INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007...

INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences

2007 Pearson Education Asia

Chapter 6 Chapter 6 Matrix AlgebraMatrix Algebra


INTRODUCTORY MATHEMATICAL ANALYSIS

0. Review of Algebra

1. Applications and More Algebra

2. Functions and Graphs

3. Lines, Parabolas, and Systems

4. Exponential and Logarithmic Functions

5. Mathematics of Finance

6. Matrix Algebra

7. Linear Programming

8. Introduction to Probability and Statistics


9. Additional Topics in Probability

10. Limits and Continuity

11. Differentiation

12. Additional Differentiation Topics

13. Curve Sketching

14. Integration

15. Methods and Applications of Integration

16. Continuous Random Variables

17. Multivariable Calculus

INTRODUCTORY MATHEMATICAL ANALYSIS


• Concept of a matrix.

• Special types of matrices.

• Matrix addition and scalar multiplication operations.

• Express a system as a single matrix equation using matrix multiplication.

• Matrix reduction to solve a linear system.

• Theory of homogeneous systems.

• Inverse matrix.

• Use a matrix to analyze the production of sectors of an economy.

Chapter 6: Matrix Algebra

Chapter ObjectivesChapter Objectives


Matrices

Matrix Addition and Scalar Multiplication

Matrix Multiplication

Solving Systems by Reducing Matrices

Solving Systems by Reducing Matrices (continued)

Inverses

Leontief’s Input—Output Analysis

6.1)

6.2)

6.3)

6.4)


Chapter OutlineChapter Outline

6.5)

6.6)

6.7)



6.1 Matrices6.1 Matrices• A matrix consisting of m horizontal rows and n

vertical columns is called an m×n matrix or a matrix of size m×n.

• For the entry aij, we call i the row subscript and j the column subscript.

mnmm

n

n

aaa

aaa

aaa

...

......

......

......

...

...

21

21221

11211


a. The matrix has size .

b. The matrix has size .

c. The matrix has size .

d. The matrix has size .


6.1 Matrices

Example 1 – Size of a Matrix

021 31

49

15

61

23

7 11

11126

865119

42731

53



6.1 Matrices

Example 3 – Constructing Matrices

Equality of Matrices

• Matrices A = [aij ] and B = [bij] are equal if they have the same size and aij = bij for each i and j.

Transpose of a Matrix

• A transpose matrix is denoted by AT.

If , find .

Solution:

Observe that .

654

321A

63

52

41TA

AATT

TA



6.2 Matrix Addition and Scalar Multiplication6.2 Matrix Addition and Scalar Multiplication

Example 1 – Matrix Addition

Matrix Addition

• Sum A + B is the m × n matrix obtained by adding corresponding entries of A and B.

a.

b. is impossible as matrices are not of the same

size.

68

83

08

0635

4463

2271

03

46

27

63

52

41

1

2

43

21



6.2 Matrix Addition and Scalar Multiplication

Example 3 – Demand Vectors for an Economy

Demand for the consumers is

For the industries is

What is the total demand for consumers and the industries?

Solution:

Total:

1264 1170 523 321 DDD

0530 8020 410 SEC DDD

182571264 1170523321 DDD

1265005308020410 SEC DDD

3031571265018257




Scalar Multiplication

• Properties of Scalar Multiplication:

Subtraction of Matrices

• Property of subtraction is AA 1




Example 5 – Matrix Subtraction

a.

b.

13

08

84

3203

1144

2662

30

14

26

23

14

62

52

80

42

66

10

262BAT



6.3 Matrix Multiplication6.3 Matrix Multiplication

Example 1 – Sizes of Matrices and Their Product

• AB is the m× p matrix C whose entry cij is given by

A = 3 × 5 matrix

B = 5 × 3 matrix

AB = 3 × 3 matrix but BA = 5 × 5 matrix.

C = 3 × 5 matrixD = 7 × 3 matrixCD = undefined but DC = 7 × 5 matrix.

njinji

n

kjikjikij babababac

...221

11



6.3 Matrix Multiplication

Example 3 – Matrix Products

a.

b.

c.

d.

32

6

5

4

321

183

122

61

61

3

2

1

1047

0110

11316

212

312

201

401

122

031

2222122121221121

2212121121121111

2221

1211

2221

1211

babababa

babababa

bb

bb

aa

aa




Example 5 – Cost Vector

Given the price and the quantities, calculate the total cost.

Solution:

The cost vector is

432P

C of units

B of units

Aof units

11

5

7

Q

73

11

5

7

432

PQ




Example 7 – Associative Property

If

compute ABC in two ways.

Solution 1: Solution 2:

Note that A(BC) = (AB)C.

11

20

01

211

103

43

21CBA

196

94

43

12

43

21

11

20

01

211

103

43

21BCA

196

94

11

20

01

1145

521

11

20

01

211

103

43

21CAB




Example 9 – Raw Materials and Cost

Find QRC when

Solution:

975Q

1358256

21912187

17716205

R

1500

150

800

1200

2500

C

71650

81550

75850

1500

150

800

1200

2500

1358256

21912187

17716205

RC

900,809,1

71650

81550

75850

1275

RCQQRC




Example 11 – Matrix Operations Involving I and O

If

compute each of the following.

