Stewart J. College Algebra: Concepts and Contexts

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604 CHAPTER 7

■

Systems of Equations and Data in Categories

(a) Express each entry in the matrix in Example 1 as a proportion of the column

total. Name the resulting matrix P.

(b) What does the (1, 1) entry in P represent? What about the (3, 2) entry?

example

Categorical Data and Proportions

Brown

Blue

Green

Eye

color

example

Tabulating Categorical Data

The results of the class survey described above are tabulated in the following data matrix.

Hair color

Dark Blond Red

(a) How many students have dark hair and blue eyes?

(b) How many students have dark hair?

(d) How many students were surveyed?

Solution

(a) This is the entry in the “dark hair” column and the “blue eyes” row (the (2, 1)

entry in matrix A). We see that there are 0 students with dark hair and blue eyes.

(b) We add the numbers in the “dark hair” column.

There are 12 students with dark hair.

(d) We add all the entries in the matrix.

Twenty-six students were surveyed.

■ NOW TRY EXERCISE 7 ■

It is often more convenient to make a matrix in which the entries are proportions

with specific characteristics rather than the actual number of individuals. For exam-

ple, we can report the proportion of dark-haired students who have brown eyes, the

proportion of red-haired students with green eyes, and so on. To do this, we first cal-

culate the total number in each column of the matrix.

Hair color

Dark Blond Red

12 10 4 Column totals

A = £

9 3 0

0 6 3

3 1 1

TTT

9 + 3 + 0 + 0 + 6 + 3 + 3 + 1 + 1 = 26

0 + 6 + 3 = 9

9 + 0 + 3 = 12

A = £

9 3 0

0 6 3

3 1 1

TTT

Brown

Blue

Greend

Eye

color

SECTION 7.4

■

Matrices and Data in Categories 605

■ Adding Matrices

If we make the same hair and eye color survey with a different group of students, we

can combine the results into one large survey using matrix addition. We add two ma-

trices of the same dimension by adding corresponding entries.

example

Adding Matrices

A class of history students is surveyed as to hair color (dark, blond, red) and eye

color (brown, blue, green). The results are tabulated in the following data matrix.

Hair color

Dark Blond Red

(a) To combine the results of this survey and the one in Example 1, add matrix B

and matrix A (of Example 1).

(b) What is the total number of students (from both surveys) who have blond hair

and blue eyes?

Solution

(a) We add matrices A and B to combine the results of the two surveys. The total

in each category is simply the sum of the corresponding entries in matrices

A and B.

A + B = £

930

063

311

§+ £

12 4 1

2104

0 22

§= £

21 7 1

2167

3 33

B = £

12 4 1

2 10 4

0 2 2

TTT

Brown

Blue

Greend

Eye

color

Solution

(a) The total of the first column of matrix A is 12. So the entries in the first

column of matrix P are

The other entries in matrix P are calculated similarly.

Hair color

Dark Blond Red

(b) The (1, 1) entry tells us that of the students with dark hair, 75% have brown

eyes. The (3, 2) entry tells us that of the students with blond hair, 10% have

green eyes.

■ NOW TRY EXERCISE 11 ■

P = £

0.75 0.30 0.00

0.00 0.60 0.75

0.25 0.10 0.25

TTT

= 0.75

= 0.00

= 0.25

Brown

Blue

Green

Eye

color

606 CHAPTER 7

■

Systems of Equations and Data in Categories

example

Scalar Multiplication of a Matrix

The following questions refer to matrix A of Example 1.

(a) Find the scalar product 3A.

(b) In this new data matrix, what is the proportion of the students with dark hair

who have brown eyes?

Solution

(a) To perform the scalar multiplication, we multiply each entry in the data

matrix by 3.

(b) The total in the “dark hair” column is . So the proportion of

students with dark hair who have brown eyes is

Notice that this is the same proportion as in the original matrix. You can check that

all the other proportions are the same as in the original matrix (see Example 2).

■ NOW TRY EXERCISE 15 ■

= 0.75

27 + 0 + 9 = 36

3A = 3 £

930

063

311

§= £

27 9 0

0189

933

■ Multiplying a Matrix Times a Column Matrix

We define what it means to multiply a matrix A times a column matrix

C. The product AC is a column matrix that is obtained as follows.3 * 1

3 * 13 * 3

a b c

££§§§

The first entry in the product matrix ( ) is calculated by multiplying the entries in the

first row of matrix A by the entries in the column of matrix C and adding the results.

