , cos  and tan  are all positive. These definitions for sin , cos  and tan  can be extended to any values of . If P is in the second quadrant,  is between 90° and 180°. In this case, the x coordinate of P is negative, but the y coordinate remains positive. So

P (x, y) 1

y



sin  is positive cos  is negative tan  is negative.

x

O

If P is in the third quadrant,  is between 180° and 270°. In this case, both the x coordinate and the y coordinate of P are negative. So

sin  is negative cos  is negative tan  is positive.



x

O y 1 P (x, y)

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Further Pure Mathematics 1 If P is in the fourth quadrant,  is between 270° and 360°. In this case, the x coordinate of P is positive and the y coordinate is negative. So



sin  is negative cos  is positive tan  is negative.

x O y

1

P (x, y) In the case of an clockwise rotation, the angle is negative. However, to find out whether sin , cos  and tan  are positive or negative, all you need to know is which quadrant the angle is in. So an angle of –30° is in the fourth quadrant, where sin  and tan  are negative but cos  is positive, and an angle of –150° is in the third quadrant, where sin  and cos  are negative bur tan  is positive.

Angles with the same values of sin, cos or tan Suppose you want to find an angle  for which sin  = 0.574. Using a calculator, you find that arcsin 0.574 (or sin -1 0.574) = 35°. Your calculator gives you only one answer, but in fact there are an infinite number of possible answers. The symmetry of the diagram below shows that for the angles 35°, 145°, 215° and 325° the numerical values of sin , cos  and tan  are all the same, although some are negative. (You could also include 395°, 505°, -35° and so on.)

 = 145°

 = 35°

(= 180° - 35°)

 = 215°

 = 325°

(= 180° + 35°)

(= 360° - 35°)

Since sin  is positive in the first and second quadrants, the values of  between 0° and 360° which satisfy the equation sin  = 0.574 are 35° and 145°.

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Further Pure Mathematics 1 Example Find the values of  between 0° and 360° which satisfy the following equations: (i) sin  = -0.4 (ii) tan  = 2 (iii) cos  = -0.7 Solution (i) Using a calculator, arcsin 0.4 = 23.6°. As sin  is negative in the third and fourth quadrants, we need to find the angles equivalent to 23.6° in these quadrants. In the third quadrant this is 180° + 23.6° = 203.6° In the fourth quadrant this is 360° - 23.6° = 336.4°  = 203.6° or 336.4°. (ii)

Using a calculator, arctan 2 = 63.4°. tan  is positive in the first and third quadrants. In the first quadrant the angle required is 63.4°. In the third quadrant the angle required in 180° + 63.4° = 243.4°  = 63.4° or 243.4°.

(iii)

Using a calculator, arcos 0.7 = 45.6°. cos  is negative in the second and third quadrants. In the second quadrant the angle required is 180° - 45.6° = 134.4° In the third quadrant the angle required is 180° + 45.6° = 225.6°.  = 134.4° or 225.6°.

Now try these. 1.

Find the values of  between 0° and 360° which satisfy the following equations. (i) sin  = 0.5 (ii) cos  = 0.8 (iii) tan  = 0.6 (iv) sin  = -0.3 (v) cos  = -0.65 (vi) tan  = -1.5 (vii) cos  = -0.24 (viii) sin  = -0.9

2.

Find the values of  between -180° and 180° which satisfy the following equations. (i) cos  = 0.15 (ii) sin  = 0.75 (iii) tan  = 0.82 (iv) sin  = -0.67 (v) tan  = -0.42 (vi) cos  = -0.55 (vii) sin  = -0.38 (viii) cos  = -0.84

(Answers at the bottom of the next page).

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Further Pure Mathematics 1 Special angles You will find it useful to know the exact values of sin, cos and tan of some special angles which come up quite often. The angles 30° and 60° 60°

2

This special right-angled triangle gives you the following information:

1

30°

3 sin 30  sin 60 

1 2 3 2

3 2 1 cos 60  2 cos 30 

tan 30 

1 3

tan 60  3

Eventually you will probably be able to remember all these. If you can’t, all you need to remember is that sin 30  12 . Then you can draw the triangle, put in the opposite side as 1 and the hypotenuse as 2, work out that the third side is 3 , and read off the information you need. The angle 45° This is even easier. You just need to draw an isosceles triangle like this. This gives the information:

sin 45 

1 2

cos 45 

1 2

2

45°

1

45° 1 tan 45  1

Again, if you can’t remember these, just draw the isosceles triangle with two sides 1, and work out that the hypotenuse is 2 .

Answers to questions 1. (i)  = 30° or 150° (iii)  = 31.0° or 211.0° (v)  = 130.5° or 229.5° (vii)  = 103.9° or 256.1° 2.

(i) (iii) (v) (vii)

 = 81.4° or –81.4°  = 39.4° or –140.6°  = 157.2° or –22.8°  = -22.7° or –157.7°

(ii) (iv) (vi) (viii)

 = 36.9° or 323.1°  = 197.5° or 342.5°  = 123.7° or 303.7°  = 244.2° or 295.8°

(ii) (iv) (vi) (viii)

 = 48.6° or 131.4°  = -42.1° or –137.9°  = 123.4° or –123.4°  = 147.1° or –147.1°

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Further Pure Mathematics 1 Matrices Section 1: Introduction to matrices Crucial points 1. Check answers carefully It’s easy to make careless mistakes in matrix arithmetic. 2. Make sure that you are familiar with the matrices for simple transformations You need to know the matrices for reflection in the x axis, the y axis and the lines y = x and y = -x, and the matrices for rotation through 90° or 180° about the origin. All these look a bit similar, with 0s, 1s and – 1s, so make sure that you can work out what they are with a quick diagram if you’re not sure. 3. Make sure that you are also familiar with the matrices for enlargement and two-way stretches These are quite easy to remember, with the numbers on the leading diagonal giving you the scale factors, and zeros in the other two positions. 4. Make sure that you know the general rotation matrix The matrix for a rotation of θ anticlockwise about the origin is ⎛ cos θ − sin θ ⎞ ⎜ ⎟ , and that you can recognise it even if cos θ and / or sin ⎝ sin θ cos θ ⎠ θ are negative. Check that you can find the angle of rotation from the matrix, including cases where the angle is not in the first quadrant. 5. Remember the useful result about the columns of a matrix The image of the point I (1, 0) gives the first column of the matrix, and the image of the point J (0, 1) gives the second column of the matrix.

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Further Pure Mathematics 1 Matrices Section 1: Introduction to matrices Exercise 1. This diagram shows a map of the roads linking 3 towns A, B and C. The corresponding ‘direct route’ matrix is shown beside it. A B C B A ⎡0 1 2⎤ B ⎢1 0 1 ⎥ A C ⎢ ⎥ C ⎢⎣2 1 0⎥⎦ For each of the following diagrams construct the direct route matrix. (i)

A

B

(ii)

B C

C

A

2. From the following direct route matrix construct a corresponding route map.

A B C D

A ⎡0 ⎢2 ⎢ ⎢0 ⎢ ⎣0

B C D 2 0 0⎤ 0 1 0⎥⎥ 1 0 2⎥ ⎥ 0 2 0⎦

3. A café sells 3 main meals A, B, and C each day. On two days the sales of each type are shown in the matrix below. M T A ⎡4 6⎤ ⎢ ⎥ B ⎢3 5⎥ C ⎢⎣7 2⎥⎦ If meal A costs £4, meal B costs £5 and meal C costs £3 construct a matrix showing the amount taken for each of the meals on each of the two days. Hence state the total amount taken for each mean over the two days.

4. Five football teams took part in a league competition during which they each played each other in the first half of the season and again in the second half of the season. The results matrix for the first half of the season is shown in the matrix below. W D L

A B C D E

⎡2 ⎢2 ⎢ ⎢0 ⎢ ⎢1 ⎢⎣1

1 1 4 1 1

1⎤ 1⎥⎥ 0⎥ ⎥ 2⎥ 2⎥⎦

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Further Pure Mathematics 1 The results from the second half of the season are shown below: B3:A4 C1:B0 E1:D0 D3:C3 E0:A6 E 2 : C2 D2:A0 E5:B2 C3:A1 D2:B4 Construct a results matrix for the whole season. If a win gains 3 points and a draw 1 point, determine the league positions at the end of the season.

5. Find the values of x and y in each of the following matrix equations: ⎛ 3 − 5⎞ ⎛ 1 y ⎞ ⎛ 4 6 ⎞ ⎟⎟ + ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ (i) ⎜⎜ ⎝ 2 x ⎠ ⎝ 3 2 ⎠ ⎝ 5 − 2⎠ ⎛ 3 ⎞ ⎛ − 2⎞ ⎛ − 5⎞ (ii) ⎜⎜ ⎟⎟ + x⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ − 1⎠ ⎝ y ⎠ ⎝ 11 ⎠

6. Plot the object and image for each of the following on the same diagram and describe each as a single transformation. Object

Matrix

(i) P(4, 2) Q(4, 4) R(0, 4)

0 ⎞ ⎛ − 0.5 ⎟ ⎜⎜ − 0.5 ⎟⎠ ⎝ 0

(ii) P(-6, 8) Q(-2, 8) R(-2, 6)

⎛ 0 − 1⎞ ⎜⎜ ⎟⎟ ⎝−1 0 ⎠

7. Draw a quadrilateral with vertices A(3, 4) B(4, 0) C(3, 1) D (0, 0) and find its ⎛− 2 0 ⎞ ⎟⎟ . Describe the transformation, and find image under the transformation ⎜⎜ ⎝ 0 − 2⎠ the ratio of the image area to object area.

8. Find the images of A(3, 1) B(3, 3) C(6, 3) D(6, 1) under the transformation ⎛ 1 0⎞ ⎜⎜ ⎟⎟ . Illustrate the effect of the transformation. ⎝ − 2 1⎠

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Further Pure Mathematics 1 Matrices Section 1: Introduction to matrices Solutions to Exercise 1. (i)

A B C

A ⎛0 1 0⎞ ⎜ ⎟ B ⎜1 0 2⎟ C ⎜⎝ 0 2 0 ⎟⎠ 2.

A B C

(ii)

A ⎛0 1 2⎞ ⎜ ⎟ B ⎜1 0 1⎟ C ⎜⎝ 2 1 0 ⎟⎠

A

B

D

C

3. Multiply the A row by 4, the B row by 5 and the C row by 3. M

T

M

A⎛ 4× 4 4×6 ⎞ A ⎛ 16 ⎜ ⎟ = ⎜ B⎜5 ×3 5 ×5 ⎟ B ⎜ 15 C ⎜⎝ 3 × 7 3 × 2 ⎟⎠ C ⎜⎝ 21

T 24 ⎞ ⎟ 25 ⎟ 6 ⎟⎠

The total taken for meal A over the two days is £16 + £24 = £40. The total taken for meal B over the two days is £15 + £25 = £40. The total taken for meal C over the two days is £21 + £6 = £27. 4.

In the second half of the season: A win two and lose two B win one and lose three C win two and draw two D win one, draw one and lose two E win two, draw one and lose one The results matrix for the whole season is

W L

D

A⎛2 0 2⎞ B⎜ 1 0 3⎟ ⎜ ⎟ C ⎜2 2 0⎟ D ⎜⎜ 1 1 2 ⎟⎟ E ⎜⎝ 2 1 1 ⎟⎠

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Further Pure Mathematics 1 W L

D

W L

D

W L

D

A⎛2 1 1 ⎞ A⎛2 0 2⎞ A⎛4 1 B⎜2 1 1⎟ B⎜ 1 0 3⎟ B⎜3 1 ⎜ ⎟ + ⎜ ⎟ = ⎜ C ⎜0 4 0⎟ C ⎜2 2 0⎟ C ⎜2 6 ⎜ ⎟ ⎜ ⎟ D⎜1 1 2⎟ D⎜1 1 2⎟ D ⎜⎜ 2 2 E ⎜⎝ 1 1 2 ⎟⎠ E ⎜⎝ 2 1 1 ⎟⎠ E ⎜⎝ 3 2

3⎞ 4⎟ ⎟ 0⎟ 4 ⎟⎟ 3 ⎟⎠

A’s points are (4 × 3) + (1 × 1) = 13 B’s points are (3 × 3) + (1 × 1) = 10 C’s points are (2 × 3) + (6 × 1) = 12 D’s points are (2 × 3) + (2 × 1) = 8 E’s points are (3 × 3) + (2 × 1) = 11 The league positions are A, C, E, B, D. ⎛ 3 −5 ⎞ ⎛ 1 y ⎞ ⎛ 4 6 ⎞ 5. (i) ⎜ ⎟+⎜ ⎟=⎜ ⎟ ⎝ 2 x ⎠ ⎝ 3 2 ⎠ ⎝ 5 −2 ⎠ ⎛ 4 −5 + y ⎞ ⎛ 4 6 ⎞ = ⎜5 x + 2 ⎟⎠ ⎜⎝ 5 −2 ⎟⎠ ⎝ −5 + y = 6 ⇒ y = 11 x + 2 = −2 ⇒ x = −4 ⎛3⎞ ⎛ −2 ⎞ ⎛ −5 ⎞ (ii) ⎜ ⎟ + x ⎜ ⎟ = ⎜ ⎟ ⎝ −1 ⎠ ⎝ y ⎠ ⎝ 11 ⎠ ⎛ 3 − 2 x ⎞ ⎛ −5 ⎞ ⎜ −1 + xy ⎟ = ⎜ 11 ⎟ ⎝ ⎠ ⎝ ⎠

3 − 2 x = −5 ⇒ x = 4 −1 + xy = 11 ⇒ −1 + 4 y = 11

⇒y=3

6. (i) The transformation is x ′ = −0.5 x

y ′ = −0.5 y The image of P (4, 2) is P′ (-2, -1) The image of Q (4, 4) is Q′ (-2, -2) The image of R (0, 4) is R′ (0, -2)

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Further Pure Mathematics 1 R

Q

P

P′ R′

Q′

The transformation is an enlargement, scale factor -0.5, centre the origin. (ii) The transformation is x ′ = − y

y′ = − x The image of P (-6, 8) is P′ (-8, 6) The image of Q (-2, 8) is Q′ (-8, 2) The image of R (-2, 6) is R′ (-6, 2) P P′

Q′

Q R

R′

The transformation is a reflection in the line y = -x. 7. The transformation is x ′ = −2 x

y ′ = −2 y The image of the quadrilateral is A′ (-6, -8), B′ (-8, 0), C′(-6, -2), D′(0, 0).

