Matrices 1

MatricesMM1 module 3, lecture 1

David Godfrey

Slide number 2

Matrices lecture 1 objectives

• After this lecture you should have a clear understanding of:• What matrices are;• Performing basic operations on matrices;• Special forms of matrices;• The matrix determinant;• How to calculate the inverse of a 2 2 matrix.

Slide number 3

What is a matrix?

• A matrix is a rectangular array of elements

• The elements may be of any type (e.g. integer, real, complex, logical, or even other matrices).

• In this course we will only consider matrices that have integer, real, or complex elements.

5 0 1 23 4 9 23 1 4 2

Slide number 4

Order of matrices…

• Order 4 3:

• Order 3 4:

3 columns

4 rows

5 0 1

2 3 4

9 2 6

3 1 4

4 columns

3 rows

5 0 1 2

3 4 9 2

3 1 4 2

Slide number 5

…Order of matrices

• Order 2 4:

• Order 1 6:

• Order 3 1:

1 columns

3 rows

3

1

5

6 columns1 rows

2 1 1 2 1 5

4 columns2 rows

23 0.5 4.3 12

8 2 8 1

Slide number 6

Specifying matrix elements

• aij denotes the element of the matrix A on the ith row and jth column.

A

column j

row i

5 0 12 3 4 9 2 63 1 4

• a12 = 0

• a21 = 2

• a23 = -4

• a32 = 2

• a41 = 3

• a43 = 4

Slide number 7

What are matrices used for?

• Transformations• Transitions• Linear equations

Slide number 8

Matrix operations: scalar multiplication

• Multiplying an m n matrix by a scalar results in an m n matrix with each of its elements multiplied by the scalar.

• e.g.

826

12418

864

2010

413

629

432

105

2

1239

18627

1296

3015

413

629

432

105

3

Slide number 9

Matrix operations: addition…

• Adding or subtracting an m n matrix by an m n matrix results in an m n matrix with each of its elements added or subtracted.

• e.g.

431

8112

113

136

024

213

541

231

413

629

432

105

417

436

971

334

024

213

541

231

413

629

432

105

Slide number 10

…Matrix operations: addition

• Note that matrices being added or subtracted must be of the same order.

• e.g.

invalid! 113

201

413

629

432

105

Slide number 11

Matrix operations: multiplication…

• Multiplying an m n matrix by an n p matrix

results in an m p matrix

wsrom

columnsp

wsron

columnsp

wsrom

columnsn

Slide number 12

…Matrix operations: multiplication…

• Example 1…

1 0 2

3 1 1

2 1

3 2

1 4

0

( 12) (0 3) (2 1) 0

1 0 2

3 1 1

2 1

3 2

1 4

0 7

( 1 1) (0 2) (2 4) 7

Slide number 13


• …Example 1

1 0 2

3 1 1

2 1

3 2

1 4

0 7

10

(32) (13) (11) 10

1 0 2

3 1 1

2 1

3 2

1 4

0 7

10 9

(3 1) (1 2) (1 4) 9

Slide number 14


• Example 2…

3)13()32(

3

21

13

41

32

8)23()12(

83

21

13

41

32

Slide number 15


• …Example 2

1)14()31(

1

83

21

13

41

32

9)24()11(

91

83

21

13

41

32

Slide number 16


• Example 3

• i.e. the number of columns in the first matrix must equal the number of rows in the second matrix!

invalid!

113

201

124

862

351

Slide number 17


• Example 4

4 2 1

5

6

(4 1) ( 2 5) 6

Slide number 18


• Example 5…

4)41(

424

5

1

2)21(

2424

5

1

Slide number 19


• …Example 5

20)45(

20

2424

5

1

10)25(

1020

2424

5

1

Slide number 20

…Matrix operations: multiplication

• Matrix multiplication is NOT commutative• In general, if A and B are two matrices then

A B ≠ B A• i.e. the order of matrix multiplication is

important!• e.g.

10

22

10

02

10

21

10

42

10

21

10

02

Slide number 21

Matrix operations: transpose…

• If B = AT, then bij = aji

• i.e. the transpose of an m n matrix is an n m matrix with the rows and columns swapped.

• e.g.

4641

1230

3925

413

629

432

105T

3925

3

9

2

5

T

Slide number 22

…Matrix operations: transpose

• (A B)T = BT AT

• Note the reversal of order.• Justification (not a proof):

e.g. if A is 3 2 and B is 2 4then AT is 2 3 and BT is 4 2so AT BT cannot be multipliedbut BT AT can be multiplied.

Slide number 23

Special matrices: row and column

• A 1 n matrix is called a row matrix.e.g.

• An m 1 matrix is called a column matrix.e.g.

1 columns

3 rows

3

1

5

6 columns1 rows

2 1 1 2 1 5

Slide number 24

Special matrices: square

• An n n matrix is called a square matrix.i.e. a square matrix has the same number of rows and columns.e.g.

4705

7350

0523

5031

211

010

210

21

321

Slide number 25

Special matrices: diagonal

• A square matrix is diagonal if non-zero elements only occur on the leading diagonal.i.e. aij = 0 for i ≠ je.g.

