Matrices Part One:
How can you construct and use a rectangular array of numbers (a matrix) to organize, display, and analyze
information?
How can you interpret and operate on a matrix to reveal useful information in analyzing data?
Shoe Sales
1. The information provided in the matrix shows monthly sales for three popular athletic shoes at a
local athletic store.
a. The size of a matrix is written m x n where m is the number of rows and n is the number of
columns. What is the size of the Monthly Sales matrix?
b. Which month had the highest number of total shoe sales? How did you arrive at this answer?
c. Which brand was the “best seller” for this store for this year? How did you arrive at this
answer?
Sociology
2. Classmates at a local high school were asked who you would go to a movie with and who would
you loan money too. Each matrix presented in read from row to column. “1” means yes and “0”
means no. For example, the “1”in the first row and fourth column of the movie matrix means
student A would like to go to a movie with student D.
a. Would student A go to a move with student B? Would student B go to a movie with student A?
b. With whom would student D like to go to a movie with?
c. What does a “0” in the fourth row, third column of the loan matrix indicate?
d. To whom will student A loan money to?
Adapted from Core Plus Mathematics Project, Course 2, Unit 2 © 2008
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. What
values are on the main diagonal for both the movie and loan matrix? Does this make sense, why?
Would some other representation be better? Justify your choice.
3. Use the information provided in the matrix to justify your responses to the following questions.
a. Using the movie matrix, determine who is “friends.” Who is the most popular student?
Justify your choices.
b. Using the loan matrix, determine who is “trustworthy.” Who is the most trustworthy?
Justify your choices.
Vehicle Production
4. The two matrices show the production of passenger vechiles (PV), light commercial vehicles (LCV),
and heavy trucks (HT) in three regions for 2004 and 2005. The numbers shown are in thousands.
a. How many vehicles are represented by the entry in the first row and first column of the
2004 matrix?
b. According to these data, how many heavy trucks were produced in South America in 2005?
c. Explain the meaning of the entry in the third row and second column of the 2004 matrix.
d. By how much did passenger vehicle production increase in South America from 2004 to
2005?
5. Construct a new matrix with the same row and column titles that shows how much vehicle
production changed from 2004 to 2005. What is surprising about this new matrix?
Adapted from Core Plus Mathematics Project, Course 2, Unit 2 © 2008
6. Construct a new matrix with the same row and column titles that shows total production in 2004
and 2005.
7. The auto industry predicted that production in 2006 would be a 10% increase from 2005.
Construct a new matrix with same row and column titles that would show the 2006 projections
based off of this prediction.
Football
8. The two matrices show the 2004 and 2005 regular season
passing statistics for the top three NFL quarterbacks. “Att” is
an abbreviation for passes attempted; “cmp” refers to passed
completed; “td” refers to passes thrown for a touchdown; and
“int” refers to intercepted passes.
Let A represent the 2005 matrix and B represent the 2004
matrix.
a. Compute A + B. What does this 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?What does a negative number in the “cmp” column of A – B tell you about the
trend in completions from 2004 to 2005?
c. Compute B – A. How do the numbers in B – A differ from A – B? What does a negative
number in the “cmp” column of B – A tell you about the trend in completions from 2004 to
2005?
Adapted from Core Plus Mathematics Project, Course 2, Unit 2 © 2008
d. Compute (1/2)A. What does this mean in the situation?
9. Practice these matrix operations:
Matrices Part Two:
How can you multiple matrices to make predictions in analyzing data trends?
Shoe Business
1. Each entry in the brand-switching matrix is
the percentage of customers who will buy a
certain brand of shoe, given the brand they
currently own. We can use this information to
make estimates about future purchases. For
example, the entry in the second row and third
column means that we can estimate that 30%
of the people who now own Reebok will buy
Fila as their next pair of shoes.
a. What percentage of people who now own Nike will buy Reebok next?
b. What percentage of people who now own Reebok will stay with Reebok on their next shoe
purchase?
c. Based on these data, to which shoe brand do you think the customers are most loyal?
Explain.
Adapted from Core Plus Mathematics Project, Course 2, Unit 2 © 2008
2. Assume that buyers purchase a new pair of shoes every year; suppose that this year 700 people
bought Nike, 500 people bought Reebok, and 400 people bought Fila. Answer the following
questions by using those assumptions and the brand-switching matrix above.
a. How many people will buy Nike next year?
b. How many people will buy Reebok next year?
3. One way to answer the questions in Problem 2 is to use a new matrix operation. A one-row matrix
for this year’s numbers is written to the left of the brand-switching matrix. A one-row matrix is
often more simply called a row matrix.
a. Complete the computation below for the number of people who will buy Fila next year.
Future Fila buyers = 700*(_____) + 500*(_____) + 400*(_____) = ________
b. To which column of the brand-switching matrix do the numbers in the blanks correspond?
c. Which column of the brand-switching matrix can you combine with the row matrix to get
the number of people who will buy Reebok next year? Do this calculation to predict next
year’s Reebok buyers.
d. Which column of the brand-switching matrix can you combine with the row matrix to get
the number of people who will buy Nike next year? Do this calculation to predict next
year’s Nike buyers.
