# Analyzing Matrices

Matrices can be used to organize all sorts of data, not just sales data. In
this investigation,

you will explore three different situations in which matrices are used to

help make sense of data.

Archeologists study ancient people and their cultures. One way they study

these people is by exploring sites where they have lived and analyzing objects

which they have made. Archeologists use

matrices to classify and then compare the

objects they find at various archeological

sites. For example, suppose that pieces of

pottery are found at five different sites. The

pottery has certain characteristics: it is either

glazed or not glazed, ornamented or not, colored

or natural, thin or thick.

1. Information about the characteristics of the

pottery at all five sites is organized in the

matrix below. A “1” means the pottery has

the characteristic and a “0” means it does

not have the characteristic.

**Pottery Characteristics**

Glaze | Orn | Color | Thin | |

Site A | 0 | 1 | 0 | 0 |

Site B | 1 | 0 | 0 | 0 |

Site C | 1 | 0 | 1 | 0 |

Site D | 1 | 1 | 1 | 1 |

Site E | 0 | 1 | 1 | 1 |

a. What does it mean for pottery to be “glazed”? “Ornamented”?

b. What does the “1” in the third row and the first column mean?

c. Is the pottery at site E thick or thin?

d. Which site has pottery that is glazed and thick, but is not ornamented or

colored?

e. How many of the sites had glazed pottery? Explain how you used the rows

or columns of the matrix to answer the question.

2. You can use the matrix to determine how much the pottery differs between

sites. For example, the pottery found at sites A and B differ on exactly two

characteristics—glaze and ornamentation. So you can say that the degree of

difference between the pottery at sites A and B is 2.

a. Explain why the degree of difference between pottery at sites A and C is
3.

b. Find the degree of difference between sites D and E.

3. You can build a new matrix that summarizes all the degree of difference
information.

a. What number would best describe the difference between site A and site A?

b. What number should be placed in the third row, fourth column? What does

this number tell you about the pottery at these two sites ?

c. Complete the degree of difference matrix shown below.

**Degree of Difference**

d. Describe at least one pattern you see in the degree of difference matrix.

4. Archeologists want to learn about the civilizations that existed at the
sites. For

instance, they would like to know whether different sites represent different

civilizations and whether one civilization was more advanced than another. A

lot of evidence is needed to make such decisions. However, make some conjectures

based just on the pottery data in the matrices from Activities 1 and 3.

a. Find two sites that you think might be from the same civilization. Explain

how the pottery evidence supports your choice.

b. Find two sites that you think might be from different civilizations. Give an

argument defending your choice.

c. Give an argument supporting the claim that the civilization at site D was

more advanced than the others. What assumptions are you making about

what it means for a civilization to be “advanced”?

5. Matrices also are used frequently by sociologists in their study of social
relations.

For example, a sociologist may be studying friendship and trust among

five classmates at a certain high school. The classmates are asked to indicate

with whom they would like to go to a movie and to whom they would loan $5.

Their responses are summarized in the following two matrices. (“1” means

“yes” and “0” means “no”.) Each matrix is read from row to column. For

example, the “1” in the first row and fourth column of the movie matrix means

that student A would like to go to a movie with student D.

**Movie Matrix**

Would Like to Go to a Movie |

**Loan Matrix**

Would Loan Money |

a. Would student Alike to go to a movie with student B? Would student B like

to go to a movie with student A?

b. With whom would student D like to go to a movie?

c. What does the “0” in the fourth row, third column of the loan matrix mean?

d. To whom would student A loan $5?

e. A square matrix has the same number of rows and columns. The main

diagonal of a square matrix is the diagonal line of entries running from the

top left corner to the bottom right corner. Why do you think there are zeroes

for each entry in the main diagonals of the matrices above?

6. Now consider further information conveyed by the matrices above.

a. Explain why the movie matrix could be used to describe friendship, while

the loan matrix could describe trust.

b. Write two interesting statements about friendship and trust among these

five students, based on the information in the matrices.

7. Discuss with your group how you can use the rows or columns of the movie

and loan matrices to answer the following questions.

a. How many students does student C name as friends?

b. How many students name student C as a friend?

c. Who seems to be the most trustworthy student?

d. Who seems to be the most popular student?

**Checkpoint**

In the previous investigations , you performed computations on the row or

column entries of a matrix to get useful information about the situation

being modeled. Give three examples from your analysis of pottery, shoe

sales, or friendship and trust showing how you operated on the entries of

the given matrix to get additional information . For each example, describe

the situation, the computation, and the information obtained.

**Be prepared to share your group’s examples with the class.**

**On Your Own**

In 2001, the University of Notre Dame won the NCAA women’s basketball

championship. They completed a 34-2 season by defeating Purdue University in

the championship game.

a. Senior Ruth Riley was the high scorer in the championship game for Notre

Dame, with 28 points. Teammate Erika Haney scored 13 points. What other

factors besides points scored should be taken into account when deciding

which player contributed most to the victory?

The matrix below shows some of the non-shooting performance statistics for the

seven Notre Dame players who played in the championship game.