++=

The other entries in the product matrix are calculated similarly, as is illustrated in the

next example.

■ Scalar Multiplication of Matrices

Suppose we double each entry in the data matrix in Example 1. This would not

change any of the proportions calculated in Example 2. Doubling each entry in a data

matrix simply “scales up” the size of the data. In general, multiplying a matrix by a

constant is called scalar multiplication. The next example explains the notation.

(b) From the (2, 2) entry in the matrix , we see that 16 students have blond

hair and blue eyes.

■ NOW TRY EXERCISE 13 ■

A + B

SECTION 7.4

■

Matrices and Data in Categories 607

example

Multiplying a Matrix Times a Column Matrix

An organic farmer sells produce at an open-air market three days a week: Thursdays,

Fridays, and Saturdays. He sell oranges, broccoli, and beans. Matrix A tabulates the

number of pounds of produce he sold in a certain week. Matrix C gives the prices per

pound (in dollars) he charged that week.

(a) Find the product AC.

(b) What was the total revenue on Friday?

Produce

Oranges Broccoli Beans

Price

C = £

0.90

1.20

1.50

§A = £

50 20 10

70 35 30

45 15 25

TTT

Thursday

Friday

Saturday

Day

Oranges

Broccoli

Beans

Produce

Pounds of

oranges

Pounds of

broccoli

Pounds of

beans

Price per

pound

Price per

pound

Price per

pound

7.4 Exercises

Fundamentals

1–4 ■ Determine whether the data described is numerical or categorical data.

1. The height of each child in a third grade class.

2. The number of laundry items tabulated by color (white, pastel, dark).

3. The number of gulls seen at a beach on a given day, tabulated by species (California,

Western, Ring-billed) and maturity (immature, juvenile, adult).

4. The number of birds observed at a bird feeder each day.

CONCEPTS

Solution

(a) The entries in the product matrix are obtained as explained above.

(b) The (2, 1) entry of matrix AC is calculated as follows:

This means that the (2, 1) entry of matrix AC is the sum of Friday’s revenue

from oranges, broccoli, and beans. So the total revenue on Friday was $150.00.

the week.

So the total revenue from all three days was $330.00.

■ NOW TRY EXERCISE 17 ■

84.00 + 150.00 + 96.00 = 330.00

70 * 0.90 + 35 * 1.20 + 30 * 1.50 = 150.00

AC = £

50 20 10

70 35 30

45 15 25

§£

0.90

1.20

1.50

§= £

0.90 + 20

1.20 + 10

1.50

0.90 + 35

1.20 + 30

1.50

0.90 + 15

1.20 + 25

1.50

§= £

84.00

150.00

96.00

608 CHAPTER 7

■

Systems of Equations and Data in Categories

5–6

■ Organize the given set of categorical data into a data matrix.

5. Education and Income A women’s club takes a survey to determine the education

and income of its members. Of the women with no postsecondary education, 5 have a

yearly income below $50,000, 2 have an income between $50,000 and $100,000, and 0

has an income above $100,000. Of the women with up to four years of postsecondary

education, 2 have a yearly income below $50,000, 10 have an income between $50,000

and $100,000, and 4 have an income above $100,000. Of the women with more than

four years of postsecondary education, 0 has a yearly income below $50,000, 3 have an

income between $50,000 and $100,000, and 2 have an income above $100,000.

Postsecondary education (years)

None 0–4 More than 4

6. Exam Scores A physics class takes a survey of the number of hours the students slept

before the exam and the exam score. Of the students who slept less than four hours, six

students got a score below 60, three got a score between 60 and 80, and one got a score

above 80. Of the students who slept between four and seven hours, two students got a

score below 60, seven got a score between 60 and 80, and four got a score above 80. Of

the students who slept more than seven hours, three students got a score below 60, nine

got a score between 60 and 80, and six got a score above 80.

Hours of sleep

Less than 4 4–7 More than 7

7. Service Ratings A Cincinnati car dealership conducted a customer satisfaction

survey, and the results are tabulated in the data matrix A.

(a) How many of the customers that had a bill of less than $50 gave a fair service

rating?

(b) How many customers had a bill above $100?

(d) How many customers were surveyed?

Amount of bill

Less than More than

$50 $50–100 $100

8. Service Ratings An Akron car dealership conducted a customer satisfaction survey,

and the results are tabulated in the data matrix B.

