An n x m matrix has n rows and m columns

Matrices

A matrix is an array of numbers, the size of which is described by its dimensions:

Eg write down the dimensions of these matrices:

01

3222 315 31

4

312

34

20

5123

01

32A

25

63B

This can only take place between matrices of the same dimension, by simply adding or subtracting values in corresponding positions in the matrix array.

and , find A + B and A – B

Matrix addition & subtraction

Eg if

2051

6332BA

26

91

2051

6332BA

24

35

xy

yx 2C

yy

xx

2

3Dand , find C + D and C – DEg if

yx

xyxx

20

23DC

yxy

xyx

22

22DC

Now try Ex4A

01

32A

Multiply every number in the array by the scalar.

, find 3A and -2A

Scalar multiplication

Eg if

03

963A

02

642-

xy

yx 2C , find 5C and -CEg if

xy

yx

55

1055C

xy

yx 2-C

Now try Ex4B

C1 C2

R1

R2

dc

baP

hg

feQ

Requires a form of ‘cross multiplication’ and careful calculation:

and , then

Matrix multiplication

If

dhcfdgce

bhafbgaePQ

ae bg

ce dg

af bh

cf dh

hg

fe

dc

ba

Eg if and NB: Matrix multiplication is commutative - order matters. AB does not equal BA.

6 153 0

12 6

6 0

01

32A

25

63B

AB

63

189

6 610 2

9 0

15 0

1512

90BA

25

63

01

32

01

32

25

63BAAB

Eg for the possible products in the example above,

The product PQ will have the number of rows of P and the number of columns of Q

Matrix multiplication can only take place if the number of columns of the matrix on the left is equal to the number of rows of the matrix on the right.

Product will be 2 x 1

a) write down the dimensions of the products

b) Evaluate the products

01

32A 315 B

4

3C

34

20

51

D

Eg for the matrices given, which pairs can be multiplied?

Possible: AC , BD , CB , DA , DC

4

3

01

32AC

3

18 Product will be 1 x 2

34

20

51

315BD 367

Product will be 2 x 3

3154

3

CB

12420

9315

Product will be 3 x 2

01

32

34

20

51

DA

125

02

37

Now try Ex4C

Eg if , and

NB: Matrix multiplication is associative, meaning A(BC) = (AB)C

01

32A

25

63B

Matrix multiplication

41

12C

41

12

25

63

01

32BCA

38

2712

01

32

2712

630

41

12

25

63

01

32CAB

41

12

63

189

2712

630

Inverse matrices

Inverse of is where

dc

baM bcadM )det(

ac

bd

MM

)det(

11

Eg Find M–1.

53

24M

3254 )det(M 620

43

25

26

11M

26

A matrix M is described as singular if det(M) = 0

Eg Show that M is singular

31

62M

1632 )det(M 0 A singular matrix has no inverse

Inverse of is

WB22

(a) Given that a = 2, find M–1.

, where a is a real constant.

(b) Find the values of a for which M is singular.

a

aM

6

32

where

dc

baM bcadM )det(

ac

bd

MM

)det(

11

26

34M 6324 )det(M 10

46

32

10

11M

Singular 0 )det(M

0182 2 a

92 a3 a

Given that a = 0,(c) find A–1.

WB24

42

5

a

aA , where a is real.

(a) Find det A in terms of a.

(b) Show that the matrix A is non-singular for all values of a.

104 aaAdet 1042 aa

dc

baM bcadM )det(

01042 aa

Singular 0 Adet

01042 2 a

62 2 a Has no real roots

Inverse of is

dc

baM

ac

bd

MM

)det(

11

42

50A 2540 Adet 10

02

54

10

11A

ac

bd

Mdc

ba

det

1

dabcdccd

baabbcad

Mdet

1

1MM

ac

bd

dc

ba

Mdet

1

bcadbcad

bcadbcad

0

0

10

01

Why do inverse matrices work?

Eg Ex 4G Q4a) Given that ABC = I, prove that B-1 = CA

ABC = I

BC = A-1

CA = B-1

BCA = I

Multiply on the left by A-1

Multiply on the right by A

Multiply on the left by B-1

b) Given that and , find B

61

10A

13

12C

Using (a), CA = B-1

61

10

13

12

31

41

Then B is the inverse of this

11 Bdet

11

43B

Now try Ex4G, Q5, 6, 7

Using inverse matrices

Q5a) Given that AB = C, find an expression for B

AB = C

B = A-1CMultiply on the left by A-1

b) Given that and , find B

34

12A

221

63C

Using (a), B = A-1C

221

63

24

13

10

1

2010

4010

10

1

10Adet

24

13

10

11A

21

41

Q6a) Given that BAC = B, where B is a non-singular matrix, find an expression for A