• Premultiplying a matrix by a diagonal matrix scales each row by the diagonal element.

• Postmultiplying a matrix by a diagonal matrix scales each column by the diagonal element.

4000

0300

0020

0001

200

010

000

20

021

Slide number 26

Special matrices: triangular

• A lower triangular matrix is a square matrix having all elements above the leading diagonal zero.e.g.

• An upper triangular matrix is a square matrix having all elements below the leading diagonal zero.e.g.

4705

0350

0023

0001

1 20 1

Slide number 27

Special matrices: null

• The null matrix, 0, behaves like 0 in arithmetic addition and subtraction.

• Null matrices can be of any order and have all of their elements zero.

0000

0000

0000

0000

00

00

00

00

00

0000

00

00

Slide number 28

Special matrices: identity…

• The identity matrix, I, behaves like 1 in arithmetic multiplication.

• Identity matrices are diagonal. They have 1s on the diagonal and 0s elsewhere.e.g.

• In the world of the matrix the identity truly is ‘the one’.

1000

0100

0010

0001

100

010

001

10

011

II

II

Slide number 29

…Special matrices: identity

• The identity matrix multiplied by any compatible matrix results in the same matrix.i.e. I A = Ae.g.

• Any matrix multiplied by a compatible identity matrix results in the same matrix.i.e. A I = Ae.g.

• Multiplication by the identity matrix is thus commutative.

51

13

51

13

10

01

51

13

10

01

51

13

Slide number 30

Determinant of a 22 matrix…

• Matrices can represent geometric transformations, such as scaling, rotation, shear, and mirroring.

• 2 2 matrices can represent geometric transformations in a 2–dimensional space, such as a plane.

• Determinants of 2 2 matrices give us information about how such transformations change the area of shapes.

• Determinants are also useful to define the inverse of a matrix.

Slide number 31

…Determinant of a 22 matrix…

• The determinant of a 2 2 matrix is the product of the 2 leading diagonal terms minus the product of the cross- diagonal.

• i.e. if A is a 2 2 matrix, then the determinant of A is denoted by det(A) = |A| = a11 a22 – a21 a12

• e.g.

det3 1

2 6

3 1

2 6 36 2 1 20

det 2 5

1 3

2 5

1 3 2 3 15 11

Slide number 32

…Determinant of a 22 matrix

• |A B| = |A| |B|• Note the order is not important.• Justification (not a proof):

We will shortly see that A and B can represent geometric transformations, and A B represents the combined transformation of B followed by A. The determinant represents the factor by which the area is changed, so the combined transformation changes area by a factor |A B|. Looking at the individual transformations, the area of the first is changed by a factor |B|, and the second by |A|. The overall transformation is thus changed by a factor |B| |A|, which is the same as |A| |B|.

Slide number 33

Inverse of a matrix

• In arithmetic multiplication the inverse of a number c is 1/c sincec 1/c = 1 and 1/c c = 1

• For matrices the inverse of a matrix A is denoted by A-1

• A A-1 = IA-1 A = Iwhere I is the identity matrix.

• Multiplication of a matrix by its inverse is thus commutative.

• We shall only consider the inverse of 2 2 matrices.

Slide number 34

Inverse of a 22 matrix…

• The inverse of a 2 2 matrix A is given by

• Note:The leading term is 1/determinant;The diagonal elements are swapped;The cross-diagonal elements change their sign.

a11 a12

a21 a22

1

1

a11 a12

a21 a22

a22 a12

a21 a11

Slide number 35

…Inverse of a 22 matrix…

• Example 1

• Note that A A-1 = I (right inverse)

and A-1 A = I (left inverse)

11

24

6

1

11

24

41

211

41

211

10

01

11

24

6

1

41

21

10

01

41

21

11

24

6

1

Slide number 36


• Example 2

• Note that A A-1 = I (right inverse)

and A-1 A = I (left inverse)

2 2

2 3

1

1

2 2

2 3

3 2

2 2

1

10

3 2

2 2

0.3 0.2

0.2 0.2

10

01

2.02.0

2.03.0

32

22

10

01

32

22

2.02.0

2.03.0

Slide number 37


• Example 3

• Note that the determinant is zero so the inverse does not exist for this matrix.

• Matrices with zero determinant can have no inverse.

• Such matrices are called singular.

2 2

3 3

1

1

2 2

3 3

3 2

3 2

1

0

3 2

2 2

invalid!

Slide number 38

…Inverse of a 22 matrix

• (A B)-1 = B-1 A-1

• Note the reversal of order.• Justification (not a proof):

B-1 A-1 A B = B-1 (A-1 A) B = B-1 B = Iso B-1 A-1 is the inverse of A Bi.e. (A B)-1 = B-1 A-1

Slide number 39

Matrices lecture 1 objectives

• After this lecture you should have a clear understanding of:• What matrices are;• Performing basic operations on matrices;• Special forms of matrices;• The matrix determinant;• How to calculate the inverse of a 2 2 matrix.

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Matrices 1