Solution:

00

00

10

01

41

23

103

101

51

52

OIBA

31

22

41

23

10

01 a.

AI

63

63

20

02

41

233

10

012

41

23323 b. IA

OAO

00

00

41

23 c.

IAB

10

01

41

23 d.

103

101

51

52




Example 13 – Matrix Form of a System Using Matrix Multiplication

Write the system

in matrix form by using matrix multiplication.

Solution:

If

then the single matrix equation is

738

452

21

21

xx

xx

7

4

38

52

2

1 Bx

xXA

7

4

38

52

2

1

x

x

BAX



6.4 Solving Systems by Reducing Matrices6.4 Solving Systems by Reducing Matrices

Elementary Row Operations

1. Interchanging two rows of a matrix

2. Multiplying a row of a matrix by a nonzero number

3. Adding a multiple of one row of a matrix to a different row of that matrix



6.4 Solving Systems by Reducing Matrices6.4 Solving Systems by Reducing Matrices

Properties of a Reduced Matrix

• All zero-rows at the bottom.

• For each nonzero-row, leading entry is 1 and the rest zeros.

• Leading entry in each row is to the right of the leading entry in any row above it.



6.4 Solving Systems by Reducing Matrices

Example 1 – Reduced Matrices

For each of the following matrices, determine whether it is reduced or not reduced.

Solution:

a. Not reduced b. Reduced

c. Not reduced d. Reduced

e. Not reduced f. Reduced

0000

2100

3010

f.

010

000

001

e. 000

000d.

01

10c.

010

001b.

30

01 a.




Example 3 – Solving a System by Reduction

By using matrix reduction, solve the system

Solution:Reducing the augmented coefficient matrix of the system,

We have

1

5

1

11

12

32

1

52

132

yx

yx

yx

0

3

4

00

10

01

3

4

y

x




Example 5 – Parametric Form of a Solution

Using matrix reduction, solve

Solution:Reducing the matrix of the system,

We have and x4 takes on any real value.

9

2

10

6303

1210

6232

9633

22

06232

431

432

4321

xxx

xxx

xxxx

1

0

4

100

0010

001

21

25

421

3

2

425

1

1

0

4

xx

x

xx



6.5 Solving Systems by Reducing Matrices 6.5 Solving Systems by Reducing Matrices (continued)(continued)

Example 1 – Two-Parameter Family of Solutions

Using matrix reduction, solve

Solution:

The matrix is reduced to

The solution is

32

143

3552

4321

4321

4321

xxxx

xxxx

xxxx

0

2

1

0000

1210

3101

sx

rx

srx

srx

4

3

2

1

22

31



6.5 Solving Systems by Reducing Matrices (Continue)

• The system

is called a homogeneous system if c1 = c2 = … = cm = 0.

• The system is non-homogeneous if at least one of the c’s is not equal to 0.

mnmnmm

nn

cxaxaxa

cxaxaxa

...

.

.

.

.

...

2211

11212111

Concept for number of solutions:

1. k < n infinite solutions

2. k = n unique solution



6.5 Solving Systems by Reducing Matrices (Continue)

Example 3 – Number of Solutions of a Homogeneous System

Determine whether the system has a unique solution or infinitely many solutions.

Solution:

2 equations (k), homogeneous system, 3 unknowns (n).

The system has infinitely many solutions.

0422

02

zyx

zyx



6.6 Inverses6.6 Inverses

Example 1 – Inverse of a Matrix

• When matrix CA = I, C is an inverse of A and A is invertible.

Let and . Determine whether C is

an inverse of A.

Solution:

Thus, matrix C is an inverse of A.

73

21A

13

27C

ICA

10

01

73

21

13

27



6.6 Inverses

Example 3 – Determining the Invertibility of a Matrix

Determine if is invertible.

Solution: We have

Matrix A is invertible where

Method to Find the Inverse of a Matrix

• When matrix is reduced, ,

- If R = I, A is invertible and A−1 = B.

- If R I, A is not invertible.

BRIA

22

01A

10

01

22

01IA BI

2

11

01

10

01

21

1

1

01A



6.6 Inverses

Example 5 – Using the Inverse to Solve a System

Solve the system by finding the inverse of the coefficient matrix.

Solution:

We have

For inverse,

The solution is given by X = A−1B:

1102

2 24

1 2

321

321

31

xxx

xxx

xx

1021

124

201

A

115

4

229

29

2411A

4

17

7

1

2

1

115

4

229

29

241

3

2

1

x

x

x



6.7 Leontief’s Input-Output Analysis6.7 Leontief’s Input-Output Analysis

Example 1 – Input-Output Analysis

• Entries are called input–output coefficients.• Use matrices to show inputs and outputs.

Given the input–output matrix,

suppose final demand changes to be 77 for A, 154 for B, and 231 for C. Find the output matrix for the economy. (The entries are in millions of dollars.)



6.7 Leontief’s Input-Output Analysis

Example 1 – Input-Output Analysis

Solution:

Divide entries by the total value of output to get A:

Final-demand matrix:

Output matrix is

231

154

77

D

495

380

5.6921DAIX

INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007...

Documents

Transcript of INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007...