Download How is information organized in a matrix?

Question 1: How is information organized in a matrix?
When we use a matrix to organize data, all of the definitions we developed in Chapter 2
still apply. You’ll recall that a matrix can have any number of rows and columns and is
typically named with a capital letter. A matrix with m rows and n columns named A would
look like
 a11
a
A   21
 

 am1
a12
a22

am 2
a13  a1n 
a23  a2 n 
   

am 3  amn 
Notice that this matrix does not contain a dashed vertical line in front of the last column.
The dashed line is unique to augmented matrices and is used to separate the
coefficients from the constants. The dots in the matrix indicated a pattern in the matrix.
In this case, the dots indicate the arbitrary number of rows m and columns n in the
matrix.
The individual entries (also called elements) of the matrix are symbolized with
lowercase letters, like amn , and these symbols represent numbers. The subscript on the
lowercase letter indicates the location of the entry in the matrix. The symbol a23
represents the number in the second row, third column of the matrix.
The size of a matrix (also called the dimensions of the
matrix) is the number of rows and columns in a matrix. For
the matrix A with m rows and n columns, we would say the
size of the matrix is m x n (read m by n).
Several sizes of matrices are given special names. A matrix with the same number of
rows and column is called a square matrix. An example of a square matrix is the 2 x 2
matrix
2
1 3
0 7 


The exact number of rows and columns in a square matrix is not important, only the fact
that the number of rows and columns is the same.
Matrices with a single row or a single column are also given special names. Row
matrices like 1 0 3 or  4 2.5 7 12 are matrices with only a single row, but any
 3.1 
0.04 
2
 are matrices with any number
number of columns. Column matrices like   or 
 4 
 3


 10 
of rows, but a single column.
The size of a matrix is an important prerequisite in determining if two matrices are
equal.
Two matrices are equal if they have the same size and each
entry in one matrix is equal to the corresponding entry in the
other matrix.
Example 1
Matrix Terminology
The matrices
 1 1 2 
A   2 3 4 
 3 4 5 
 1 1 0 
B   2 3 4 
 3 4 5 
are 3 x 3 square matrices.
a. What is the value of the entry b32 ?
3
Solution The subscript on b32 refers to the entry in the third row, second
column of the matrix B. Therefore, b32  4 .
b. Is a23  b23 ?
Solution The subscripts on a23 and b23 refer to the corresponding
entries in the second row, third column of A and B. Since the entry in
that location is 4 in both matrices, a23  b23 and is equal to 4.
c. Is A  B ?
Solution For the matrices to be equal, they must have the same size
and each entry in A must be equal to the corresponding entry in B. Both
matrices are 3 x 3. In part b, we determined that the entries in the
second row, third column of each matrix were equal. However, a13 in
 1 1 2 
A   2 3 4 
 3 4 5 
and b13 in
 1 1 0 
B   2 3 4 
 3 4 5 
are not the same so A  B . This is in spite of the fact that every other
set of corresponding entries are equal.
Ed Magazine is a fictional magazine that publishes four issues each year. It has a loyal
base of subscribers and twice a year it conducts subscription drives for new
subscribers. At the same time they are acquiring new subscribers, the current
4
subscriber’s subscriptions are expiring. Some of these expiring subscriptions belong to
first time subscribers and others are long time subscribers who have renewed their
subscriptions in the past. The table below shows the numbers of new and expiring
subscribers by quarter.
Expiring Subscribers
New
Subscribers
First Time
Long Time
Quarter Ending 3/31
5200
6000
15000
Quarter Ending 6/30
3000
2000
2600
Quarter Ending 9/30
5600
6500
12000
Quarter Ending 12/31
2500
1500
3600
Although this information could be placed in a matrix in several different ways, two
approaches stand out. Since the rows in the table correspond to the four different
quarters during the year and the columns correspond to numbers of subscribers, we
could use a matrix with four rows and three columns:
New
Q1
Q2
Q3
Q4
5200
3000

5600

 2500
First
Long
6000 15000 
2000 2600 
6500 12000 

1500 3600 
Normally we don’t include the red labels on a matrix. However, they are often included
to help clarify how the information in the matrix is organized. To name this matrix of
subscribers, we could use the letter S and write
5200
3000
S
5600

