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3Matrix Algebra and Applications

3.1Matrix Addition and

Scalar Multiplication

3.2Matrix Multiplication

3.3Matrix Inversion

3.4Game Theory

3.5Input-Output Models

Key Concepts

Review Exercises

Case Study Exercises

Technology Guides

CASE STUDYThe Japanese Economy

A senator walks into your cubicle in the Congressional Budget Office. "Look here," she says, "I don't see why the Japanese trade representative is getting so upset with my proposal to cut down on our use of Japanese finance and insurance. He claims that it'll hurt Japan's mining operations. But just look at Japan's input-output table. The finance sector doesn't use any input from the mining sector. How can our cutting down demand for finance and insurance hurt mining?" How should you respond? 173

Jose Fuste Raga/Zefa/Corbis

Online you will find:

•Section by section tutorials •A detailed chapter summary •A true/false quiz •Additional review exercises •A matrix algebra tool, game theory utility, and other resources

16314_04_ch3_p173-208.qxd 7/17/06 4:24 PM Page 173

Introduction

We used matrices in Chapter 2 simply to organize our work. It is time we examined them as interesting objects in their own right. There is much that we can do with matrices besides row operations: We can add, subtract, multiply, and even, in a sense, "divide" matrices. We use these operations to study game theory and input-output models in this chapter, and Markov chains in a later chapter. Many calculators, electronic spreadsheets, and other computer programs can do these matrix operations, which is a big help in doing calculations. However, we need to know how these operations are defined to see why they are useful and to understand which to use in any particular application.

174Chapter 3 Matrix Algebra and Applications

quickExamples

Matrix, Dimension, and Entries

An m◊nmatrixAis a rectangular array of real numbers with mrows and ncolumns. We refer to mand nas the dimensionsof the matrix. The numbers that appear in the ma- trix are called its entries.We customarily use capital letters A, B, C, ...for the names of matrices. 1.A= 201
is a 2◊3matrix because it has 2 rows and 3 columns. 2.B= 23
10 44 83
is a 4◊2matrix because it has 4 rows and 2 columns.

3.1Matrix Addition and Scalar Multiplication

LetÕs start by formally deÞning what a matrix is and introducing some basic terms. Hint:Remember that the number of rows is given Þrst and the number of columns second. An easy way to remember this is to think of the acronym "RC" for "Row then Column." quickExample

Referring to the Entries of a Matrix

There is a systematic way of referring to particular entries in a matrix. If iand jare num- bers, then the entry in the ith row and jth column of the matrix Ais called the ijth entry of A. We usually write this entry as a ij or A ij . (If the matrix was called B, we would write its ijth entry as b ij or B ij .) Notice that this follows the ÒRCÓ convention: The row number is specified first and the column number second.

With A=

201
a 13 =1First row, third column a 21
=33Second row, first column

16314_04_ch3_p173-208.qxd 7/17/06 4:24 PM Page 174

3.1 Matrix Addition and Scalar Multiplication175

According to the labeling convention, the entries of the matrix Aabove are A= a 11 a 12 a 13 a 21
a 22
a 23
In general, the m◊nmatrix Ahas its entries labeled as follows: A= a 11 a 12 a 13 ...a 1n a 21
a 22
a 23
...a 2n a m1 a m2 a m3 ...a mn We say that two matrices Aand Bare equalif they have the same dimensions and the corresponding entries are equal. Note that a 3◊4matrix can never equal a 3◊5 matrix because they do not have the same dimensions. using Technology

See the Technology Guides at

the end of the chapter to see how matrices are entered and used in a TI-83/84or Excel.For the authors' web-based utility, follow:

Chapter 3

?Tools ?Matrix Algebra Tool

There you will find a computa-

tional tool that allows you to do matrix algebra. Use the following format to enter the matrix Aon the previous page (spaces are optional):

A=[2, 0,1

To display the matrix A, type Ain

the formula box and press "Compute."

Example1Matrix Equality

Let A=

79x
and B= 790
.Find the values of xand ysuch that A=B. SolutionFor the two matrices to be equal, we must have corresponding entries equal, so x=0a 13 =b 13 y+1=11 ory=10a 23
=b 23
quickExamples

Row Matrix, Column Matrix, and Square Matrix

A matrix with a single row is called a row matrix,or row vector.A matrix with a sin- gle column is called a column matrixor column vector.A matrix with the same num- ber of rows as columns is called a square matrix.

The 4◊1matrix D=

2 10 8 is a column matrix.

The 3◊3matrix E=

014 is a square matrix. Before we go on...Note in Example 1 that the matrix equation 79x
790
used only the two that were interesting.