BAC = B

AC = IMultiply on the left by B-1

b) When , find A

23

35C

Using (a), A = C-1

53

32A = C-1Multiply on the right by C-1

7) The matrix and , . Find the matrix B

34

12A

18138

874AB

B = A-1AB

24

13

2

11A

18138

874

24

13

2

1B

420

684

2

1

210

342

Matrix transformations

Eg transform the shape T shown below using the matrix

20

02A

231

521

20

02AT

462

1042

231

521T

T

Original point

Image

(1,1) (2,2)

(2,3) (4,6)

(5,2) (10,4)

k

k

0

0

The geometrical transformation represented by the matrix

is an enlargement about the origin, scale factor k

AT

You can express coordinates as column vectors

Use matrix multiplication to transform the shape

Giving:

Matrix transformations

The effect of a matrix transformation can be established by considering what happens to just the coordinates (1,0) and (0,1).

01

10CEg Given that

describe fully the geometrical transformation represented by C,

0

1

1

0

1

0

0

1

Rotation 90o anticlockwise about the origin

Before After

The coordinates (1,0) and (0,1) are represented by the identity matrix

10

01

Hence the effect of a matrix can be found by taking the 1st and 2nd columns as the positions taken by the coordinates (1,0) and (0,1) respectively, and interpreting a diagram of this.

Applying any matrix to this will give the matrix itself

(1,0) becomes (0,1)

(0,1) becomes (-1,0)

Matrix transformations

Describe fully the geometrical transformation represented by the given matrices:

10

01

01

10

10

01

10

01

01

10

01

10

Rotation 180o about (0,0)Reflection in the x - axis

Reflection in the y - axis

Reflection in the line y=x

Reflection in the line y=-x

0

1

1

0Compare with the coordinates (1,0) and (0,1)

Rotation 90o clockwise about the origin

Matrix transformations

Describe fully the geometrical transformation represented by the given matrices:

10

01

01

10

10

01

10

01

01

10

01

10

0

1

1

0Compare with the coordinates (1,0) and (0,1)

Rotations by multiples of 45o

Eg Describe fully the geometrical transformation represented by the matrix A.

0

1

1

0

212

1

21

21

Rotation 135o anticlockwise about the origin

Before After

2

12

12

12

1

A

Now try Ex4EDescribe the transformation represented by:

Two 135o rotations gives a 270o rotation

b) A-1

a) A2

The inverse of an anticlockwise rotation is a clockwise rotation

The transformation given by a power of a matrix can be understood by considering its geometric representation

WB21

(a) Find AB.

(c) write down C100.

35

02A

25

13B

10

01CGiven that

(b) describe fully the geometrical transformation represented by C,

Product will be 2 x 2

25

13

35

02AB

10

26

0

1

1

0

1

0

0

1Reflection in y - axis

Before After

Doing C again will return the images to their original position, as will any even power of C

10

01100 IC

WB23(a) Given that

(ii) describe fully the geometrical transformation represented by A2.

12

21A

(i) find A2,

12

21

12

212A

30

03

An enlargement about the origin, scale factor 3

(b) Given that

where k is a constant, find the value of k for which the matrix C is singular.

01

10B

describe fully the geometrical transformation represented by B.

(c) Given that

9

121

k

kC

0

1

1

0

1

0

0

1

Reflection in line xy

Before After

Singular 0 )det(C

01219 kk

039 k 3 k

dc

baM bcadM )det(

WB28

21

21

21

21

M (a) Describe fully the geometrical transformation represented by the matrix M.

0

1

1

0

212

1

21

21

Rotation 45o anticlockwise about the origin

Before After

q

p

21

21

21

21

2

2qp

qp

24

23 232

qp

242

qp

6 qp

8 qp

)(1

)(2

)()( 21 142 p 7 p

)()( 12 22 q 1 q

The transformation represented by M maps the point A with coordinates (p, q) onto the point B with coordinates (3√2, 4√2).(b) Find the value of p and the value of q.

The point B is mapped onto the point C by the transformation represented by M2.(e) Find the coordinates of C.

(c) Find, in its simplest surd form, the length OA, where O is the origin.

(d) Find M2.