(a) How many of the customers who had a bill of more than $100 gave an excellent

service rating?

(b) How many customers had a bill less than $50?

(d) How many customers were surveyed?

A = £

10 8 5

5 7 6

1 3 5

TTT

A = £

TTT

A = £

TTT

CONTEXTS

Less than $50,000

$50,000–$100,000

More than $100,000d

Income

level

Below 60

60–80

Above 80d

Exam

score

Excellent

Good

Fair

Service

rating

SECTION 7.4

■

Matrices and Data in Categories 609

Amount of bill

Less than More than

$50 $50–100 $100

9. Health Insurance Residents of the small town of Springfield were surveyed to obtain

information on the type of health insurance and the number of emergency room visits

within the past year. The data are tabulated in matrix C.

(a) How many of those surveyed are members of an HMO and had four or more

emergency visits in the past year?

(b) How many of those surveyed had no health insurance?

(d) How many people were surveyed?

Type of insurance

HMO Non-HMO None

10. Health Insurance Residents of a Chicago suburb were surveyed to obtain

information on the type of health insurance and the number of emergency room visits

within the past year. The data are tabulated in matrix D.

(a) How many of those surveyed are members of an HMO and had four or more

emergency visits in the past year?

(b) How many of those surveyed had no health insurance?

(d) How many people were surveyed?

Type of insurance

HMO Non-HMO None

11. Sales A tasting booth at Joe’s Specialty Foods offers pesto pizza on Monday, spinach

ravioli on Tuesday, and macaroni and cheese on Wednesday. The sales distribution for

these products is tabulated in matrix A.

(a) Express each entry in the matrix as a proportion of the column total. Call the

resulting matrix P.

(b) What does the (1, 3) entry in P represent? What about the (3, 3) entry?

Specialty food

Pesto Spinach Macaroni and

pizza ravioli cheese

A = £

50 20 15

40 75 20

35 60 100

TTT

D = £

76 82 54

32 49 79

21 32 87

TTT

C = £

11 8 3

5 6 5

3 2 7

TTT

B = £

9 15 12

8 10 6

5 2 7

TTT

Excellent

Good

Faird

Service

rating

0–1

2–3

4 or mored

Emergency

visits

0–1

2–3

4 or more

Emergency

visits

Monday

Tuesday

Wednesday

Day

610 CHAPTER 7

■

Systems of Equations and Data in Categories

12. Politics A history class makes a survey of the students in the class, and the results are

tabulated in matrix B.

(a) Express each entry in the matrix as a proportion of the column total. Call the

resulting matrix P.

(b) What does the (2, 2) entry in P represent? What about the (1, 3) entry?

Political Affiliation

Democrat Republican Independent

13. Service Ratings The following questions refer to matrices A and B from Exercises

7 and 8.

(a) Find a data matrix for the results of the survey in Exercise 7 combined with the

results of the survey in Exercise 8.

(b) What is the total number of customers (from both surveys) who had a bill between

$50 and $100 and gave an excellent service rating?

14. Health Insurance The following questions refer to matrices C and D from Exercises

9 and 10.

(a) Find a data matrix for the results of the survey in Exercise 9 combined with the

results of the survey in Exercise 10.

(b) What is the total number of those surveyed (from both surveys) that had no health

insurance and had four or more emergency visits in the past year?

15. Sales The following questions refer to matrix A in Exercise 11.

(a) Find the scalar product 4A.

(b) In this new data matrix, what proportion of the pesto pizza sales occurs on

Wednesday?

16. Politics The following questions refer to matrix B in Exercise 12.

(a) Find the scalar product 5B.

(b) In this new data matrix, what proportion of the female students are Republican?

17. Car-Manufacturing Profits A specialty car manufacturer has plants in Auburn,

Biloxi, and Chattanooga. Three models are produced, with daily production given in

matrix A. The profit (in dollars) per car is tabulated by model in matrix C.

(a) Find the product matrix AC.

(b) Assuming that all cars produced are sold, what is the daily profit from the Biloxi

plant?

Cars produced each day

Model K Model R Model W

Profit

C = £

1000

2000

1500

A = £

12 10 0

4 4 20

8 9 12

TTT

B = c

7 9 3

8 10 2

TTT

Male

Femaled

Gender

Auburn

Biloxi

Chattanoogad

City

Model K

Model R

Model W

SECTION 7.5

■

Matrix Operations: Getting Information from Data 611

18. Sandwich Revenue A sandwich shop sells three types of sandwiches: Bite-size,

Foot-long, and Mega-long. The sales distribution is given in matrix A. The price (in

dollars) per sandwich is tabulated in matrix C.