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Further Pure Mathematics 1 8

y

6

A

4

C

2

D′

B′ –8

–6

–4

–2

D

2

B 4

x 6

8

–2

C′

–4 –6 –8

A′

The transformation is an enlargement, scale factor -2, centre the origin. The area of the object = 16 − 6 − 2 − 2 = 6 . The area of the image = 64 − 24 − 8 − 8 = 24 . The ratio of the image area to the object area is 4 : 1. 8. The transformation is x ′ = x

y ′ = −2 x + y The image of A (3, 1) is A′ (3, -5) The image of B (3, 3) is B′ (3, -3) The image of C (6, 3) is C′ (6, -9) The image of D (6, 1) is D′ (6, -11).

B

C

A

D

B′ A′ C′ D′

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Matrices Open the Matrices investigation and select a type of transformation (“General” is not appropriate for this investigation).

Investigating Press the view buttons on the Transformation, Matrix and Image boxes so that everything is visible. Change the transformation* and observe the relationship between it and the matrix of the transformation. You should check that any relationship still holds if you move any of the points A, B, C or D. (The red points can be moved by dragging them).

Setting yourself some questions 1. Finding matrices and images from transformations • Press the hide buttons on the Matrix and Image boxes so that only the transformation is visible. • Move the points A, B, C or D to create a new quadrilateral. • Set a new transformation. (i) (ii) (iii)

Write down the 2 × 2 matrix of the transformation. Draw the position of the image and write down the 2 × 4 matrix representing the coordinates of the image (A’, B’, C’, D’). Check that the result of multiplying the matrix of the transformation (i) by the 2 × 4 matrix representing the coordinates of the original shape (A, B, C, D) gives the matrix in (ii). Use the view buttons to check your answers to (i) & (ii).

2. Finding transformations from matrices and images • Press the hide button on the transformation box so that only the Matrix and Image boxes are visible. • Move the points A, B, C or D to create a new quadrilateral. • Set a new transformation. (i)

Write down the transformation. Use the view button to check your answer.

*Changing the transformations Enlargement: Reflection: Rotation: 2-way stretch: Shear: General:

The scale factor can be varied with the slider-bar. By selecting the relevant graph either y = x, y = -x, x = 0 or y = 0 can be chosen as the line of reflection. By selecting the relevant direction either anti-clockwise or clockwise can be chosen. The angle of rotation can be varied with the slider-bar. The factors of the stretches in either direction can be varied using the slider-bars. By selecting the relevant direction either horizontal or vertical can be chosen. The factor of the shear can be varied with the slider-bar. The four values of the transformation matrix can be varied with the sliderbars.

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Matrix matchings Cut out the diagrams, descriptions of transformations and matrices and match up each diagram with the appropriate transformation and the matrix which describes it. In the diagrams, the blue triangle is the object and the pink one is the image.

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Matrix matchings 

Rotation by 90 clockwise Rotation by 90 anticlockwise Reflection in y-axis

Reflection in x-axis Reflection in the line

Reflection in the line

yx y  x

Rotation by 180 Enlargement with scale factor 2, centre the origin Stretch with scale factor 2, in the direction of the x-axis Stretch with scale factor 2, in the direction of the y-axis

1 0  0 1  2 0  1 0  2 0   1 0   1 0  0 1  0  1  0  1 

© MEI & Susan Whitehouse, 20/02/07

0 2  1 0  0 1  0 1 0 2  0 1  0 1 1 0  1 0  1 0  2/2

Further Pure Mathematics 1 Matrices Section 1: Introduction to matrices Multiple Choice Test ⎛ 3 1) The order of the matrix ⎜⎜ −2 ⎜ ⎝ 3

(a) 3 × 5 (c) 15 (e) I don’t know

8

6

4 0

−1 0

0⎞ ⎟ 0 2 ⎟ is 2 −4 ⎠⎟ 2

(b) 5 × 3 (d) 8

⎛ −1 2 ⎞ In Questions 2 – 4, A is the matrix ⎜ ⎟ and B is the matrix ⎝ 1 3⎠

⎛2 4 ⎞ ⎜ ⎟. ⎝ 0 −1⎠

2) A + B = ⎛1 6 ⎞ (a) ⎜ ⎟ ⎝1 2 ⎠ ⎛ 6 1⎞ (c) ⎜ ⎟ ⎝ 2 1⎠ (e) I don’t know

⎛ −3 −2 ⎞ (b) ⎜ ⎟ ⎝1 4⎠ ⎛ −2 −3 ⎞ (d) ⎜ ⎟ ⎝ 4 1⎠

3) A – B = ⎛ −1 − 2 ⎞ (a) ⎜ ⎟ ⎝1 2⎠ ⎛1 6 ⎞ (c) ⎜ ⎟ ⎝1 2 ⎠ (e) I don’t know

⎛ −3 −2 ⎞ (b) ⎜ ⎟ ⎝1 4⎠ ⎛1 2 ⎞ (d) ⎜ ⎟ ⎝1 2 ⎠

4) 3A – 2B = ⎛ −2 −1⎞ (a) ⎜ ⎟ ⎝ 2 6⎠ ⎛ −7 − 2 ⎞ (c) ⎜ ⎟ ⎝ 3 11 ⎠ (e) I don’t know

⎛ 1 14 ⎞ (b) ⎜ ⎟ ⎝3 7 ⎠ ⎛ 6 11⎞ (d) ⎜ ⎟ ⎝6 6 ⎠

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Further Pure Mathematics 1 ⎛ −1 0 ⎞ 5) The matrix ⎜ ⎟ represents the transformation ⎝ 0 −1 ⎠ (a) Reflection in the line y = -x (b) Rotation through 180° about the origin (c) Reflection in the line y = x (d) Rotation through 90° clockwise about the origin (e) I don’t know ⎛ 0 −1 ⎞ 6) The matrix ⎜ ⎟ represents the transformation ⎝1 0 ⎠ (a) Reflection in the x axis (b) Reflection in the y axis (c) Rotation through 90° anticlockwise about the origin (d) Rotation through 90° clockwise about the origin (e) I don’t know

7) A two-way stretch, scale factor 4 in the x direction and scale factor 2 in the y direction is represented by the matrix ⎛ 0 4⎞ (a) ⎜ ⎟ ⎝ 2 0⎠ ⎛ 0 2⎞ (c) ⎜ ⎟ ⎝ 4 0⎠ (e) I don’t know

⎛2 (b) ⎜ ⎝0 ⎛4 (d) ⎜ ⎝0

0⎞ ⎟ 4⎠ 0⎞ ⎟ 2⎠

8) A reflection in the x axis is represented by the matrix ⎛1 0 ⎞ (a) ⎜ ⎟ ⎝ 0 −1 ⎠ ⎛ 0 −1 ⎞ (c) ⎜ ⎟ ⎝1 0 ⎠ (e) I don’t know

⎛ −1 (b) ⎜ ⎝0 ⎛0 (d) ⎜ ⎝ −1

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Further Pure Mathematics 1 9) Which of these matrices represent rotations? ⎛ 0.8 −0.6 ⎞ A=⎜ ⎟ ⎝ 0.6 0.8 ⎠ ⎛ −0.8 0.6 ⎞ D=⎜ ⎟ ⎝ 0.6 −0.8 ⎠

⎛ −0.8 0.6 ⎞ B=⎜ ⎟ ⎝ −0.6 −0.8 ⎠ ⎛ 0.8 0.6 ⎞ E=⎜ ⎟ ⎝ −0.6 0.8 ⎠

(a) A and E (c) A, B, D and E (e) I don’t know

⎛ 0.8 0.6 ⎞ C=⎜ ⎟ ⎝ 0.6 −0.8 ⎠

(b) A only (d) A, B and E

10) A transformation maps a point P to a point P′ as follows: P is mapped to P′ on the line y = 2x in such a way that the line joining P to P′ is parallel to the y axis, as shown in the diagram. y P′

x y = 2x

P

The matrix describing the transformation is ⎛1 0⎞ (a) ⎜ ⎟ ⎝0 2⎠ ⎛1 0⎞ (c) ⎜ ⎟ ⎝ 2 0⎠ (e) I don’t know

⎛1 (b) ⎜ ⎝0 ⎛2 (d) ⎜ ⎝0

2⎞ ⎟ 0⎠ 0⎞ ⎟ 1⎠

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MEI Further Pure Mathematics 1 Matrices Section 2: Matrix multiplication Study Plan Background This section covers the multiplication of matrices and its application to transformations. It also looks at composite transformations (one transformation followed by another) and how this is represented by matrices.

Detailed work plan 1. Read the pages 14 – 17 carefully, and study the examples to make sure that you understand the technique. There are some further examples of matrix multiplication in the Notes and Examples. Look carefully at the application of matrix multiplication to transformations, and also Example 1.5, which looks at a different kind of application. 2. You can practise matrix multiplication using the interactive questions Multiplying 2 by 2 matrices. 3. Read pages 18 – 20. The Activities are useful, and full solutions to these are given in the Notes and Examples. Make sure that you understand the meaning of the identity matrix. 4. Exercise 1C Attempt questions 1, 2, 3, 5, 7 and 8. For a challenge you could also try question 9. 5. Read pages 23 – 25. Make sure you understand the “Notation” section. The enrichment Activity 1.10 gives some useful Trigonometry results which are covered in C4. 6. Exercise 1D Attempt questions 1, 2, 3, 4, 6, 8 and 9. For a challenge you could also try questions 10 and 11.

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MEI Further Pure Mathematics 1 Matrices Section 2: Multiplying matrices Notes and Examples These notes contain subsections on  Multiplying matrices  Properties of matrix multiplication  Composition of transformations

Multiplying matrices Multiplying matrices is an important skill which you must master. It takes a bit of getting used to, but after plenty of practice you will find it quite straightforward. Read Example 1.4 and the two other multiplications which follow it carefully and make sure that you understand each step. The important points to remember are:  Use each row of the first matrix with each column of the second.  When you are using row a of the first matrix with column b of the second matrix, the result gives you the element in row a, column b of the product matrix.  To multiply matrices, the number of columns in the first matrix must be the same as the number of rows in the second matrix. If this is not the case, the matrices do not conform and cannot be multiplied.

For some practise in multiplying matrices, try the interactive questions Multiplying 2 by 2 matrices.