4. In #3 you learned a matrix multiplication procedure. The term matrix multiplication is a specific
procedure that applies both in procedure and term to matrices only. Practice that procedure on
these two problems.
Adapted from Core Plus Mathematics Project, Course 2, Unit 2 © 2008
School Dance Planning
5. To prepare for a dance, a school needs to rent 40 chairs, 3 large
tables, and 6 punch bowls. There are two rental shops nearby
that rent all of these items, but they have different prices as
shown in the matrix below. Prices shown are per item.
a. What is the size of the rental-prices matrix?
b. Create a row matrix based off the quantity of chairs, tables, and bowls needed.
c. Use matrix multiplication to find a matrix that shows the total cost of renting all equipment
from each of the two shops.
d. Who should you use as a rental service provider? Explain.
Sports Uniforms
6. A little league baseball teams are considering two suppliers for
their uniform purchase. The Cost per Uniform matrix shows the
total pricing options for the two local suppliers. Each team needs
a variety of different sizes, this is shown in the Quantity of
Uniforms matrix. The teams want to purchase all uniforms from
one supplier but need help in determining the best option that will
save the most money.
Use a similar method to multiplying a row matrix by another matrix to
complete the matrix multiplication problem.
a. The entries in the product matrix provide very useful information. Explain what the number in the
first row and second column of your product matrix means. What does the number in the second
row and third column mean?
Adapted from Core Plus Mathematics Project, Course 2, Unit 2 © 2008
b. Label each of your rows and each of the columns for the product matrix. Additionally, title this
new matrix appropriately so that readers can easily infer what the matrix displays.
c. Who should the little league use as a supplier? Justify your choice.
7. Call the cost matrix C and the quantity matrix Q. So far, you have multiplied C x Q. Try multiplying Q x
C. Can you do it? Explain why or why not?
Toy Production
8. A toy company in Seattle makes stuffed toys, including crabs, ducks, and cows. The owner designs the
toys, and then they are cut out, sewn, and stuffed by independent contractors. For the months of
September and October, each contractor agrees to make the number of stuffed toys shown in the
following matrix. Two of the contractors, Elise and Harvey, know from experience how many minutes
it takes them to make each type of toy, as shown in the matrix.
a. Use matrix multiplication to find a matrix that shows the total number of minutes each of the two
contractors will need in order to fulfill their contracts for each of the two months.
b. Convert the minute totals to total hours. What matrix operation could you use to do this
conversion?
Adapted from Core Plus Mathematics Project, Course 2, Unit 2 © 2008
9. Perform the following matrix multiplications:
Matrices Part Three:
How can you represent a vertex-edge graph with a matrix?
If you multiply a matrix by itself, what information do you get about the vertex-edge graph and the situation
represented by the graph?
Tournament Rankings
Powers of adjacency matrices give you information about certain lengths of paths in a corresponding vertex-edge
graph. This connection between graphs and matrices is useful for solving a variety of problems. For example, it is
often very difficult to rank players or teams in a tournament accurately and systematically. A vertex-edge graph
can give you a good picture of the status of the tournament. The corresponding adjacency matrix can help
determine the ranking of the players or teams. Consider the following tournament situation.
1. The second round of a city tennis tournament involved six girls, each of whom was to play every other girl.
However, the tournament was rained out after each girl had played only four matches. The results of play
were the following:
 Erina beat Keadra
 Akiko beat Julia
 Keadra beat Akiko and Julia
 Julia beta Erina and Maria
 Maria beta Erina, Cora and Akiko
 Cora beat Erina, Keadra, and Akiko.
Adapted from Core Plus Mathematics Project, Course 2, Unit 2 © 2008
a. Represent the status of the tournament by completing a copy of the diagraph and adjacency matrix
below.
b. Rank the girls as clearly as you can. Explain
c. If you did not use row sums, what additional information would row sums provide?
d. Compute the square of the adjacency matrix and discuss what the entries tell you about the
tournament. How could you use this information to help rank the girls?
e. Use further operations on the adjacency matrix to rank the players with no ties. Explain the ranking
system you used. Compare your method and results with others.
Adapted from Core Plus Mathematics Project, Course 2, Unit 2 © 2008
Social Dominance
2. In any groups of people, some are leaders and some are followers. This
relationship of leaders and followers is called social dominance. The
following diagraph shows social dominance within a group of five
people in the advertising agency. An arrow from one vertex to another
means that the first person is socially dominant (is the “leader”) over
the other.
a. Construct an adjacency matrix M for this digraph.
b. Using your adjacency matrix, identify an overall leader of this group. Can you rank, with no ties, all
five people in terms of social dominance?
c. Compute M3.
i. Explain what the “1” for the A-E entry means in terms of social dominance.
ii. Trace the paths in the digraph that correspond to the “2” for the C-A entry.
d. Use powers of your adjacency matrix and sums of rows to break any ties and rank the five people in
terms of social dominance.
e. What do the arrows between A, B, and E indicate about these three people? Explain how this could
be possible in a social group.
Adapted from Core Plus Mathematics Project, Course 2, Unit 2 © 2008