16314_04_ch3_p173-208.qxd 7/17/06 4:24 PM Page 175

Matrix Addition and Subtraction

The Þrst matrix operations we discuss are matrix addition and subtraction. The rules for these operations are simple.

176Chapter 3 Matrix Algebra and Applications

quickExamples

Matrix Addition and Subtraction

Two matrices can be added (or subtracted) if and only if they have the same dimensions. To add (or subtract) two matrices of the same dimensions, we add (or subtract) the cor- responding entries. More formally, if Aand Bare m◊nmatrices, then A+Band (A+B) ij =A ij +B ij ijth entry of the sum =sum of the ijth entries ij =A ij ij ijth entry of the difference =difference of the ijth entries

Visualizing Matrix Addition

Example2Sales

The A-Plus auto parts store chain has two outlets, one in Vancouver and one in Quebec. Among other things, it sells wiper blades, windshield cleaning fluid, and floor mats. The monthly sales of these items at the two stores for two months are given in the follow- ing tables:

Vancouver Quebec

Wiper Blades20 15

Cleaning Fluid (bottles)10 12

Floor Mats84

January Sales

Vancouver Quebec

Wiper Blades23 12

Cleaning Fluid (bottles)812

Floor Mats45

February Sales

10 11 1. 10 013 113

Corresponding entries added

2. 10 013 00

Corresponding entries subtracted

16314_04_ch3_p173-208.qxd 7/17/06 4:24 PM Page 176

Scalar Multiplication

A matrix Acan be added to itself because the expression A+Ais the sum of two ma- trices that have the same dimensions. When we compute A+A, we end up doubling every entry in A. So we can think of the expression 2Aas telling us to multiply every element in A by2. In general, to multiply a matrix by a number, multiply every entry in the matrix by that number. For example, 6 5 2 10 5 6 60
It is traditional when talking about matrices to call individual numbers scalars.For this reason, we call the operation of multiplying a matrix by a number scalar multiplication.

3.1 Matrix Addition and Scalar Multiplication177

Use matrix arithmetic to calculate the change in sales of each product in each store from

January to February.

SolutionThe tables suggest two matrices:

J= 20 15 10 12 84
andF= 23 12
812
45
To compute the change in sales of each product for both stores, we want to subtract cor- responding entries in these two matrices. In other words, we want to compute the differ- ence of the two matrices: 23 12
812
45
20 15 10 12 84
Thus, the change in sales of each product is the following:

Vancouver Quebec

using Technology

See the Technology Guides at

the end of the chapter to see how to add and subtract matrices using a TI-83/84or Excel.

Alternatively, use the Matrix

Algebra Tool at

Chapter 3

?Tools ?Matrix Algebra Tool

There, first enter the two matri-

ces you wish to add or subtract (subtract, in this case) as shown:

J=[20, 15, 10

12, 8,4]

F=[23, 12, 8

12, 4,5]

To compute their difference, type

F-Jin the formula box and press

"Compute." (You can enter multi- ple formulas separated by com- mas in the formula box. For in- stance, F+J, F-Jwill compute both the sum and difference.)

Example3Sales

The revenue generated by sales in the Vancouver and Quebec branches of the A-Plus auto parts store (see Example 2) was as follows:

Vancouver Quebec

Wiper Blades140.00 105.00

Cleaning Fluid30.00 36.00

Floor Mats96.00 48.00

January Sales in Canadian Dollars

16314_04_ch3_p173-208.qxd 7/17/06 4:24 PM Page 177

Formally, scalar multiplication is defined as follows:

178Chapter 3 Matrix Algebra and Applications

If the Canadian dollar was worth $0.65 U.S. at the time, compute the revenue in U.S. dollars. SolutionWe need to multiply each revenue figure by 0.65. Let Abe the matrix of rev- enue figures in Canadian dollars: A=

140.00 105.00

30.00 36.00

96.00 48.00

The revenue Þgures in U.S. dollars are then given by the scalar multiple

0.65A=0.65

140.00 105.00

30.00 36.00

96.00 48.00

91.00 68.25

19.50 23.40

62.40 31.20

In other words, in U.S. dollars, $91 worth of wiper blades was sold in Vancouver, $68.25 worth of wiper blades was sold in Quebec, and so on. using Technology

See the Technology Guides at the

end of the chapter to see how to compute scalar multiples using a

TI-83/84or Excel.Alternatively,

go to the online Matrix Algebra

Tool at

Chapter 3

?Tools ?Matrix Algebra Tool

There, enter the January sales in

U.S. Dollars:

A=[140, 105

30, 36

quotesdbs_dbs11.pdfusesText_17