Example 1

2 A is the matrix   1  3 B is the matrix  1

3 . 5 2 0 . 4 2 

 1 1    C is the matrix  0 3  . 2 4    Find where possible (i) AB (ii) (iv) CB (v)

BA AC

(iii) (vi)

BC CA

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MEI FP1 Matrices Section 2 Notes and Examples Solution (i) A is a 2 × 2 matrix and B is a 2 × 3 matrix, so these matrices conform.  2 3  3 2 0   2  3  3 1 2  2  3  4 2  0  3  2  AB =      1 5  1 4 2   1 3  5 1 1 2  5  4 1 0  5  2 

(ii) (iii)

(iv)

(v) (vi)

 3 16 6     8 18 10  B is a 2 × 3 matrix and A is a 2 × 2 matrix, so these matrices do not conform. It is not possible to find the product BA. B is a 2 × 3 matrix and C is a 3 × 2 matrix, so these matrices conform.  1 1  3 2 0     3 1  2  0  0  2 3  1  2  3  0  4  BC =  0 3      1 4 2   2 4   11  4  0  2  2 1 1  4  3  2  4     3 9     3 3  C is a 3 × 2 matrix and B is a 2 × 3 matrix, so these matrices conform.  1 1 1 3  11 1 2  1 4 1 0  1 2     3 2 0    CB =  0 3      0  3  3  1 0  2  3  4 0  0  3   2   2 4   1 4  2   2  3  4  1 2  2  4  4 2  0  4   2       4 2 2      3 12 6   2 20 8    A is a 2 × 2 matrix and C is a 3 × 2 matrix, so these matrices do not conform. It is not possible to find the product AC. C is a 3 × 2 matrix and A is a 2 × 2 matrix, so these matrices conform.  1 1  1  2  1   1 1  3   1  5     2 3   CA =  0 3      0  2  3  1 0  3  3  5   1 5   2  2  4  1 2  3  4  5   2 4       3 2      3 15   0 26   

Note that it is not necessary to write out the calculations in full, as in the Example above. It is shown here so that you can see what is being done, and it is also written out like this in the Solutions on the website to Question 1 of Exercise 1C, but not in any subsequent Solutions. You may like to write it out in full until you feel confident, or you may feel able to miss out that step from the beginning. Being able to multiply matrices gives you a shorthand for dealing with transformations. Look back to Question 5 of Exercise 1B. Here you had to find out the coordinates of the images of O (0, 0), A (1, 0), B (1, 1) and C (0, 1)

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MEI FP1 Matrices Section 2 Notes and Examples  4 3 under the transformation represented by the matrix   . To do this you  5 4 needed to write out the equations represented by the matrix: x  4 x  3 y y  5 x  4 y and apply these to each of the points O, A, B and C. Using matrix multiplication you can do this whole process in one go:

 4 3  0 1 1 0   0 4 7 3       5 4  0 0 1 1   0 5 9 4  and simply read off the coordinates of the image points from the matrix: O′ = (0, 0), A′ = (4, 5), B′ = (7, 9), C′ = (3, 4). Notice that the transformation matrix must come first in the multiplication and the matrix of coordinates second. In this case of course the matrices would not conform if they were not in that order, but the same rule applies whatever the order of the matrices.

Properties of matrix multiplication Make sure that you know the important properties of matrix multiplication:  Matrices must be conformable for multiplication  Matrix multiplication is not commutative  Matrix multiplication is associative You have already seen in Example 1 above that matrix multiplication is not commutative. In that case, AB exists but BA does not, BC and CB both exist but are different (in fact they have different orders) and AC does not exist but CA does. Activity 1.5 gives you the chance to prove that matrix multiplication is associative for any 2 × 2 matrices. The full solution to this Activity is given below.

Activity 1.5 a c   e g i k Using P =  , Q =   and R =   , find b d   f h j l (i) PQ (ii) (PQ)R (iii) QR (iv) P(QR) and so demonstrate that matrix multiplication is associative.

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MEI FP1 Matrices Section 2 Notes and Examples Solution

 a c  e g   ae  cf ag  ch  PQ =      b d  f h   be  df bg  dh   ae  cf ag  ch  i k  (ii) (PQ)R      be  df bg  dh  j l   aei  cfi  agj  chj aek  cfk  agl  chl     bei  dfi  bgj  dhj bek  dfk  bgl  dhl   e g  i k   ei  gj ek  gl  (iii) QR =      f h  j l   fi  hj fk  hl   a c  ei  gj ek  gl  (iv) P(QR)      b d  fi  hj fk  hl   aei  agj  cfi  chj aek  agl  cfk  chl     bei  bgj  dfi  dhj bek  bgl  dfk  dhl  (PQ)R = P(QR) so matrix multiplication is associative for all 2 × 2 matrices. (i)

You could carry out a similar proof for matrices of any order, provided they were conformable, i.e. their orders were p × q, q × r and r × s respectively. The enrichment material covers a further property of matrix multiplication: that matrix multiplication is distributive over matrix addition. Although not required for the syllabus, it is not difficult to prove in Activity 1.6. The full solution to the Activity is given below.

Activity 1.6 a c   e g i k Using P =  , Q =   and R =   , find b d   f h j l (i) P(Q + R) (ii) PQ + PR (iii) (P + Q)R (iv) PR + QR and so demonstrate the distributive property of matrix multiplication over matrix addition. Solution (i)

 a c  e P(Q + R)     b d  f

g   i k    h   j l   a c  e  i g  k      b d  f  j h  l   a (e  i )  c ( f  j ) a ( g  k )  c ( h  l )     b (e  i )  d ( f  j ) b( g  k )  d ( h  l ) 

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MEI FP1 Matrices Section 2 Notes and Examples (ii)

(iii)

(iv)

 a c  e g   a c  i k  PQ + PR       b d  f h   b d  j l   ae  cf ag  ch   ai  cj ak  cl      be  df bg  dh   bi  dj bk  dl   ae  cf  ai  cj ag  ch  ak  cl     be  df  bi  dj bg  dh  bk  dl 

 a c   e g  i k  (P + Q)R        b d   f h  j l   a  e c  g  i k      b  f d  h  j l   (a  e)i  (c  g ) j (a  e)k  (c  g )l     (b  f )i  (d  h) j (b  f )k  (d  h)l   a c  i k   e g  i k  PR + QR       b d  j l   f h  j l 

 ai  cj ak  cl   ei  gj ek  gl      bi  dj bk  dl   fi  hj fk  hl   ai  cj  ei  gj ak  cl  ek  gl     bi  dj  fi  hj bk  dl  fk  hl  So P(Q + R) = PQ + PR and (P + Q)R = PR + QR

Composition of transformations The important thing to notice here is the order in which transformations take place is read right to left. If you have done the work on functions in C3, then you will know that this also happens when you write down composite functions. QX(P) means “carry out the transformation X on the point P, then carry out the transformation Q on the result”. If you want to find the matrix representing the transformation “X followed by Q”, you must work out the matrix product QX. In Activity 1.9, you can prove that if two transformations can be represented by matrices, then the composite transformation can be represented by the product of the matrices of the two transformations, in the correct order. The full solution to this Activity is given below.

Solution to Activity 1.9  a c  x   ax  cy  (i) p = Tp =        b d  y   bx  dy   p r  ax  cy   apx  cpy  brx  dry  (ii) p = Sp′ =      q x  bx  dy   aqx  cqy  bsx  dsy 

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MEI FP1 Matrices Section 2 Notes and Examples (iii)

p U = ST =  q  ap  br Up =   aq  bs

r  a c   ap  br cp  dr     s  b d   aq  bs cq  ds  cp  dr  x   apx  brx  cpy  dry       = p cq  ds  y   aqx  bsx  cqy  dsy 

The enrichment Activity 1.10 proves some trigonometry results called the compound angle formulae, which are covered in C4. The solution to the Activity is given in the Answers in the textbook. These formulae are useful in the Complex Number work you meet in chapter 2.

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Further Pure Mathematics 1 Matrices Section 2: Matrix multiplication Crucial points 1. Make sure that you can do matrix multiplication confidently This is fundamental to the whole chapter. 2. Remember that matrix multiplication is not commutative In general, AB ≠ BA. This is an easy mistake to make as we are all used to ordinary multiplication being commutative. 3. Make sure you multiply matrices in the correct order for composite transformations Remember that “transformation A followed by transformation B” is represented by the matrix BA.

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Further Pure Mathematics 1 Matrices Section 2: Matrix multiplication Exercise 1. A square has vertices at (0, 0) (1, 1) (0, 2) (-1, 1). (i) Write down a matrix S that describes this information. 3 ⎞ ⎛ 4 ⎟⎟ by matrix (ii) Find the image of the square under the transformation ⎜⎜ ⎝ − 3 − 2⎠ multiplication. (iii) Draw both the object and the image on the same diagram. 2. Using matrix multiplication find and draw the image of the square O(0, 0) A(1, 0) ⎛ 3 0⎞ ⎟⎟ . B(1, 1) C(0, 1) under the transformation matrix ⎜⎜ ⎝ 0 2⎠ Describe the transformation. ⎛1 0 ⎞ ⎛ − 2 1⎞ ⎟⎟ and B = ⎜⎜ ⎟⎟ , 3. If A = ⎜⎜ ⎝ 2 −1⎠ ⎝ 3 0⎠ (i) Find (A + B)2 (ii) Find AB and BA (iii) Show that (A + B)2 = A2 + AB + BA + B2 ⎛ 3 1⎞ ⎟⎟ and B = 4. If A = ⎜⎜ ⎝ x 2⎠

⎛6 2⎞ ⎜⎜ ⎟⎟ find the values of x and y given that AB = BA. ⎝4 y⎠

⎛ 2 0⎞ ⎟⎟ and h and k are numbers such that A2 = hA + kI, 5. The matrix A = ⎜⎜ ⎝ 1 2⎠ where I is the identity matrix. Find h and k. 6. Three garages G1, G2, G3 sell cars of two types A and B. The sales in one week are shown below together with a matrix showing the price to be paid for each type. Work out the total value of sales for the three garages in that week. A B⎞ ⎛ ⎜ ⎟ ⎛ A 8000 ⎞ ⎜ G1 3 1 ⎟ ⎟ Sales ⎜ Prices ⎜⎜ ⎟ G2 2 0 B 10500 ⎟⎠ ⎝ ⎜ ⎟ ⎜G 4 1 ⎟ ⎝ 3 ⎠ 7. Find 2 × 2 matrices to represent the transformation P, which is a reflection in the y axis and the transformation Q, which is a rotation of 90° clockwise about the origin. Hence find a single matrix to represent a reflection in the y axis followed by a rotation of 90° clockwise about the origin. Describe this as a single

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Further Pure Mathematics 1 transformation.

8. Find the effect on the square with vertices O(0, 0) A(1, 0) B(1, 1) C(0, 1) of the ⎛ 0 1⎞ ⎛ 1 − 1⎞ ⎟⎟ and S = ⎜⎜ ⎟⎟ . matrices R, S, RS and SR given that R = ⎜⎜ ⎝ − 1 0⎠ ⎝0 1 ⎠

9. (i)

Draw triangle ABC such that A(2, 1) B(7, 1) and C(2, 4) ⎛1 − 1⎞ ⎟⎟ and plot the image on the (ii) Find the image of ABC under the matrix ⎜⎜ ⎝1 1 ⎠ same graph. (iii) The transformation is a rotation followed by an enlargement. Calculate the angle of rotation and the scale factor of the enlargement.

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Further Pure Mathematics 1 Matrices Section 2: Matrix multiplication Solutions to Exercise ⎛ 0 1 0 −1 ⎞ 1. (i) S = ⎜ ⎟ ⎝0 1 2 1 ⎠ 6 −1 ⎞ ⎛ 4 3 ⎞ ⎛ 0 1 0 −1 ⎞ ⎛ 0 7 = (ii) ⎜ −3 −2 ⎟ ⎜ 0 1 2 1 ⎟ ⎜ 0 −5 −4 1 ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ The image of the square has vertices at (0, 0), (7, -5), (6, -4) and (-1, 1). (iii)

⎛ 3 0 ⎞⎛0 1 1 0⎞ ⎛0 3 3 0 ⎞ 2. ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎝ 0 2 ⎠⎝0 0 1 1 ⎠ ⎝0 0 2 2 ⎠

The transformation is a two-way stretch, with scale factor 3 parallel to the xaxis, and scale factor 2 parallel to the y-axis.

