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3 Matrix Algebra and Applications
3.1 Matrix Addition and
Scalar Multiplication
3.2 Matrix Multiplication CASE STUDY The Japanese Economy
3.3 Matrix Inversion A senator walks into your cubicle in the Congressional Budget Office. “Look here,” she says,
3.4 Game Theory “I don’t see why the Japanese trade representative is getting so upset with my proposal to cut
3.5 Input-Output Models 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
Key Concepts input from the mining sector. How can our cutting down demand for finance and insurance
Review Exercises hurt mining?” How should you respond?
Case Study Exercises
Technology Guides
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
173
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174 Chapter 3 Matrix Algebra and Applications
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.
3.1 Matrix Addition and Scalar Multiplication
Let’s start by formally defining what a matrix is and introducing some basic terms.
Matrix, Dimension, and Entries
An m×nmatrixAis a rectangular array of real numbers with m rows and n columns.
We refer to m and n as the dimensions of 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.
quick Examples
A= 2012×3
1. 33 −22 0 is a matrix because it has 2 rows and 3 columns.
23
10 44
B = 4×2
2. is a matrix because it has 4 rows and 2 columns.
−13
83
−22
The entries of A are 2, 0, 1, 33, , and 0. The entries of B are the numbers 2, 3, 10,
−1
44, , 3, 8, and 3.
Hint:Remember that the number of rows is given first and the number of columns second.
An easy way to remember this is to think of the acronym “RC” for “Row then Column.”
Referring to the Entries of a Matrix
There is a systematic way of referring to particular entries in a matrix. If i and j are num-
bers, then the entry in the ith row and jth column of the matrix A is called the ijth entry
a A
of A. We usually write this entry as ij or ij. (If the matrix was called B, we would
ij b B
write its th entry as ij or ij.) Notice that this follows the “RC” convention: The row
number is specified first and the column number second.
quick Example
A= 201
With 33 −22 0 ,
a13 = 1 First row, third column
a21 = 33 Second row, first column
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3.1 Matrix Addition and Scalar Multiplication 175
using Technology According to the labeling convention, the entries of the matrix A above are
See the Technology Guides at A=a11 a12 a13
the end of the chapter to see a21 a22 a23
how matrices are entered and m×n
used in a TI-83/84 or Excel. For In general, the matrix A has its entries labeled as follows:
the authors’ web-based utility, a11 a12 a13 ... a1n
follow: a21 a22 a23 ... a2n
Chapter 3 A= . . . . .
. . . .. .
Tools . . . .
Matrix Algebra Tool am1 am2 am3 ... amn
There you will find a computa- We say that two matrices A and B are equal if they have the same dimensions and
3×4 3×5
tional tool that allows you to do the corresponding entries are equal. Note that a matrix can never equal a
matrix algebra. Use the following matrix because they do not have the same dimensions.
format to enter the matrix A on
the previous page (spaces are
optional): Example 1 Matrix Equality
A=[2, 0,1
79x 790
33, −22, 0] Let A = and B = . Find the values of x and y such
0 −1 y+1 0 −111
To display the matrix A, type A in that A = B.
the formula box and press
“Compute.” Solution For the two matrices to be equal, we must have corresponding entries equal, so
x =0 a13 = b13
y + 1 = 11 or y = 10 a23 = b23
Before we go on... Note in Example 1 that the matrix equation
+
79x = 790
0 −1 y+1 0 −111
is really six equations in one: 7 = 7, 9 = 9, x = 0, 0 = 0, −1 =−1. and y + 1 = 11. We
■
used only the two that were interesting.
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 matrix or column vector.A matrix with the same num-
ber of rows as columns is called a square matrix.
quick Examples 1×5 C =[ ]
The matrix 3 −401−11 is a row matrix.
2
10
4×1 D=
The matrix −1 is a column matrix.
8
1 −20
3×3 E =
The matrix is a square matrix.
014
−4321
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176 Chapter 3 Matrix Algebra and Applications
Matrix Addition and Subtraction
The first matrix operations we discuss are matrix addition and subtraction. The rules for
these operations are simple.
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 A and B are m × n matrices, then A + B and
A−B m×n
are the matrices whose entries are given by:
(A+B) = A +B ij ij
ij ij ij th entry of the sum = sum of the th entries
(A−B) = A −B ij ij
ij ij ij th entry of the difference = difference of the th entries
Visualizing Matrix Addition
2 −3 11 3 −2
+ =
10 −21 −11
quick Examples 2 −3 9 −5 11 −8
1. + = Corresponding entries added
10 013 113
−13 −13 −26
2 −3 9 −5 −72
2. − = Corresponding entries subtracted
10 013 1 −13
−13 −13 00
Example 2 Sales
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:
January Sales
Vancouver Quebec
Wiper Blades 20 15
Cleaning Fluid (bottles) 10 12
FloorMats 84
February Sales
Vancouver Quebec
Wiper Blades 23 12
Cleaning Fluid (bottles) 812
FloorMats 45
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