(a) Find the product matrix AC.

(b) What is the total revenue on Friday?

Sandwich

Bite-size Foot-long Mega-long

Price

C = £

4.00

5.00

9.00

A = £

15 30 20

10 50 30

20 45 25

TTT

Friday

Saturday

Sunday

Day

Bite-size

Foot-long

Mega-longd

7.5 Matrix Operations: Getting Information from Data

■

Addition, Subtraction, and Scalar Multiplication

■

Multiplying Matrices

■

Getting Information from Categorical Data

IN THIS SECTION… we study addition, subtraction, scalar multiplication, and

multiplication of matrices. We’ll see how these operations allow us to get useful information

from categorical data.

■ Addition, Subtraction, and Scalar Multiplication

Two matrices can be added or subtracted if they have the same dimension. As we saw

in Section 7.4, we add matrices by adding corresponding entries. To multiply a ma-

trix by a number (a scalar), we multiply each element of the matrix by that number.

example

Algebraic Operations on Matrices

Let

Perform the indicated operation, or explain why it cannot be performed.

(a) (b) (c) (d) 5AC + AC - DA + B

A = £

2 - 3

§B = £

1 0

- 3 1

2 2

§C = c

7 - 30

015

dD = c

60- 6

81 9

612 CHAPTER 7

■

Systems of Equations and Data in Categories

Solution

(a) Since A and B are matrices with the same dimension , we can add

them.

(b) Since C and D are matrices with the same dimension , we can

subtract them.

(d) To find 5A, we multiply each entry in A by 5.

■ NOW TRY EXERCISES 7 AND 15 ■

5A = 5 £

2 - 3

0 5

§= £

10 - 15

0 25

C + A

C - D = c

7 - 30

0 15

d- c

60- 6

81 9

d= c

1 - 3 6

- 8 0 - 4

12 * 3 2

A + B = £

2 - 3

§+ £

- 31

§= £

3 - 3

- 36

13 * 2 2

■ Matrix Multiplication

In Section 7.4 we learned how to multiply a matrix times a column matrix. We now

describe how to use this to define matrix multiplication in general (when the second

factor has more than just one column).

First, the product AB of two matrices A and B is defined only when the number

of columns in A is equal to the number of rows in B. This means that if we write their

dimensions side by side, the two inner numbers must match.

Matrices

Dimensions

Columns in A Rows in B

If the dimensions of A and B match in this way, then the product AB is a matrix of

dimension . Before describing the procedure for obtaining the elements of AB,

we define the inner product of a row of A and a column of B.

If is a row of A and if is a column of B, then their

inner product is the number . For example taking the inner

product, we have

32 -1 0 44≥

- 3

¥= 2

5 + 1- 12

6 + 0

1- 32+ 4

2 = 12

+ a

≥

¥3a

a



m * k

n * km * n

SECTION 7.5

■

Matrix Operations: Getting Information from Data 613

If A is an matrix and B is an matrix, then their product AB is an

matrix C.

The entries of the matrix C are calculated as follows: The entry in the ith row

and jth column of the matrix C is the inner product of the ith row of A and the

jth column of B.

AB = C

m * k

n * km * n

The following diagram shows that the entry in the ith row and jth column of

the product matrix AB is obtained by multiplying the entries in the ith row of A by

the corresponding entries in the jth column of B and then adding the results.

ⵧ

d £

ⵧ

§= c c

example

Multiplying Matrices

Calculate, if possible, the products AB and BA.

Solution

Since matrix A has dimension and matrix B has dimension , the product

AB is defined and has dimension . We can write

where the question marks must be filled in by using the rule defining the product of

two matrices. If we define , then the entry is the inner product of

the first row of A and the first column of B.

1 3

- 10

- 1 52

0 47

1

1- 12+ 3

0 =-1

C = AB = 3c

AB = c

- 10

- 152

047

d= c

2 * 3

2 * 32 * 2

A = c

- 10

dB = c

- 152

047

2 * 22 * 3

Matrix Multiplication

ith row of A

jth column of B Entry in ith row and

jth column of AB

The next example illustrates the process.

Inner numbers match,

so product is defined.

Outer numbers give dimension

of product: .

2 * 3

? ? ?