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Further Pure Mathematics 1 3. (a) (i)

(ii) (iii)

⎡⎛ 1 ( A + B ) = ⎢⎜ ⎣⎝ 2 ⎛ −1 =⎜ ⎝5 ⎛ 6 =⎜ ⎝ −10 2

0 ⎞ ⎛ −2 1 ⎞ ⎤ + −1 ⎟⎠ ⎜⎝ 3 0 ⎟⎠ ⎥⎦ 1 ⎞ ⎛ −1 1 ⎞ −1 ⎟⎠ ⎜⎝ 5 −1 ⎟⎠ −2 ⎞ 6 ⎟⎠

2

⎛ 1 0 ⎞ ⎛ −2 1 ⎞ ⎛ −2 1 ⎞ AB = ⎜ ⎟ ⎟⎜ ⎟=⎜ ⎝ 2 −1 ⎠ ⎝ 3 0 ⎠ ⎝ −7 2 ⎠ ⎛ −2 1 ⎞ ⎛ 1 0 ⎞ ⎛ 0 −1 ⎞ BA = ⎜ ⎟ ⎟⎜ ⎟=⎜ ⎝ 3 0 ⎠ ⎝ 2 −1 ⎠ ⎝ 3 0 ⎠

⎛ 1 0 ⎞⎛ 1 0 ⎞ ⎛ 1 0⎞ (b) A 2 = ⎜ ⎟ ⎟⎜ ⎟=⎜ ⎝ 2 −1 ⎠ ⎝ 2 −1 ⎠ ⎝ 0 1 ⎠ ⎛ −2 1 ⎞ ⎛ −2 1 ⎞ ⎛ 7 −2 ⎞ B2 = ⎜ ⎟ ⎟⎜ ⎟=⎜ ⎝ 3 0 ⎠ ⎝ 3 0 ⎠ ⎝ −6 3 ⎠ ⎛ 1 0 ⎞ ⎛ −2 1 ⎞ ⎛ 0 −1 ⎞ ⎛ 7 −2 ⎞ A 2 + AB + BA + B 2 = ⎜ ⎟+⎜ ⎟+⎜ ⎟+⎜ ⎟ ⎝ 0 1 ⎠ ⎝ −7 2 ⎠ ⎝ 3 0 ⎠ ⎝ −6 3 ⎠ ⎛ 6 −2 ⎞ =⎜ ⎟ ⎝ −10 6 ⎠ = ( A + B)2

6+y ⎞ ⎛ 3 1 ⎞ ⎛ 6 2 ⎞ ⎛ 22 4. AB = ⎜ = ⎟⎜ ⎟ ⎜ ⎟ ⎝ x 2 ⎠⎝ 4 y ⎠ ⎝6x + 8 2 x + 2y ⎠ 10 ⎞ ⎛ 6 2 ⎞ ⎛ 3 1 ⎞ ⎛ 18 + 2 x BA = ⎜ = ⎟⎜ ⎟ ⎜ ⎟ ⎝ 4 y ⎠ ⎝ x 2 ⎠ ⎝ 12 + xy 4 + 2 y ⎠ 6 + y ⎞ ⎛ 18 + 2 x ⎛ 22 AB = BA ⇒ ⎜ ⎟=⎜ ⎝ 6 x + 8 2 x + 2 y ⎠ ⎝ 12 + xy 22 = 18 + 2x 6 + y = 10 Check:

10 ⎞ 4 + 2 y ⎟⎠

⇒x=2 ⇒y=4

6x + 8 = 12 + 8 = 20 2x + 2y = 4 + 8 = 12

12 + xy = 12 + 8 = 20 4 + 2y = 4 + 8 = 12

⎛2 0⎞ 5. A = ⎜ ⎟ ⎝1 2⎠ ⎛ 2 0 ⎞⎛ 2 0 ⎞ ⎛ 4 0 ⎞ A² = ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎝ 1 2 ⎠⎝ 1 2 ⎠ ⎝ 4 4⎠

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Further Pure Mathematics 1 ⎛4 A² = hA + kI ⇒ ⎜ ⎝4 ⎛4 ⇒⎜ ⎝4

0⎞ ⎛2 0⎞ ⎛1 0⎞ = + h k ⎜ 1 2⎟ ⎜0 1⎟ 4 ⎟⎠ ⎝ ⎠ ⎝ ⎠ 0 ⎞ ⎛ 2h + k 0 ⎞ =⎜ ⎟ 4⎠ ⎝ h 2h + k ⎟⎠

⇒ h = 4, k = −4

6. Total value of sales for each garage is given by G1 ⎛ 3 1 ⎞ G 1 ⎛ 3 × 8000 + 1 × 10500 ⎞ G 1 ⎛ 34500 ⎞ ⎟ ⎜ ⎟ ⎛ 8000 ⎞ ⎜ ⎟ ⎜ G2 ⎜ 2 0 ⎟ ⎜ ⎟ = G 2 ⎜ 2 × 8000 + 0 × 10500 ⎟ = G 2 ⎜ 16000 ⎟ 10500 ⎠ G ⎜ 4 × 8000 + 1 × 10500 ⎟ G ⎜ 42500 ⎟ G 3 ⎜⎝ 4 1 ⎟⎠ ⎝ 3⎝ 3⎝ ⎠ ⎠

7. Under P, the point (1, 0) is mapped to the point (-1, 0) and the point (0, 1) is unchanged. ⎛ −1 0 ⎞ So P is represented by ⎜ ⎟. ⎝ 0 1⎠ Under Q, the point (1, 0) is mapped to the point (0, -1) and the point (0, 1) is mapped to the point (1, 0). ⎛ 0 1⎞ So Q is represented by ⎜ ⎟. ⎝ −1 0 ⎠ ⎛ 0 1 ⎞ ⎛ −1 0 ⎞ ⎛ 0 1 ⎞ The single matrix is QP = ⎜ ⎟⎜ ⎟=⎜ ⎟. ⎝ −1 0 ⎠ ⎝ 0 1 ⎠ ⎝ 1 0 ⎠ This transformation is a reflection in the line y = x.

1 1⎞ ⎛ 0 1 ⎞⎛0 1 1 0⎞ ⎛0 0 = 8. ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎝ −1 0 ⎠ ⎝ 0 0 1 1 ⎠ ⎝ 0 −1 −1 0 ⎠

R is a rotation clockwise about the origin through 90°. ⎛ 1 −1 ⎞ ⎛ 0 1 1 0 ⎞ ⎛ 0 1 0 −1 ⎞ ⎜ 0 1 ⎟⎜0 0 1 1 ⎟ = ⎜0 0 1 1 ⎟ ⎠ ⎝ ⎠⎝ ⎠ ⎝

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Further Pure Mathematics 1

S is a shear parallel to the x-axis. ⎛ 0 1 ⎞ ⎛ 1 −1 ⎞ ⎛ 0 1 ⎞ RS = ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎝ −1 0 ⎠ ⎝ 0 1 ⎠ ⎝ −1 1 ⎠

⎛ 1 −1 ⎞ ⎛ 0 1 ⎞ ⎛ 1 1 ⎞ SR = ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎝ 0 1 ⎠ ⎝ −1 0 ⎠ ⎝ − 1 0 ⎠

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Further Pure Mathematics 1 ⎛ 1 −1 ⎞ ⎛ 2 9. (i), (ii) ⎜ ⎟⎜ ⎝ 1 1 ⎠⎝ 1

7 1

2 ⎞ ⎛ 1 6 −2 ⎞ = 4 ⎠⎟ ⎝⎜ 3 8 6 ⎠⎟

⎛ cos θ Rotation anticlockwise through θ is represented by ⎜ ⎝ sin θ ⎛k 0 ⎞ Enlargement scale factor k is represented by ⎜ ⎟ ⎝0 k⎠

(iii)

− sin θ ⎞ cos θ ⎟⎠

Rotation followed by enlargement is represented by ⎛ k 0 ⎞ ⎛ cos θ − sin θ ⎞ ⎛ k cos θ −k sin θ ⎞ ⎜ 0 k ⎟ ⎜ sin θ cos θ ⎟ = ⎜ k sin θ k cos θ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ 1 −1 ⎞ Comparing this with ⎜ ⎟ gives: ⎝1 1 ⎠ k cos θ = 1 ⇒ tan θ = 1 ⇒ θ = 45° k sin θ = 1 1 cos θ = ⇒k = 2 2 The angle of rotation is 45° anticlockwise and the scale factor of the enlargement is 2 .

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Further Pure Mathematics 1 Matrices Section 2: Matrix multiplication Multiple Choice Test In Questions 1-5,

⎛2 A is the matrix ⎜ ⎝1 ⎛2 ⎜ B is the matrix ⎜ 4 ⎜1 ⎝

3⎞ ⎟. 3⎠

3⎞ ⎟ 6⎟ 2 ⎟⎠ ⎛ −1 0 4 ⎞ C is the matrix ⎜ ⎟ ⎝ 3 1 −2 ⎠ ⎛ 3 0 4⎞ ⎜ ⎟ D is the matrix ⎜ −2 1 0 ⎟ ⎜ ⎟ ⎝ 2 3 −1 ⎠

1) BA is the matrix ⎛ 7 15 ⎞ ⎜ ⎟ (a) ⎜ 14 30 ⎟ ⎜ 4 9⎟ ⎝ ⎠ ⎛16 24 ⎞ (c) ⎜ ⎟ ⎝14 21 ⎠ (e) I don’t know

⎛ 7 14 (b) ⎜ ⎝ 15 30

4⎞ ⎟ 9⎠

(d) Not defined

2) AB is the matrix (a) Not defined

⎛ 7 14 4 ⎞ (c) ⎜ ⎟ ⎝ 15 30 9 ⎠ (e) I don’t know

⎛ 7 ⎜ (b) ⎜ 14 ⎜ 4 ⎝ ⎛16 (d) ⎜ ⎝14

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15 ⎞ ⎟ 30 ⎟ 9 ⎟⎠ 24 ⎞ ⎟ 21 ⎠

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Further Pure Mathematics 1 3) CB is the matrix ⎛ 7 3 14 ⎞ (a) ⎜ ⎟ ⎝ 14 6 28 ⎠ (c) Not defined

⎛7 ⎜ (b) ⎜ 14 ⎜5 ⎝ ⎛2 (d) ⎜ ⎝8

3 14 ⎞ ⎟ 6 28 ⎟ 2 8 ⎟⎠ 5⎞ ⎟ 11⎠

⎛5 (b) ⎜ ⎝3 ⎛13 ⎜ (d) ⎜ 2 ⎜2 ⎝

12 −8 ⎞ ⎟ −5 14 ⎠ 1⎞ ⎟ −5 ⎟ 11 ⎟⎠

(e) I don’t know 4) CD is the matrix ⎛13 2 2 ⎞ (a) ⎜ ⎟ ⎝ 1 −5 11⎠ ⎛5 3⎞ ⎜ ⎟ (c) ⎜ 12 −5 ⎟ ⎜ −8 14 ⎟ ⎝ ⎠ a. I don’t know 5) DB is the matrix ⎛ 10 0 15 ⎞ (a) ⎜ ⎟ ⎝17 0 22 ⎠ ⎛10 17 ⎞ ⎜ ⎟ (c) ⎜ 0 0 ⎟ ⎜ 15 22 ⎟ ⎝ ⎠ (e) I don’t know

⎛0 1 ⎞ ⎜ ⎟ (b) ⎜ 7 12 ⎟ ⎜ 7 10 ⎟ ⎝ ⎠ ⎛0 7 7 ⎞ (d) ⎜ ⎟ ⎝ 1 12 10 ⎠

6) A is a 2 × 2 matrix, B is a 2 × 3 matrix and C is a 3 × 2 matrix. Which of the following calculations are possible? (i) BC + A (ii) CA + B (a) (ii) only (c) (i) only (e) I don’t know

(b) neither (d) both

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Further Pure Mathematics 1 7) A triangle has vertices A (2, 0), B (3, 1) and C (-1, 3) ⎛2 1⎞ The triangle is transformed using the matrix ⎜ ⎟. ⎝ 3 4⎠ The vertices of the image are (a) A′ = (2, 4), B′ = (7, 9), C′ = (11, 7) (c) A′ = (6, 4), B′ = (13, 7), C′ = (9, 1) (e) I don’t know

(b) A′ = (4, 2), B′ = (9, 7), C′ = (7, 11) (d) A′ = (4, 6), B′ = (7, 13), C′ = (1, 9)

⎛ 2 0⎞ In Questions 8 – 10, the transformation A is represented by the matrix A = ⎜ ⎟ ⎝ −1 3 ⎠ ⎛1 2⎞ and the transformation B is represented by the matrix B = ⎜ ⎟ ⎝ −2 −3 ⎠ 8) The composite transformation “A followed by B” is represented by the matrix ⎛ 2 −4 ⎞ (a) ⎜ ⎟ ⎝ 5 −7 ⎠ ⎛0 6⎞ (c) ⎜ ⎟ ⎝ −1 −9 ⎠ (e) I don’t know

4 ⎞ ⎛ 2 (b) ⎜ ⎟ ⎝ −7 −11⎠ ⎛ 0 7⎞ (d) ⎜ ⎟ ⎝ −6 −9 ⎠

9) The composite transformation “B followed by A” is represented by the matrix ⎛0 6⎞ (a) ⎜ ⎟ ⎝ −1 −9 ⎠ ⎛ 2 −4 ⎞ (c) ⎜ ⎟ ⎝ 5 −7 ⎠ (e) I don’t know

4 ⎞ ⎛ 2 (b) ⎜ ⎟ ⎝ −7 −11⎠ ⎛ 0 7⎞ (d) ⎜ ⎟ ⎝ −6 −9 ⎠

10) The image of the point (3, 2) under the transformation “A followed by B followed by A again” is (a) (0, -49) (c) (28, -35) (e) I don’t know

(b) (18, -15) (d) (24, -75)

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MEI Further Pure Mathematics 1 Matrices Section 3: Inverse matrices Study Plan Background This section looks at finding the inverse of a 2 × 2 matrix. If a matrix represents a transformation, the inverse matrix represents the inverse transformation: i.e. the transformation which “undoes” the original transformation. Finding an inverse matrix involves finding the determinant of the matrix. The second half of the section looks at the significance of the determinant of a transformation matrix, and what happens if the determinant is zero.

