Chapter 6 - Matrix Algebra

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 Algebra7. Linear Programming

8. Introduction to Probability and Statistics


9. Additional Topics in Probability10. Limits and Continuity11. Differentiation12. Additional Differentiation Topics13. Curve Sketching14. Integration15. Methods and Applications of Integration16. Continuous Random Variables17. 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

aaaaaa

.....................

...

...

21

21221

11211


a. The matrix has size .

b. The matrix has size .

c. The matrix has size .

d. The matrix has size .

Chapter 6: Matrix Algebra6.1 Matrices

Example 1 – Size of a Matrix 021 31

491561

23

7 11

1112686511942731

53


Chapter 6: Matrix Algebra6.1 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 .

654321

A

635241

TA

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.

688308

063544632271

034627

635241

12

4321


Chapter 6: Matrix Algebra6.2 Matrix Addition and Scalar Multiplication

Example 3 – Demand Vectors for an EconomyDemand 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.

130884

320311442662

301426

231462

52

804266

1026

2BAT



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 matrixB = 5 × 3 matrixAB = 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


Chapter 6: Matrix Algebra6.3 Matrix Multiplication

Example 3 – Matrix Products

a.

b.

c.

d.

32654

321

18312261

61321

1047011011316

212312201

401122031

2222122121221121

2212121121121111

2221

1211

2221

1211

babababababababa

bbbb

aaaa



Example 5 – Cost VectorGiven the price and the quantities, calculate the total cost.

Solution:The cost vector is

432PC of unitsB of units Aof units

1157

Q

731157

432

PQ



Example 7 – Associative Property

If

compute ABC in two ways.

Solution 1: Solution 2:

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

112001

211103

4321

CBA

19694

4312

4321

112001

211103

4321

BCA

19694

112001

1145521

112001

211103

4321

CAB



Example 9 – Raw Materials and CostFind QRC when

Solution:

975Q

13582562191218717716205

R

1500150800

12002500

C

716508155075850

1500150800

12002500

13582562191218717716205

RC

900,809,1716508155075850

1275

RCQQRC



Example 11 – Matrix Operations Involving I and OIf

compute each of the following.

Solution:

0000

1001

4123

103

101

51

52

OIBA

3122

4123

1001

a.

AI

6363

2002

4123

3 1001

24123

323 b. IA

OAO

0000

4123

c.

IAB

1001

4123

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

738452

21

21

xxxx

74

3852

2

1 Bxx

XA

74

3852

2

1

xx

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


Chapter 6: Matrix Algebra6.4 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.


Chapter 6: Matrix Algebra6.4 Solving Systems by Reducing Matrices

Example 1 – Reduced MatricesFor each of the following matrices, determine whether it is reduced or not reduced.

Solution:a. Not reduced b. Reducedc. Not reduced d. Reducede. Not reduced f. Reduced

000021003010

f. 010000001

e. 000000

d.

0110

c. 010001

b. 3001

a.



Example 3 – Solving a System by ReductionBy using matrix reduction, solve the system

Solution:Reducing the augmented coefficient matrix of the system,

We have

151

111232

152

132

yxyxyx

03

4

001001

34

yx



Example 5 – Parametric Form of a SolutionUsing matrix reduction, solve

Solution:Reducing the matrix of the system,

We have and x4 takes on any real value.

92

10

630312106232

96332206232

431

432

4321

xxxxxxxxxx

104

1000010

001

21

25

421

3

2

425

1

104

xxx

xx



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

Example 1 – Two-Parameter Family of SolutionsUsing matrix reduction, solve

Solution:The matrix is reduced to

The solution is

32 143 3552

4321

4321

4321

xxxxxxxxxxxx

02

1

000012103101

sxrx

srxsrx

4

3

2

1

2231


Chapter 6: Matrix Algebra6.5 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


Chapter 6: Matrix Algebra6.5 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.

042202

zyxzyx



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.

7321

A

1327

C

ICA

1001

7321

1327


Chapter 6: Matrix Algebra6.6 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

2201

A

1001

2201

IA BI

2

1101

1001

21

1

101

A


Chapter 6: Matrix Algebra6.6 Inverses

Example 5 – Using the Inverse to Solve a SystemSolve the system by finding the inverse of the coefficient matrix.

Solution:We have

For inverse,

The solution is given by X = A−1B:

1102 2 241 2

321

321

31

xxxxxx

xx

1021124201

A

1154

229

29

2411A

4177

121

1154

229

29

241

3

2

1

xxx



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.)


Chapter 6: Matrix Algebra6.7 Leontief’s Input-Output AnalysisExample 1 – Input-Output Analysis

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

Final-demand matrix:

Output matrix is

23115477

D

495380

5.6921DAIX