A(7, 1)22 17 OA 50 25

21

21

21

21

21

21

21

21

2M

01

10

24

2301

102BM

23

24

Matrix transformations using the formulae booklet

cossin

sincos

22

22

cossin

sincos

Anticlockwise rotation through θ about 0:

Reflection in the line : xy tan

The formulae booklet provides you with general rules:

Eg to rotate 90o anticlockwise about 0 90

01

10

Eg to reflect in the x-axis 0

10

010 y 0 tan

Eg to reflect in the y = x 45

01

101 tan

Now try Ex4E, Q4-5

T

Combined transformations

01

10A

10

01B

01

10

A represents a rotation 90o clockwise about the origin.B represents a reflection in the y – axis.Hence describe fully the geometrical transformation represented by AB

Eg Find AB

10

01

01

10AB

Consider a triangle T

BT

ABT

AB represents a reflection in the line y = x

Eg using the same A and B above, write down:

a) B7

b) A20

7 reflections in the y-axis is the same as 1

10

017 BB

20 rotations of 90o will get back to where you started

10

0120 IA

Now try Ex4F, Q2 onwards

WB26 Write down the 2 × 2 matrix that represents(a) an enlargement with centre (0, 0) and scale factor 8,

Given that AB represents the same transformation as T,(e) find the value of k and the value of c.

(b) a reflection in the x-axis.

(c) Hence, or otherwise, find the matrix T that represents an enlargement with centre (0, 0) and scale factor 8, followed by a reflection in the x-axis.

80

08

10

01Change sign of y coordinate only

80

08

10

01T

80

08

24

16A

6

1

c

kB

where k and c are constants.

and

(d) Find AB.

6

1

24

16

c

kAB

824

06

ck

ck

86 ck

024 ck

)(1

)(2

80

08

824

06

ck

ck

21 )( 16212 ck )(3

)()( 23 168 k 2 k)(1 in Sub 812 c 4 c

Using inverse matrices in geometric situations

Eg A triangle P is transformed to a triangle Q by the matrix M

The triangle Q has vertices at the points (3, -1), (7, -9) and (24, 2). Find the coordinates of the vertices of P.

291

2473Q QMP 1

291

2473

42

13

10

1

405010

703010

10

1

451

731Coordinates are(1,-1), (3,-5), (7,-4)

32

14

dc

baM bcadM )det( 10Mdet

ac

bd

MM

)det(

11

42

13

10

11M

You can apply M-1 to the image to find the original points

1100

1010

dc

ba

dcdc

baba

0

0

ca,

db,

dcba ,

cbabdcdcbaArea

bcacbdbcbdbcadac

bcad

bdcArea 21

cbaArea 21

Mdet

dc

baMIf then

1Area

MArea det

kArea

MkArea det

If a shape is transformed by a matrix M, then: )det(MArea factor =

Determining the area

where bcadM )det(

Given that the matrix A maps the point with coordinates (4, 6) onto the point with coordinates (2, −8),(a) find the value of a and the value of b.

WB25, where a and b are constants.

A quadrilateral R has area 30 square units.It is transformed into another quadrilateral S by the matrix A.Using your values of a and b,(b) find the area of quadrilateral S.

2

4

b

aA

6

4

2

4

b

a

124

616

b

a

8

2 2616 a

8124 b

3 a

1 b

)det(MArea factor of transformation under M =

21

34A 5 Adet units square 150530S of Area

(b) Find A–1.

WB27

The triangle S has vertices at the points (0, 4), (8, 16) and (12, 4).(d) Find the coordinates of the vertices of R.

The triangle R is transformed to the triangle S by the matrix A.Given that the area of triangle S is 72 square units,(c) find the area of triangle R.

(a) Find det A.

31

22A

dc

baM bcadM )det(

4Adet

Inverse of is

dc

baM

ac

bd

MM

)det(

11

21

23

4

11A

)det(MArea factor = Area of R units square 18472

4164

1280S

SAR 1

4164

1280

21

23

4

1

20408

44568

4

1

5102

11142

Coordinates (2,2), (14,10), (11,5)

Wb27.agg

Now try Ex4I

Now try Ex4J

Using inverse matrices to solve simultaneous equations

532 yx865 yx

Eg use an inverse matrix to solve the simultaneous equations

65

32M

25

36

3

11M

8

5

25

36

3

1

y

x

9

6

3

1

3

2

3 2 yx ,Solution

8

5

y

xM

8

51My

xMultiply on the left by M-1

Describe fully the geometrical transformation represented by B

0

1

1

0

1

0

0

1

Rotation 90o anticlockwise about (0,0)

Before

After

01

10B

20

02AUsingMatrix algebra

0 00 2

2 0

0 0ABC

02

20

4Cdet

02

20

4

11C1C

01

10

20

02

The triangle S has vertices at the points (-2,2), (-10,2) and (-8,6). Find the coordinates of the vertices of R.

The triangle R is transformed to the triangle S by the matrix C.Given that the area of triangle R is 6 square units, find the area of triangle S.

CdetArea factor = Area of S units square 2446

622

8102S

SCR 1

622

8102

02

20

4

1

16204

1244

4

1

451

311 Coordinates (1,1), (1,5), (3,4)

dc

baM bcadM det

ac

bd

MM

det

11

Compare with

10

01