Detailed work plan 1. Read pages 28 – 30, and do the Activities if possible. There are some additional notes in the Notes and Examples. 2. For practice in this work, try the interactive questions Find the inverse of a 2 by 2 matrix. 3. Exercise 1E Attempt all the questions. The syllabus only requires you to be able to find the inverse of a 2 × 2 matrix, but notice that Question 6 involves 3 × 3 matrices. Here you are given help in finding the inverses. This type of question does sometimes come up in exams, but you will not be expected to find the inverse of a 3 × 3 matrix without guidance. (Inverses of 3 × 3 matrices are covered in FP2.) 4. Read pages 31 – 34, and do the Activities if possible. There are some further notes in the Notes and Examples. 5. Exercise 1F Attempt questions 1 – 7 If you have time try the starred questions 8 and 9. Question 9 gives an interesting result: that all shears preserve area. (Question 4 shows that a particular shear preserves area).

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MEI Further Pure Mathematics 1 Matrices Section 3: Inverse matrices Notes and Examples These notes contain subsections on  Inverse matrices  Using the determinant of a 2  2 matrix

Inverse matrices This section looks at inverse transformations and their matrices. The inverse of a transformation undoes the effect of the original transformation.   

The inverse of a rotation is a rotation through the same angle about the same point but in the opposite direction. The inverse of a reflection is another reflection in the same line. The inverse of an enlargement with scale factor k is an enlargement 1 with scale factor . k

You can use these ideas in Activity 1.11. The Discussion Point and Activity 1.11 demonstrate a very important point – that the product of the matrix representing a transformation with the matrix representing the inverse of the transformation gives the identity matrix. This makes sense – the identity matrix maps a point to itself, and a transformation followed by its inverse maps a point back to its original position. So a matrix multiplied by its inverse gives the identity matrix. This is similar to a number and its reciprocal – when you multiply a number by its reciprocal, you always get 1. Activity 1.12 takes some time, but it is worth doing if you have time as it shows you how to find the “recipe” for an inverse matrix. The solution is given in the textbook Answers. In the first part of the Activity, you find the inverse of a particular matrix, M, by solving simultaneous equations. In the second part, you find the inverse of the a c  general matrix   , using the same method. It turns out that the inverse of b d   d  ad  bc this matrix is   b   ad  bc

c  ad  bc  . a   ad  bc 

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MEI FP1 Matrices Section 3 Notes and Examples This can be written more simply as

1  d c   . ad  bc  b a 

The determinant, ad – bc, is important, not only because you need it to find an inverse matrix, but for other reasons which you will discover in the next section. Notice that if the determinant is zero the matrix has no inverse. You will learn more about the kind of transformations which have no inverse in section 4 of this chapter.

For practice in finding inverses, try the interactive questions Find the inverse of a 2 by 2 matrix.

The section on the inverse of a product is important. It is easy to understand if you think of it in terms of transformations: if you want to undo two transformations, you must undo the second one before undoing the first one.

Using the determinant of a 2 × 2 matrix In Question 4 of Exercise 1E, you found that the determinant of a particular matrix was equal to the ratio of the area of image shape to the area of the object shape. You will now see that this is in fact true for all matrices. Activity 1.15 shows that this is true for two particular matrices, and Activity 1.16 shows that it is true for all matrices. The section on matrices with zero determinant is very important. You have already met some situations where the whole plane is mapped onto a straight line, such as Exercise 1B Question 6. In this section you see that all singular matrices map the plane onto a straight line. Remember that a singular matrix has no inverse. This makes sense in the context of transformations because a singular matrix transforms all points onto a single line (or onto a single point, the origin, in the case of the zero matrix). This means that each image point could be the image of an infinite number of different object points, so it is impossible to distinguish which object point each image point came from. This means it is impossible to “undo” the transformation. Note also that the area of a line or point is zero, and so det T = 0 for a singular matrix can still be thought of as the area factor. Example 1.7 shows how to find the image line of a singular matrix, and how to find the equation of the line which maps to a particular point on the image line. Activity 1.19 generalises this.

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Further Pure Mathematics 1 Matrices Section 3: Inverse matrices Crucial points 1. Remember the rule for the inverse of a matrix product For square matrices M and N, (MN)-1 = N-1M-1. 2. Make sure you understand the significance of a zero determinant for a matrix transformation For a matrix with zero determinant, all points on the plane are mapped to a straight line through the origin, and each set of object points which are mapped to a single image point all lie on a straight line. 3. Remember the physical significance of the determinant The determinant of a matrix represents the area scale factor of the associated transformation, not just the scale factor (the area scale factor is the square of the scale factor).

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Further Pure Mathematics 1 Matrices Section 3: Inverse matrices Exercise ⎛ 2 4⎞ ⎟⎟ and AB = I find A. 1. If B = ⎜⎜ ⎝ 1 3⎠ ⎛ − 2 0⎞ ⎛ 1 0⎞ ⎟⎟ = ⎜⎜ ⎟⎟ . 2. Find Y if Y ⎜⎜ ⎝ 3 1⎠ ⎝ 0 1⎠ ⎛ 2 − 2⎞ ⎟⎟ and AB = 3. Find B if A = ⎜⎜ ⎝−1 3 ⎠

⎛ 4 − 2⎞ ⎜⎜ ⎟⎟ . ⎝0 7 ⎠

⎛ 1 − 2⎞ ⎟⎟ has no inverse find x. 4. If the matrix ⎜⎜ ⎝x 4 ⎠ ⎛ 2 3 2⎞ ⎛ 1 −4 3 ⎞ ⎟ ⎟ ⎜ ⎜ 5. P = ⎜ 4 6 5 ⎟ and Q = ⎜ 1 2 − 2⎟ . ⎜ 5 7 6⎟ ⎜− 2 1 0 ⎟⎠ ⎠ ⎝ ⎝ Find PQ and QP. What can you deduce from your answers?

6 ⎞ ⎛ 3 ⎟⎟ maps all points of the x-y plane onto a straight 6. Show that the matrix ⎜⎜ ⎝ − 2 − 4⎠ line and find the equation of that line. ⎛ 2 − 3⎞ ⎟⎟ . 7. Matrix A represents a transformation T where A = ⎜⎜ ⎝ 4 − 4⎠ (i) Find the inverse of A. (ii) Find the coordinates of the point that is mapped to (9, 16) under transformation T. (iii)Find A2. ⎛d 0⎞ ⎟⎟ stating the value of d. (iv) Show that A3 = ⎜⎜ ⎝0 d⎠ (v) Give a geometrical description of the matrix A3.

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Further Pure Mathematics 1 Matrices Section 2: Determinants and inverses of 2 × 2 matrices Solutions to Exercise 1. AB = I ⇒ A = B-1 ⎛ 2 4⎞ det B = det ⎜ ⎟=6−4= 2 ⎝ 1 3⎠ 1 ⎛ 3 −4 ⎞ A = B-1 = ⎜ ⎟. 2 ⎝ −1 2 ⎠

−1 ⎛ −2 0 ⎞ ⎛ −2 0 ⎞ ⎛1 0⎞ 2. Y ⎜ ⎟ = ⎜0 1⎟ ⇒ Y = ⎜ 3 1⎟ ⎝ ⎠ ⎝ 3 1⎠ ⎝ ⎠ ⎛ −2 0 ⎞ det ⎜ ⎟ = −2 − 0 = −2 ⎝ 3 1⎠ −1 0 ⎞ ⎛ −0.5 ⎛ −2 0 ⎞ 1⎛ 1 = − Y= ⎜ ⎟ ⎜ ⎟=⎜ 2 ⎝ −3 −2 ⎠ ⎝ 1.5 ⎝ 3 1⎠

⎛ 4 −2 ⎞ ⎛ 2 −2 ⎞ 3. AB = ⎜ ⇒ ⎟ ⎜ −1 3 ⎟ B = ⎝0 7 ⎠ ⎝ ⎠ 2 − 2 ⎛ ⎞ det ⎜ ⎟=6−2 = 4 ⎝ −1 3 ⎠ −1 ⎛ 2 −2 ⎞ 1 ⎛3 2⎞ ⎜ −1 3 ⎟ = ⎜ 1 2 ⎟ 4⎝ ⎝ ⎠ ⎠ 1 ⎛ 3 2 ⎞⎛ 4 −2 ⎞ 1 ⎛ 12 B= ⎜ ⎟⎜ ⎟= ⎜ 4 ⎝ 1 2 ⎠⎝ 0 7 ⎠ 4 ⎝ 4

⎛1 4. ⎜ ⎝x

0⎞ 1 ⎟⎠

⎛ 4 −2 ⎞ ⎜0 7 ⎟ ⇒ B = ⎝ ⎠

−1 ⎛ 2 −2 ⎞ ⎛ 4 −2 ⎞ ⎜ −1 3 ⎟ ⎜ 0 7 ⎟ ⎝ ⎠ ⎝ ⎠

8 ⎞ ⎛3 2⎞ = 12 ⎟⎠ ⎜⎝ 1 3 ⎟⎠

−2 ⎞ ⎛1 is singular det ⇒ ⎜x 4 ⎟⎠ ⎝

−2 ⎞ =0 4 ⎟⎠ ⇒ 4+ 2x = 0 ⇒ x = −2

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Further Pure Mathematics 1 ⎛2 ⎜ 5. PQ = ⎜ 4 ⎜5 ⎝

3 ⎞ ⎛1 ⎟ ⎜ 6 2 −2 ⎟ = ⎜ 0 7 1 0 ⎟⎠ ⎜⎝ 0 ⎛ 1 −4 3 ⎞ ⎛ 2 3 2 ⎞ ⎛ 1 ⎜ ⎟⎜ ⎟ ⎜ QP = ⎜ 1 2 −2 ⎟ ⎜ 4 6 5 ⎟ = ⎜ 0 ⎜ −2 1 0 ⎟⎠ ⎜⎝ 5 7 6 ⎟⎠ ⎜⎝ 0 ⎝ Q is the inverse of P. 3

2 ⎞⎛ 1 ⎟⎜ 5 ⎟⎜ 1 6 ⎟⎠ ⎜⎝ −2

−4

0 0⎞ ⎟ 1 0⎟ 0 1 ⎟⎠

0 0⎞ ⎟ 1 0⎟ 0 1 ⎟⎠

6 ⎞ ⎛ x ⎞ ⎛ 3 x + 6 y ⎞ ⎛ 3( x + 2 y ) ⎞ ⎛ 3 6. ⎜ ⎟ ⎟⎜ ⎟ = ⎜ ⎟=⎜ ⎝ −2 −4 ⎠ ⎝ y ⎠ ⎝ −2 x − 4 y ⎠ ⎝ −2( x + 2 y ) ⎠ x ′ = 3( x + 2 y )

y ′ = −2( x + 2 y ) x ′ y′

= 3 −2 −2 x ′ = 3 y ′ The matrix maps all points onto the straight line 2 x + 3 y = 0 . 7. (i)

(ii)

(iii) (iv) (v)

⎛ 2 −3 ⎞ det ⎜ ⎟ = −8 + 12 = 4 ⎝ 4 −4 ⎠ 1 ⎛ −4 3 ⎞ A-1 = ⎜ ⎟ 4 ⎝ −4 2 ⎠ ⎛x⎞ ⎛ 9⎞ A⎜ ⎟=⎜ ⎟ ⎝ y ⎠ ⎝ 16 ⎠ 9 ⎞ 1 ⎛ −4 3 ⎞ ⎛ 9 ⎞ 1 ⎛ 12 ⎞ ⎛ 3 ⎞ ⎛x⎞ −1 ⎛ ⎜ y ⎟ = A ⎜ 16 ⎟ = ⎜ −4 2 ⎟ ⎜ 16 ⎟ = ⎜ −4 ⎟ = ⎜ −1 ⎟ ⎝ ⎠ ⎝ ⎠ 4⎝ ⎠⎝ ⎠ 4 ⎝ ⎠ ⎝ ⎠ The point (3, -1) is mapped to (9, 16). ⎛ 2 −3 ⎞ ⎛ 2 −3 ⎞ ⎛ −8 6 ⎞ A² = ⎜ ⎟. ⎟⎜ ⎟=⎜ ⎝ 4 −4 ⎠ ⎝ 4 −4 ⎠ ⎝ −8 4 ⎠

⎛ −8 6 ⎞⎛ 2 −3 ⎞ ⎛ 8 0 ⎞ A³ = ⎜ ⎟⎜ ⎟=⎜ ⎟ so d = 8 ⎝ −8 4 ⎠⎝ 4 −4 ⎠ ⎝ 0 8 ⎠ The matrix A³ is an enlargement, centre the origin, scale factor 8.