**Non-Shooting Performance Statistics**

Assists | Steals | Rebounds | Blocked Shots |
Turn-overs | Fouls | |

Haney | 2 | 1 | 5 | 1 | 0 | 3 |

Siemon | 6 | 0 | 9 | 0 | 7 | 3 |

Riley | 1 | 0 | 13 | 7 | 3 | 3 |

Ratay | 2 | 1 | 4 | 0 | 1 | 4 |

Ivey | 4 | 6 | 5 | 1 | 4 | 0 |

Joyce | 1 | 0 | 0 | 0 | 0 | 1 |

Barksdale | 0 | 0 | 2 | 2 | 0 | 0 |

b. A “turnover” is when an action (other than fouling or scoring a basket)
gives

the other team control of the ball. How many turnovers did Ratay have?

c. What is a “rebound”? How many rebounds were made by the Notre Dame

players during the game?

d. Which of the performance factors do you think are positive, that is, they
contribute

toward winning the game? Which performance factors do you think are

negative? For the factors that are negative, change the entries in the matrix to

negative numbers .

e. Which player had the largest number of positive performance actions?

f. Describe how you could give a “non-shooting performance score” to each

player. The score should include both positive and negative factors. Which

player do you think contributed most to the game in the area of non-shooting

performance? Explain your choice.

## Combining Matrices

You have seen that a matrix can be used to store and organize data. You also
have

seen that you can operate on the numbers in the rows or columns of a matrix to

get additional information and draw conclusions about the data. Sometimes it is

useful to combine matrices, as you will see in the next two situations.

1. Shown below are the movie and loan matrices you analyzed in the previous

investigation. Study both matrices to see how friendship and trust are related

in this group of five students.

**Movie Matrix**

Would Like to Go to a Movie |

**Loan Matrix**

Would Loan Money |

a. Who does student A consider friends and yet does not trust enough to

loan $5?

b. Do you think it is reasonable that a student could have a friend who he or

she does not trust enough to loan $5?

c. Who does student B trust and yet does not consider them to be friends? Do

you know someone who you trust but who is not a friend?

d. Who does student D trust and also consider to be friends?

2. A friend you trust is a trustworthy friend.

a. Combine the movie and loan matrices to construct a new matrix that shows

who each of the five students considers to be a trustworthy friend.

b. Write down a systematic procedure explaining how to construct the

trustworthy-friend matrix.

c. Compare your procedure with that of other groups.

d. Write two interesting observations about the information in this new matrix.

In the next situation, you will explore other ways of combining matrices and
learn

some of the ways in which matrix operations are used in business and industry.

In accordance with demand from American consumers, the Big Three automakers

in the U.S.—General Motors, Ford, and Chrysler—have been producing

many small cars. The matrices below show the production data for 1998 and

1999. Data are given in hundred thousands of small cars produced each quarter

(a quarter is three months).

**1998 Small Auto Production**

**1999 Small Auto Production**

3. Analyze the two matrices above.

a. How many cars are represented by the first entry in the matrix for 1998?

b. According to these data, how many small cars were produced by Chrysler

in the second quarter of 1999?

c. Explain the meaning of the entry in the third row and second column of the

1998 matrix.

4. Additional information can be derived by combining the two matrices.

a. According to these data, by how much did first-quarter, small-car production

increase for GM from 1998 to 1999?

b. Construct a new matrix with the same row and column labels that shows

how much small-car production increased from 1998 to 1999 for each quarter

and each manufacturer. Explain any trends or unusual entries.

5. Suppose that the auto industry projected a 10% increase in small-car
production

from 1999 to 2000, over all quarters and all manufacturers. Construct a

matrix that shows the projected 2000 production figures for each quarter and

each manufacturer, based on the data given.

6. Construct a matrix with the same row and column labels as the given
matrices

that shows the total number of small cars produced over the two-year period

1998–99.

**Checkpoint**

In this investigation, you explored how combining two matrices or multiplying

each entry of a matrix by a number helped to derive new information.

**a** Which of the activities about small-car production involved
combining

matrices by adding corresponding entries?

**b** Which of the activities about small-car production involved combining

matrices by subtracting corresponding entries?

**c** Which of the activities about small-car production involved multiplying

each entry of a matrix by a number?

**d** Consider all the situations you have analyzed so far. What other
operations

have you performed on matrices?

**Be prepared to explain your group’s selections to the
entire class.**

Several of the operations you have performed on matrices in this
investigation

are commonly used and have been given standard names. To add matrices

means to combine them by adding corresponding entries. Thus, when adding two

matrices you add the entry in the first row and first column from one matrix to

the entry in that same position in the other matrix, and then do likewise for
the

entries in each of the other positions. If A represents one matrix and B
represents

another matrix, then A + B represents the matrix found by adding the matrices

entry by entry. Subtracting matrices (A – B) is just like adding matrices,
except

you subtract the corresponding entries instead of adding them. Multiplying a

matrix, B, by a number, k, means to multiply each entry in the matrix by that

number and is represented by kB.

**On Your Own**

Below are two matrices showing the 1999 and 1998 passing statistics for three

top NFL quarterbacks. (“Att” is an abbreviation for “passes attempted”; “Comp”

refers to passes completed; “TD” refers to passes thrown for a touchdown; and

“Int” refers to passes that were intercepted .)

**1999 Passing Statistics**

**1998 Passing Statistics**

Let A represent the 1999 matrix and B represent the 1998 matrix.

a. Compute A + B. What does A + B tell you about the passing performance of

the three quarterbacks?

b. Compute A – B. What does A – B tell you about the passing performance of

the three quarterbacks?

c. Compute B – A.

•How do the numbers in B – A differ from the numbers in A – B?

•What does a negative number in the TDs column of A – B tell you about the

trend in touchdown passes from 1998 to 1999?

•What does a negative number in the TDs column of B – A tell you about the

trend in touchdown passes from 1998 to 1999?

d. Compute 1/2A. What could 1/2A mean for this situation?

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