 2500
6000 15000 
2000 2600 
6500 12000 

1500 3600 
5
This organization capitalizes on the fact that all of the numbers in the table indicate the
number of subscribers in a certain category.
Let’s look at the table differently.
Expiring Subscribers
New
Subscribers
First Time
Long Time
Quarter Ending 3/31
5200
6000
15000
Quarter Ending 6/30
3000
2000
2600
Quarter Ending 9/30
5600
6500
12000
Quarter Ending 12/31
2500
1500
3600
The rows in the table still refer to quarters, but now the shading in the table emphasizes
a difference in the numbers. The numbers in the blue region corresponds to the number
of new subscribers by quarter and the red region corresponds to subscribers whose
subscriptions are expiring.
With this difference in mind, we could define two matrices for this table,
5200 
3000 
,
N 
5600 


 2500 
 6000 15000 
 2000 2600 

E
 6500 12000 


1500 3600 
The matrix N is a 4 x 1 column matrix representing the number of new subscribers of Ed
Magazine. The matrix E is a 4 x 2 matrix representing the number of expiring
subscribers in two categories by quarter. Depending on the application, these matrices
may be more useful than the 4 x 3 matrix S.
6
Example 2
Organize Information in a Matrix
A magazine’s circulation is the number of issues it distributes. Ed
Magazine is distributed to three categories of subscribers each quarter.
New
Subscriber
Issues
Renewing
Subscriber
Issues
Non-renewing
Subscriber
Issues
Quarter Ending 3/31
5200
20600
16600
Quarter Ending 6/30
8200
20600
13800
Quarter Ending 9/30
13800
20600
2925
Quarter Ending 12/31
16300
20600
0
Use this information to define three matrices named C1 , C2 , and C3 ,
where C1 describes the number of issues distributed to new
subscribers, C2 describes the number of issues distributed to
subscribers who have renewed their subscription, and C3 describes the
number of issues distributed to subscribers who have not renewed their
subscriptions.
Solution The first column in the table corresponds to issues distributed
to new subscribers
New
Subscriber
Issues
Quarter Ending 3/31
5200
Quarter Ending 6/30
8200
Quarter Ending 9/30
13800
Quarter Ending 12/31
16300
7
If we let the rows in the matrix C1 correspond to the quarters, then can
organize the information in the table in a 4 x 1 matrix as
 5200 
 8200 

C1  
13800 


16300 
Alternatively, we could also let the quarters correspond to the columns
in a 1 x 4 matrix and define
C1  5200 8200 13800 16300
Either matrix organizes the information appropriately. Since the original
table matches each row with a quarter, we’ll follow the same principal
and let the rows of the matrices correspond to the quarters.
Renewing
Subscriber
Issues
Non-renewing
Subscriber
Issues
Quarter Ending 3/31
20600
16600
Quarter Ending 6/30
20600
13800
Quarter Ending 9/30
20600
2925
Quarter Ending 12/31
20600
0
Letting the rows match the quarters, the other columns in the table give
the entries in C2 and C3 ,
 20600 
 20600 

C2  
 20600 


 20600 
16600 
13800 

C3  
 2925 


 0 
8
In Example 2, we mentioned the fact that the data in the first column of the table could
be written as a row matrix or a column matrix. These matrices are examples of
transposes. In other words, the matrix
 5200 
 8200 


13800


16300
is the transpose of the matrix
5200
8200 13800 16300
The transpose of the matrix A, written AT , is obtained by
writing the columns of the matrix A as rows in the matrix AT .
Alternatively, we could also write the rows of the matrix A as
columns in the matrix AT .
Example 3
Find the Transpose of a Matrix
Find and label the transpose of the expiring subscriber matrix
 6000 15000 
 2000 2600 

E
 6500 12000 


1500 3600 
Solution To get the transpose of the subscriber matrix, we interchange
the rows and columns. In other words, the columns of E become the
rows of the transpose ET or the rows of E become the columns of the
transpose ET to yield
 6000 2000 6500 1500 
ET  

15000 2600 12000 3600 
9
In the original matrix E, the rows of the subscriber matrix correspond to
the quarters and the columns tell us the subscriber category.
Renewing
Non-renewing
Subscribers
Suscribers
 6000 15000 
 2000 2600 


 6500 12000 


1500 3600 
Spring Quarter
Summer Quarter
Fall Quarter
Winter Quarter
In the transpose, these roles are reversed.
Spring
Summer
Fall
Winter
Quarter
Quarter
Quarter
Quarter
 6000 2000 6500 1500 
15000 2600 12000 3600 


Renewing Subscribers
Non-renewing Subscribers
The information in each matrix is the same, but organized differently.
10