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Further Pure Mathematics 1 Matrices Section 3: Inverse matrices Multiple Choice Test ⎛ 2 −3 ⎞ 1) The determinant of the matrix ⎜ ⎟ is ⎝ 4 −5 ⎠ (a) 2 (c) -22 (e) I don’t know

(b) -2 (d) 22

2) Which of the following matrices are singular? ⎛1 1⎞ ⎛1 0⎞ ⎛ 0 1⎞ (i) ⎜ (ii) ⎜ ⎟ ⎟ (iii) ⎜ ⎟ (iv) ⎝0 0⎠ ⎝0 1⎠ ⎝ 0 1⎠ (a) (ii) and (v) (c) (i) and (iii) (e) I don’t know

⎛1 1⎞ ⎟ ⎜ ⎝1 1⎠

⎛0 1⎞ (v) ⎜ ⎟ ⎝1 0⎠

(b) (i), (iii) and (iv) (d) (ii) only

⎛3 1⎞ ⎛1 1⎞ In Questions 3 to 6, A is the matrix ⎜ ⎟ and B is the matrix ⎜ ⎟. ⎝ 4 2⎠ ⎝ 4 2⎠ 3) A-1 = ⎛ −3 4 ⎞ (a) ⎜ ⎟ ⎝ 1 −2 ⎠ 1 ⎛ −3 4 ⎞ (c) ⎜ ⎟ 2 ⎝ 1 −2 ⎠ (e) I don’t know

⎛ 2 −1⎞ (b) ⎜ ⎟ ⎝ −4 3 ⎠ 1 ⎛ 2 −1⎞ (d) ⎜ ⎟ 2 ⎝ −4 3 ⎠

4) B-1 = 1 ⎛ 1 −4 ⎞ ⎜ ⎟ 2 ⎝ −1 2 ⎠ 1 ⎛ −1 4 ⎞ (c) ⎜ ⎟ 2 ⎝ 1 −2 ⎠ (e) I don’t know (a)

1 ⎛ 2 −1⎞ ⎜ ⎟ 2 ⎝ −4 1 ⎠ 1 ⎛ −2 1 ⎞ (d) ⎜ ⎟ 2 ⎝ 4 −1⎠ (b)

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Further Pure Mathematics 1 5) (AB)-1 = 1 ⎛ −8 5 ⎞ ⎜ ⎟ 4 ⎝ 12 −7 ⎠ 1 ⎛ −8 3 ⎞ (c) ⎜ ⎟ 4 ⎝ 20 −7 ⎠ (e) I don’t know

1 ⎛ 8 −3 ⎞ ⎜ ⎟ 4 ⎝ −20 7 ⎠ 1 ⎛ 8 −5 ⎞ (d) ⎜ ⎟ 4 ⎝ −12 7 ⎠

(a)

(b)

6) (BA)-1 = 1 ⎛ 8 −3 ⎞ ⎜ ⎟ 4 ⎝ −20 7 ⎠ 1 ⎛ −8 3 ⎞ (c) ⎜ ⎟ 4 ⎝ 20 −7 ⎠ (e) I don’t know

1 ⎛ 8 −5 ⎞ ⎜ ⎟ 4 ⎝ −12 7 ⎠ 1 ⎛ −8 5 ⎞ (d) ⎜ ⎟ 4 ⎝ 12 −7 ⎠

(a)

(b)

⎛ −3 6 ⎞ In Questions 7 and 8, the plane is transformed by the matrix M = ⎜ ⎟. ⎝ 1 −2 ⎠ 7) The whole plane is mapped to the line (a) -3x + y = 0 (c) x – 3y = 0 (e) I don’t know

(b) 3x + y = 0 (d) x + 3y = 0

8) The equation of the line of points that map to the point (3, -1) is (a) x + 2y = -1 (c) x + 2y = 1 (e) I don’t know

(b) x – 2y = 1 (d) x – 2y = -1

9) A triangle has vertices (1, 2), (3, 2) and (2, 6). ⎛1 2 ⎞ The triangle is transformed by the matrix ⎜ ⎟. ⎝ 3 −1⎠ The area of the image is (a) 7 (c) 28 (e) I don’t know

(b) 20 (d) 56

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Further Pure Mathematics 1 ⎛1 0 0⎞ ⎜ 10) The matrix A = ⎜ 3 2 −7 ⎟⎟ satisfies A³ = ⎜ −4 1 −3 ⎟ ⎝ ⎠ From this information, A-1 = ⎛1 0 0⎞ ⎜ ⎟ (a) ⎜ 37 −3 7 ⎟ ⎜ 11 −1 16 ⎟ ⎝ ⎠ ⎛1 0 0⎞ ⎜ ⎟ (c) ⎜ 3 2 −7 ⎟ ⎜ −4 1 −3 ⎟ ⎝ ⎠ (e) I don’t know

⎛1 0 0⎞ ⎜ ⎟ ⎜0 1 0⎟ . ⎜0 0 1⎟ ⎝ ⎠

⎛ 1 0 0⎞ ⎜ ⎟ (b) ⎜ 37 −3 7 ⎟ ⎜ 11 −1 2 ⎟ ⎝ ⎠ ⎛1 0 0⎞ ⎜ ⎟ (d) ⎜ 9 4 49 ⎟ ⎜16 1 9 ⎟ ⎝ ⎠

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MEI Further Pure Mathematics 1 Matrices Section 4: Matrices and simultaneous equations Study Plan Background You already know how to solve a pair of simultaneous linear equations by elimination or substitution. In this section you will learn how to solve such equations by a matrix method, which can be extended to a system of any number of simultaneous linear equations. You will also look at invariant points: points which are mapped to themselves under a particular transformation.

Detailed work plan 1. Read pages 36 - 39, paying particular attention to the three possible alternatives (Case 1, Case 2 and Case 3) and their geometrical interpretations, and compare these with Example 1.8. Activity 1.20 is very important and you should do this. The full solution is given in the Notes and Examples. The enrichment Discussion Point is well worth thinking about, and a full explanation of the geometrical interpretation of three simultaneous equations is also given in the Notes and Examples. 2. Look at the PowerPoint presentation, which shows examples involving three simultaneous equations, in which help is given in finding the inverse of a 3 × 3 matrix. These examples are similar to questions 5 – 7 in Exercise 1G. 3. Exercise 1G Attempt questions 1, 2, 3 and 4, and at least two of questions 5-7. 4. Read pages 41 – 43 and make sure that you follow Example 1.9. You should also read through the brief enrichment section “Invariant lines”. 5. Exercise 1H Attempt questions 1, 2, 3 and 4.

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MEI Further Pure Mathematics 1 Matrices Section 4: Matrices and simultaneous equations Notes and Examples These notes contain subsections on  Solving simultaneous equations using matrices  Invariant points  Invariant lines (enrichment)

Solving simultaneous equations using matrices You already know how to solve a pair of simultaneous linear equations using the elimination or substitution methods. In this section you see that such equations can also be solved by writing them as a single matrix equation and solving this by using the inverse matrix. You may wonder why you should use this method when the familiar elimination method may seem easier to you. The truth is that in the case of two simultaneous equations in two unknowns, it probably is. However, if you have to deal with a larger number of equations, the matrix method becomes more efficient than using algebraic methods. The method is the same for any number of equations:  

write the equations as a matrix equation: pre-multiply by the inverse matrix M-1:

Mp = p′ p = M-1p′

The only difficulty is finding the inverse matrix. If you have n simultaneous equations in n unknowns, you need to be able to find the inverse of an n × n matrix. However, computers and some calculators can do this for you. In FP1 you need to be able to use the matrix method to solve two simultaneous equations in two unknowns. You may also be asked to use it for three equations in three unknowns, but in that case you would always be given help in finding the inverse matrix. Questions 5 – 7 in Exercise 2G are of this type. There is a PowerPoint presentation which shows examples involving 3 × 3 matrices. You may find this helpful when you are attempting questions 5 – 7 in Exercise 2G. It is very important that you understand the three possible situations with two equations in two unknowns, and their geometrical interpretations. Read “Case 1”, “Case 2” and “Case 3” in the textbook carefully and compare with Example 1.8. Notice that if the determinant of the matrix is zero, then there may be infinitely many solutions or no solution: you can decide which by looking at the equations and finding out if they are consistent or inconsistent.

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MEI FP1 Matrices Section 4 Notes and Examples Activity 1.20 is very important as it looks at the possible situations with three equations in three unknowns. The full solution to this Activity is given below.

Solution to Activity 1.20 3 x  2 y  z  5  (i) The equations  x  4 y  2 z  3 can be written as the matrix equation  2x  y  z  3   3 2 1  x   5        1 4 2  y    3  .  2 1 1  z   3      

 6  1 8    Using a calculator gives the inverse matrix as  5 1 7  .  7 1 10     x   6 1 8   5   3         So  y    5 1 7   3    1   z   7 1 10   3   2         This gives the solution x = 3, y = -1, z = -2

(ii)

x  3y  2z  7  The equations 2 x  2 y  z  3 can be written as the matrix equation  3 x  y  z  10 

 1 3 2  x   7        2 2 1  y    3   3 1 1  z  10        1 3 2    Using a calculator to find the inverse of  2 2 1  gives an error. This is  3 1 1    because the determinant of the matrix is zero. To solve the equations algebraically, eliminate one variable from two pairs of the equations. (1) x  3y  2z  7  (2) 2 x  2 y  z  3  3x  y  z  10 (3)  Eliminate z from (1) and (2): (1) x + 3y – 2z = 7 (2) × 2 4x – 4y + 2z = 6

5x – y

Adding: Eliminate z from (2) and (3):

(2) (3)

2x – 2y + z = 3 3x + y – z = 10 5x – y

Adding:

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= 13 2/6

MEI FP1 Matrices Section 4 Notes and Examples This gives two equations in x and y which are the same, so there are infinitely many solutions. The solutions of the equation 5x – y = 13 can be expressed in terms of a parameter as x = λ, y = 5λ – 13. Substituting these into one of the original equations: (3)

z  3 x  y  10  3  5  13  10  8  23

(iii)

So the equations give a line of solution points, which can be expressed as x = λ, y = 5λ – 13, z = 8λ – 23. x  3y  2z  7  The equations 2 x  2 y  z  3 can be written as the matrix equation  3 x  y  z  12   1 3 2  x   7        2 2 1  y    3   3 1 1  z  12       The matrix is the same as the one in (ii), so has no inverse. Use the same approach as in (ii) to solve the equations algebraically: Eliminate z from (1) and (2): (1) x + 3y – 2z = 7 (2) × 2 4x – 4y + 2z = 6

5x – y

Adding: Eliminate z from (2) and (3):

(2) (3)

2x – 2y + z = 3 3x + y – z = 12 5x – y

Adding:

= 13

= 15

The equations are inconsistent. There are no solutions.

The geometrical interpretation of the solution to three linear simultaneous equations is beyond the syllabus. However, you are invited to think about this in the enrichment Discussion Point following Activity 1.20, and a full explanation of the possible configurations is given below. A linear equation in two variables (x and y say) can be regarded as the equation of a line in two-dimensional space. A linear equation in three variables (x, y and z say) can be regarded as the equation of a plane in three-dimensional space. So the solution of three linear simultaneous equations in three variables can be regarded as the point or points where three planes all meet. There are seven possible configurations of three planes: one in which there is a unique solution, two in which there are infinitely many solutions, and four in which there are no solutions.

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MEI FP1 Matrices Section 4 Notes and Examples (i)

If the matrix is non-singular, then the equations have a unique solution corresponding to the point at which all three planes meet. This is the case in part (i) of Activity 1.20.

The dotted lines show where each pair of planes meet. The point where the three dotted lines cross represents the unique solution.

(ii) If the matrix is singular, then there may be an infinite number of solutions. In this case there are two possibilities: (a) the three planes have a line of common points. This arrangement is known as a sheaf or pencil of planes. In this case, eliminating one variable from any two of the equations will result in the same equation each time, as is the case in part (ii) of Activity 1.20. The solution can be given in terms of one parameter. (b) the three planes all coincide. All three equations are the same. In this case the solution is given in terms of two parameters: e.g. x = λ, y = μ, then z can be given in terms of λ and μ. (iii)

If the matrix is singular, then there may be no solutions. In this case there are four possibilities:

(a) the three planes are all parallel and distinct. For example, the planes 2x + y – z = 3 -2x – y + z = 1 4x + 2y – 2z = 5 can all be expressed in the form 2x + y – z = k, with the value of k being different in each case. (b) two of the planes coincide and the third plane is parallel but distinct. For example, in the case of the planes 2x + y – z = 3 -2x – y + z = -3 4x + 2y – 2z = 5 the first two planes are the same, and the third is parallel but distinct.

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MEI FP1 Matrices Section 4 Notes and Examples (c) two planes are parallel and distinct, and the third plane crosses them. For example, in the case of the planes 2x + y – z = 3 -2x – y + z = 1 x + 2y + z = 2 the first two planes are parallel and distinct. Eliminating z from the first and third equation gives 3x + 3y = 5, and eliminating z from the first and third equation gives 3x + 3y = 1. These are a pair of parallel lines. (d) the three planes form a triangular prism. This is the case in part (iii) of Activity 1.20. Eliminating z from each pair of the equations gives three equations in x and y which are all parallel lines.

Invariant points This is quite a straightforward section. The main point that you need to understand is that for all linear transformations, either the origin is the only invariant point, or all the invariant points lie on a straight line through the origin. All you need to do to find the invariant points for the transformation with matrix a c    is to write the matrix equation b d   a c  x   x        ,  b d  y   y  write down two equations from to this matrix equation ax + cy = x bx + dy = y and decide whether or not these two equations are equivalent to each other. If they are, then they represent a line of invariant points, and all points on the line can be represented in terms of a parameter. If they are not equivalent, then the only invariant point is the origin. Finding the invariant points of a transformation can give you useful information about the transformation. For example, if you know that the transformation is a reflection, then finding the line of invariant points gives you the equation of the mirror line.

Invariant lines (enrichment) This is an enrichment section, so is beyond the syllabus for FP1. However, it is important to read this section to avoid any confusion between a line of invariant points and an invariant line. On a line of invariant points, all points are mapped to

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MEI FP1 Matrices Section 4 Notes and Examples themselves. However, on an invariant line, all points are mapped to a point on the line, so that the line is mapped to itself, but the individual points are not necessarily mapped to themselves. Notice that a line of invariant points is an invariant line, but an invariant line is not necessarily a line of invariant points!

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Further Pure Mathematics 1 Matrices Section 4: Matrices and simultaneous equations Crucial points 1. Be careful when solving matrix equations for which the matrix has no inverse It is tempting to immediately conclude that there is no solution, but in fact what this means is that there is no unique solution, and there may be either no solution or infinitely many solutions. 2. Remember that the origin is always an invariant point for a linear transformation 3. Make sure that you know the difference between a line of invariant points and an invariant line An invariant point is a point which is mapped to itself, so a line of invariant points is a line of points each of which is mapped to itself. An invariant line is a line of points each of which is mapped to a point which is also on the line (not necessarily itself). A line of invariant points is, of course, also an invariant line.

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Further Pure Mathematics 1 Matrices Section 4: Matrices and simultaneous equations Exercise 1. Solve the following simultaneous equations by matrix methods. (a) x – y = 5 (b) 4x + 6y = 8 3x + 2y = 5 2x + 3y = 4 (c) 2x – 4y = 3 x – 2y = 1

(d)

x – 3y = 3 5x – 9y = 11

⎛ − 3 2 − 1⎞ ⎛− 4 − 3 5 ⎞ ⎟ ⎟ ⎜ ⎜ 2. If A = ⎜ 2 − 1 3 ⎟ and B = ⎜ − 5 − 4 7 ⎟ find AB. ⎜ −1 1 ⎜ 1 1 ⎟⎠ 1 − 1⎟⎠ ⎝ ⎝ Hence find values of x, y and z satisfying −4 x − 3 y + 5 z = 3

−5 x − 4 y + 7 z = 4 x+ y−z =0 ⎛3 1⎞ ⎛ 6 1⎞ ⎛ − 4 3⎞ ⎟⎟ , B = ⎜⎜ ⎟⎟ , C = ⎜⎜ ⎟⎟ and D = 3. If A = ⎜⎜ ⎝ 5 2⎠ ⎝11 3⎠ ⎝ − 5 2⎠ and Z such that AX = B, BY = C, CZ = D.

⎛ 4 7⎞ ⎜⎜ ⎟⎟ Find matrices X, Y ⎝ − 2 7⎠

4. Find the value of x given that A2 = A-1 and x 1⎞ ⎛1 ⎟ ⎜ A = ⎜ − 1 − 1 0⎟ . ⎜1 0 0 ⎟⎠ ⎝

5. Find any invariant points of the transformations given by 3 ⎞ ⎛ 4 ⎛ 0.5 − 0.5 ⎞ ⎟⎟ ⎜⎜ ⎟⎟ . (i) ⎜⎜ (ii) ⎝ − 3 − 2⎠ ⎝ 0.5 1.5 ⎠

6. M is a reflection of the plane such that the image (x’, y’) of the point (x, y) is given ⎛ x ' ⎞ ⎛ 0.6 0.8 ⎞ ⎛ x ⎞ by: ⎜ ⎟ = ⎜ ⎟⎜ ⎟ . ⎝ y ' ⎠ ⎝ 0.8 −0.6 ⎠ ⎝ y ⎠ Find a point other than the origin that is invariant under this transformation and hence find the equation of the mirror line.

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Further Pure Mathematics 1 ⎛ − 3⎞ 7. T is a translation of the plane by the vector ⎜⎜ ⎟⎟ . ⎝ 1 ⎠ Point ( X, Y) is the image of (x, y) under a combined transformation TM such ⎛ X ⎞ ⎛ 0 2 −3 ⎞⎛ x ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ that ⎜ Y ⎟ = ⎜ 1 0 1 ⎟⎜ y ⎟ . ⎜ 1 ⎟ ⎜ 0 0 1 ⎟⎜ 1 ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠ Find any invariant points of the combined transformation. ⎛ 3 ⎞ 8. S is a translation of the plane by the vector ⎜⎜ ⎟⎟ . ⎝ − 3⎠ The image of (x, y) under a combined transformation SN is such that ⎛ x' ⎞ ⎛ 2 − 1 3 ⎞⎛ x ⎞ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎜ y ' ⎟ = ⎜ 3 4 − 3 ⎟⎜ y ⎟ . ⎜ 1 ⎟ ⎜0 0 1 ⎟⎠⎜⎝ 1 ⎟⎠ ⎝ ⎠ ⎝ Find any invariant points of this combined transformation.

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Further Pure Mathematics 1 Matrices Section 4: Matrices and simultaneous equations Solutions to Exercise ⎛ 1 −1 ⎞ ⎛ x ⎞ ⎛ 5 ⎞ 1. (a) ⎜ ⎟⎜ ⎟ = ⎜ ⎟ ⎝ 3 2 ⎠⎝ y ⎠ ⎝ 5 ⎠ ⎛ 1 −1 ⎞ det ⎜ ⎟=5 ⎝3 2 ⎠ ⎛ 1 −1 ⎞ 1 ⎛ 2 1⎞ Inverse of ⎜ is ⎜ ⎟ ⎟ 5 ⎝ −3 1 ⎠ ⎝3 2 ⎠ ⎛ x ⎞ 1 ⎛ 2 1 ⎞ ⎛ 5 ⎞ 1 ⎛ 15 ⎞ ⎛ 3 ⎞ ⎜ ⎟= ⎜ ⎟⎜ ⎟ = ⎜ ⎟=⎜ ⎟ ⎝ y ⎠ 5 ⎝ −3 1 ⎠ ⎝ 5 ⎠ 5 ⎝ −10 ⎠ ⎝ −2 ⎠ There is a unique solution: x = 3, y = -2. ⎛ 4 6⎞⎛ x ⎞ ⎛ 8 ⎞ (b) ⎜ ⎟⎜ ⎟ = ⎜ ⎟ ⎝ 2 3 ⎠⎝ y ⎠ ⎝ 4⎠ ⎛4 6⎞ det ⎜ ⎟ = 0 so there is not a unique solution. ⎝2 3⎠ The equations are multiples of each other so they are consistent, and there are infinitely many solutions. 2x + 3y = 4 Let y = λ, so 2 x + 3λ = 4 ⇒ x = 2 − 23 λ The solutions are x = 2 − 23 λ , y = λ .

⎛ 2 −4 ⎞ ⎛ x ⎞ ⎛ 3 ⎞ (c) ⎜ ⎟⎜ ⎟ = ⎜ ⎟ ⎝ 1 −2 ⎠ ⎝ y ⎠ ⎝ 1 ⎠ ⎛ 2 −4 ⎞ det ⎜ ⎟ = 0 so there is not a unique solution. ⎝ 1 −2 ⎠ The first equation simplifies to x – 2y = 1.5, which is inconsistent with the second equation x – 2y = 1. There are no solutions. ⎛ 1 −3 ⎞ ⎛ x ⎞ ⎛ 3 ⎞ ⎜ 5 −9 ⎟ ⎜ y ⎟ = ⎜ 11 ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ 1 − 3 ⎛ ⎞ det ⎜ ⎟=6 ⎝ 5 −9 ⎠ ⎛ 1 −3 ⎞ 1 ⎛ −9 Inverse of ⎜ is ⎜ ⎟ 6 ⎝ −5 ⎝ 5 −9 ⎠

(d)

3⎞ 1 ⎟⎠

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Further Pure Mathematics 1 ⎛ x ⎞ 1 ⎛ −9 3 ⎞ ⎛ 3 ⎞ 1 ⎛ 6 ⎞ ⎛ 1 ⎞ ⎜ ⎟= ⎜ ⎟⎜ ⎟ = ⎜ ⎟ = ⎜ 2 ⎟ ⎝ y ⎠ 6 ⎝ −5 1 ⎠ ⎝ 11 ⎠ 6 ⎝ −4 ⎠ ⎝ − 3 ⎠ There is a unique solution: x = 1, y = − 23

⎛ −3 2 −1 ⎞ ⎛ −4 −3 5 ⎞ ⎜ ⎟⎜ ⎟ 2. AB = ⎜ 2 −1 3 ⎟ ⎜ −5 −4 7 ⎟ ⎜ −1 1 1 ⎟⎠ ⎜⎝ 1 1 −1 ⎟⎠ ⎝ ⎛1 0 0⎞ ⎜ ⎟ = ⎜0 1 0⎟ ⎜0 0 1⎟ ⎝ ⎠ Hence A is the inverse of B. ⎛ −4 −3 5 ⎞ ⎛ x ⎞ ⎛ 3 ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ − 5 −4 7 ⎟ ⎜ y ⎟ = ⎜ 4 ⎟ ⎜ 1 1 −1 ⎟⎠ ⎜⎝ z ⎟⎠ ⎜⎝ 0 ⎟⎠ ⎝

⎛ x ⎞ ⎛ −4 −3 ⎜ ⎟ ⎜ ⎜ y ⎟ = ⎜ −5 −4 ⎜z ⎟ ⎜ 1 1 ⎝ ⎠ ⎝ ⎛ −3 2 ⎜ = ⎜ 2 −1 ⎜ −1 1 ⎝

5 ⎞ ⎟ 7⎟ −1 ⎟⎠

−1

⎛3⎞ ⎜ ⎟ ⎜ 4⎟ ⎜0⎟ ⎝ ⎠

−1 ⎞ ⎛ 3 ⎞ ⎟⎜ ⎟ 3 ⎟⎜ 4⎟ 1 ⎟⎠ ⎜⎝ 0 ⎟⎠

⎛ −1 ⎞ ⎜ ⎟ =⎜ 2 ⎟ ⎜ 1 ⎟ ⎝ ⎠ The solution is x = -1, y = 2, z = 1. 3. AX = B ⇒ X = A-1B ⎛3 1⎞ det A = det ⎜ ⎟=1 ⎝5 2⎠ ⎛ 2 −1 ⎞ A −1 = ⎜ ⎟ ⎝ −5 3 ⎠ ⎛ 2 −1 ⎞ ⎛ 6 1 ⎞ ⎛ 1 −1 ⎞ X=A −1 B = ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎝ −5 3 ⎠ ⎝ 11 3 ⎠ ⎝ 3 4 ⎠ BY = C ⇒ Y = B-1C

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Further Pure Mathematics 1 ⎛ 6 1⎞ det B = det ⎜ ⎟=7 ⎝ 11 3 ⎠ 1 ⎛ 3 −1 ⎞ B −1 = ⎜ ⎟ 7 ⎝ −11 6 ⎠ 1 ⎛ 3 −1 ⎞ ⎛ −4 3 ⎞ 1 ⎛ −7 Y=B −1C = ⎜ ⎟⎜ ⎟= ⎜ 7 ⎝ −11 6 ⎠ ⎝ −5 2 ⎠ 7 ⎝ 14 CZ= D ⇒ Z = C-1D ⎛ −4 3 ⎞ det C = det ⎜ ⎟=7 ⎝ −5 2 ⎠ 1 ⎛ 2 −3 ⎞ C −1 = ⎜ ⎟ 7 ⎝ 5 −4 ⎠ 1 ⎛ 2 −3 ⎞ ⎛ 4 Z=C −1 D = ⎜ ⎟⎜ 7 ⎝ 5 −4 ⎠ ⎝ −2

7 ⎞ ⎛ −1 1 ⎞ = −21 ⎠⎟ ⎝⎜ 2 −3 ⎠⎟

7 ⎞ 1 ⎛ 14 −7 ⎞ ⎛ 2 −1 ⎞ = = 7 ⎟⎠ 7 ⎜⎝ 28 7 ⎟⎠ ⎜⎝ 4 1 ⎟⎠

4. A2 = A-1 ⇒ A³ = I ⎛ 1 x 1 ⎞⎛ 1 x 1 ⎞ ⎜ ⎟⎜ ⎟ A² = ⎜ −1 −1 0 ⎟ ⎜ −1 −1 0 ⎟ ⎜ 1 0 0⎟⎜ 1 0 0⎟ ⎝ ⎠⎝ ⎠ ⎛2 − x 0 1 ⎞ ⎜ ⎟ 1 − x −1 ⎟ =⎜ 0 ⎜ 1 x 1 ⎟⎠ ⎝ ⎛2 − x ⎜ A³ = ⎜ 0 ⎜ 1 ⎝

1 ⎞⎛ 1 x 1⎞ ⎟⎜ ⎟ 1 − x −1 ⎟ ⎜ −1 −1 0 ⎟ x 1 ⎟⎠ ⎜⎝ 1 0 0 ⎟⎠ ⎛ 3 − x 2x − x2 2 − x ⎞ ⎜ ⎟ 0 ⎟ = ⎜ −2 + x −1 + x ⎜ 2−x 0 1 ⎟⎠ ⎝ ⎛1 0 0⎞ ⎜ ⎟ When x = 2, A³ = ⎜ 0 1 0 ⎟ = I ⎜0 0 1⎟ ⎝ ⎠ 0

so the value of x is 2.

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Further Pure Mathematics 1 ⎛ 4 3 ⎞⎛ x ⎞ ⎛ x ⎞ 5. (i) ⎜ ⎟⎜ ⎟ = ⎜ ⎟ ⎝ −3 −2 ⎠ ⎝ y ⎠ ⎝ y ⎠ ⎛ 4x + 3y ⎞ ⎛ x ⎞ ⎜ −3 x − 2 y ⎟ = ⎜ y ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ 3 x + 3y ⎞ ⎛ 0 ⎞ ⎜ −3 x − 3 y ⎟ = ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠ The invariant points are the points of the form ( λ , −λ ) . ⎛ 0.5 (ii) ⎜ ⎝ 0.5 ⎛ 0.5 x ⎜ 0.5 x ⎝

−0.5 ⎞ ⎛ x ⎞ ⎛ x ⎞ = 1.5 ⎟⎠ ⎜⎝ y ⎟⎠ ⎜⎝ y ⎟⎠ − 0.5 y ⎞ ⎛ x ⎞ = + 1.5 y ⎟⎠ ⎜⎝ y ⎟⎠

⎛ −0.5 x − 0.5 y ⎞ ⎛ 0 ⎞ ⎜ 0.5 x + 0.5 y ⎟ = ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠ The invariant points are the points of the form ( λ , −λ ) .

⎛ x ' ⎞ ⎛ 0.6 0.8 ⎞ ⎛ x ⎞ ⎛ 0.6 x + 0.8 y ⎞ 6. ⎜ ⎟ = ⎜ ⎟⎜ ⎟ = ⎜ ⎟ ⎝ y ' ⎠ ⎝ 0.8 −0.6 ⎠ ⎝ y ⎠ ⎝ 0.8 x − 0.6 y ⎠ ⎛ 0.6 x + 0.8 y ⎞ ⎛ x ⎞ For invariant points, ⎜ ⎟=⎜ ⎟ ⎝ 0.8 x − 0.6 y ⎠ ⎝ y ⎠ ⎛ −0.4 x + 0.8 y ⎞ ⎛ 0 ⎞ ⎜ 0.8 x − 1.6 y ⎟ = ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠ A point which is invariant is (2, 1), or any point of the form ( 2 λ , λ ) .

The mirror line of the transformation is therefore the line 2 y = x .

⎛ X ⎞ ⎛ 0 2 −3 ⎞ ⎛ x ⎞ ⎛ 2 y − 3 ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ 7. ⎜Y ⎟ = ⎜ 1 0 1 ⎟ ⎜ y ⎟ = ⎜ x + 1 ⎟ ⎜ 1 ⎟ ⎜0 0 1 ⎟⎜ 1 ⎟ ⎜ 1 ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎛2y − 3⎞ ⎛ x ⎞ ⎜ ⎟ ⎜ ⎟ For invariant points, ⎜ x + 1 ⎟ = ⎜ y ⎟ ⎜ 1 ⎟ ⎜1⎟ ⎝ ⎠ ⎝ ⎠

⎛−x + 2y − 3⎞ ⎛0⎞ ⎜ ⎟ ⎜ ⎟ ⎜ x − y + 1 ⎟ = ⎜0⎟ ⎜ ⎟ ⎜0⎟ 0 ⎝ ⎠ ⎝ ⎠

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Further Pure Mathematics 1 Adding: y − 2 = 0

y = 2, x = 1 The point (1, 2) is an invariant point.

⎛ x ' ⎞ ⎛ 2 −1 3 ⎞ ⎛ x ⎞ ⎛ 2 x − y + 3 ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ 8. ⎜ y ' ⎟ = ⎜ 3 4 −3 ⎟ ⎜ y ⎟ = ⎜ 3 x + 4 y − 3 ⎟ ⎜ 1 ⎟ ⎜0 0 ⎟ 1 ⎟⎠ ⎜⎝ 1 ⎟⎠ ⎜⎝ 1 ⎝ ⎠ ⎝ ⎠ ⎛ 2x − y + 3 ⎞ ⎛ x ⎞ ⎜ ⎟ ⎜ ⎟ For invariant points, ⎜ 3 x + 4 y − 3 ⎟ = ⎜ y ⎟ ⎜ ⎟ ⎜1⎟ 1 ⎝ ⎠ ⎝ ⎠

⎛ x − y + 3 ⎞ ⎛0⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 3 x + 3y − 3 ⎟ = ⎜ 0 ⎟ ⎜ ⎟ ⎜0⎟ 0 ⎝ ⎠ ⎝ ⎠

x −y+3=0 x +y −1 =0 Adding: 2 x + 2 = 0 x = −1, y = 2

The point (-1, 2) is an invariant point.

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Further Pure Mathematics 1 Matrices Section 4: Matrices and simultaneous equations Multiple Choice Test 1) Which of the following sets of equations are consistent? (i) 3x – 2y = 6 (ii) -3x + 2y = -1 2x + y = 4 9x – 6y = -3 (a) neither (c) (i) only (e) I don’t know

(b) (ii) only (d) both

2) The equations 3x – y = 4 2y – 6x = 1 have (a) no solutions (c) infinitely many solutions (e) I don’t know

(b) a unique solution (d) two solutions

3) The solution points of the equations 2x + 4y = 8 3x + 6y = 12 can be expressed in terms of a parameter as (a) (λ, 2λ - 4) (c) (2λ - 4, λ) (e) I don’t know

(b) (λ, 4 - 2λ) (d) (4 - 2λ, λ)

4) The solution of the equations 3x – 2y = 4 2x + 5y = 9 is (a) (λ, 2λ) (c) (2λ, λ) (e) I don’t know

(b) x = 2, y = 1 (d) x = 1, y = 2

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Further Pure Mathematics 1 Questions 5 and 6 are about the matrices 8−k 1⎞ ⎛ −2 ⎛3 1 4⎞ ⎜ ⎟ ⎜ ⎟, A = ⎜2 0 1⎟ B = ⎜ 1 − 2k 3k − 4 5 ⎟ ⎜ 4 ⎜1 2 k ⎟ −5 −2 ⎟⎠ ⎝ ⎝ ⎠ 5) By calculating the matrix AB, find the value of k for which the matrix A has no inverse matrix. (a) 5.5 (c) 5 (e) I don’t know

(b) 0 (d) 6

6) In the case for which k = 5, the solution of the matrix equation ⎛ x⎞ ⎛ 1 ⎞ ⎜ ⎟ ⎜ ⎟ A⎜ y⎟ = ⎜ 0 ⎟ ⎜ z ⎟ ⎜ −2 ⎟ ⎝ ⎠ ⎝ ⎠ is (a) x = -4, y = -19, z = 8 (c) x = -5, y = 0, z = -9 (e) I don’t know

(b) x = 1, y = -3, z = -6 (d) no solution

⎛ 2 3⎞ 7) The invariant points under the transformation ⎜ ⎟ are ⎝2 7⎠ (a) all points on the line 2x + 3y = 0 (c) all points on the line 2x + 7y = 0 (e) I don’t know

(b) all points on the line x + 3y = 0 (d) (0, 0) only

⎛2 0 ⎞ 8) The invariant points under the transformation ⎜ ⎟ are ⎝ 1 −1 ⎠ (a) only all points on the line y = 2x (c) (0, 0) and (1, 2) (e) I don’t know

(b) (0, 0) and ( 12 , 1) (d) (0, 0)

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Further Pure Mathematics 1 ⎛ 3 −1⎞ 9) The invariant points under the transformation ⎜ ⎟ are ⎝ −4 3 ⎠ (a) (2λ, -λ) (c) (λ, 2λ) (e) I don’t know

(b) (2λ, λ) (d) (λ, -2λ)

⎛ − 12 23 ⎞ 10) The matrix ⎜ ⎟ represents a reflection of the plane. ⎜ 3 1 ⎟ 2 ⎠ ⎝ 2 The equation of the mirror line is (a) x = 3 y

(b) y = − 3x

(c) y = 3x (e) I don’t know

(d) x = − 3 y

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Further Pure Mathematics 1 Matrices Chapter Assessment ⎛ −0.28 0.96 ⎞ 1. The matrix ⎜ ⎟ corresponds to a rotation R in the x-y plane. ⎝ −0.96 −0.28 ⎠ (i) State the centre of the rotation R, and find the angle of rotation (stating whether it is clockwise or anticlockwise).

[6]

The transformation S is the rotation R followed by reflection in the x-axis. (ii) Write down the matrix corresponding to reflection in the x-axis, and show that ⎛ −0.28 0.96 ⎞ the matrix corresponding to the transformation S is ⎜ [6] ⎟. ⎝ 0.96 0.28 ⎠ (iii)Show that the invariant points of the transformation S lie on a straight line, and find the equation of this line. [6] ⎛ 3 −2 ⎞ 2. The matrix ⎜ ⎟ represents a transformation, T. The inverse of transformation ⎝ 2 −1 ⎠ T is W. (i) Find the matrix representing W.

[3]

(ii) Show that TW = WT = I.

[3]

(iii)Find the invariant points for the transformation T.

[5]

(iv) T is a shear. The line of shear is the line of invariant points for the shear. The factor of a shear gives the distance a point is moved as a multiple of its perpendicular distance from the line of shear. What is the factor of the shear T? [6] ⎛ 6 −3 ⎞ 3. The plane is transformed by means of the matrix M = ⎜ ⎟. ⎝ −4 k ⎠ (i) In the case where k = 3 (a) find the determinant of M (b) find the area of the image of a triangle with area 4 square units.

[2] [2]

(ii) In the case where k = 2 (a) show that det M = 0, and that the whole plane is mapped on to the line 2x + 3y = 0. [6] (b) find the equation of the line of points which map to (3, -2). [5]

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Further Pure Mathematics 1 ⎛ k 1 −3 ⎞ ⎜ 4. A is the matrix ⎜ 4 −2 1 ⎟⎟ and B is the matrix ⎜ 5 −3 2 ⎟ ⎝ ⎠

⎛ −1 7 −5 ⎞ ⎜ ⎟ ⎜ −3 17 −13 ⎟ . ⎜ ⎟ ⎝ −2 8 −6 ⎠

(i) Find the matrix product AB.

[4]

(ii) For the case k = 1, write down the inverse matrix A-1 and hence solve the equation ⎛ 1 1 −3 ⎞⎛ x ⎞ ⎛ 2 ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟ [6] ⎜ 4 −2 1 ⎟⎜ y ⎟ = ⎜ 13 ⎟ ⎜ 5 −3 2 ⎟⎜ z ⎟ ⎜17 ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠

Total 60 marks

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