Modeling Real-World Data
with Matrices
Section 2-3
Before finishing this section
you should be able to:
• Model data using matrices
• Add, subtract, and multiply
matrices
Remember: Your textbook is your friend! This
presentation is just a supplement to the text.
BEFORE you view this, make sure you read this
section in your textbook and look at all the
great examples that are also worked there for
you.
Intro to Matrices
A matrix is a rectangular array of terms
called elements arranged in rows and
columns. These rows and columns
make up the dimensions of the matrix.
A matrix with m rows and n columns is a
matrix.
 3 0 5


 1 6 8 
This matrix has 2 rows and 3 columns.
The dimensions are 2 x 3
There are several special matrices.
A matrix with only one row is called a row matrix.
For example:
1
3 7
A matrix with only one column is a column matrix.
For example:
6
 2 
 
A matrix with the same number of rows as
columns is called a square matrix,
also called matrices of the nth order,
where n is the number of rows and columns.
For example:
 4 5
 3 3


This is a 2 x 2 matrix, so it is of order 2.
Two matrices are equal matrices if they have the
same dimensions and are identical, element by
element.
Matrices may be easily added and subtracted by
simply adding or subtraction the corresponding
elements.
The dimensions of the two matrices must be the
same.
A matrix whose elements are all zeros
is a zero matrix. A zero matrix is an additive
identity matrix because adding a zero matrix to any
matrix will not change that matrix.
zero matrix (identity matrix)
 0  1   0  1   1 
0   1  0  1   1
    
  
A number called a scalar may also multiply matrices.
Each element is multiplied by the scalar.
2 6
A 


4
5


 6 18 
3A  


12
15


Multiply each
element in matrix
A by the scalar 3.
You can also find the product of two
matrices. This is only possible when the
number of columns in the first is equal to
the number of rows in the second. Each
element in the first row of the first matrix
is multiplied by each corresponding
element in the first column of the second
matrix. Then add all of these products
together.
3x3
2x2
2x3
 4 1 2
 4 2
1 2 3


A  0 1 0  B  
C



2
3
3
1
0




 3 2 4
Find the product AB.
AB is impossible because the dimensions of matrix A are
3 x 3 and the dimensions of matrix B are 2 x 2.
In order to multiply two matrices the number of columns in
the first matrix must match the number of rows in the
second matrix.
A has 3 columns and B has 2 rows so we have a dimension
mismatch and we cannot multiply these two matrices.
Find the product BC.
B is a 2 x 2 matrix and C is a 2 x 3 matrix. The number of
columns in B is 2 and the number of rows in C is 2,
therefore we can multiply these two matrices.
 4 2 1 2 3
BC  
  3 1 0 

2
3

 

Multiply each row
in B by each
column in C.
 (4  1)  (2  3) (4  2)  (2  1) (4  3)  (2  0) 
10 10 12 
BC  

BC

 7 1 6 
( 2  1)  (3  3) ( 2  2)  (3  1) ( 2  3)  (3  0) 


DESSERT Jessica does a survey on the cost of
four different desserts at three local
restaurants. At restaurant A, a slice of apple
pie is priced at $2.25, a brownie sundae is
priced at $2.95, a slice of apple cobbler is
priced at $1.95, and an ice cream cone is
priced at $1.10. At restaurant B, apple pie is
$2.75, a brownie sundae is $3.45, apple
cobbler is $2.50, and an ice cream cone is
$1.65. At restaurant C, apple pie is $2.40, a
brownie sundae is $2.70, apple cobbler is
$2.35, and an ice cream cone is $1.15.
a. Use a matrix to represent the data.
b. Use a symbol to represent the price of a
brownie sundae at restaurant C.
a.
To represent data using a matrix, choose
which category will be represented by the columns
and which will be represented by the rows. Let’s use
the columns to represent the prices at each
restaurant and the rows to represent the prices of
each dessert. Then write each data piece as you
would if you were placing the data in a table.
A
apple pie
$2.25
brownie sundae $2.95
$1.95
apple cobbler
$1.10
ice cream
B
$2.75
$3.45
$2.50
$1.65
Notice that the category names appear
outside of the matrix.
C
$2.40 
$2.70 
$2.35
$1.15
b. The price of a brownie sundae at restaurant C
is found in row 2, column 3 of the matrix.
This element is represented by the symbol a23.
Find the values of x and y for which the
matrix equation is true.
 y   3 x + 16 
 x   3 y

  

Since the corresponding elements are equal, we
can express the equality of the matrices as two
equations.
y = 3x + 16
x = 3y
Solve the system of equations by using
substitution.
y = 3x + 16
y = 3(3y) + 16
y = -2 Solve for y.
x = 3(-2)
x = -6
Substitute 3y for x.
Substitute –2 for y in the
second equation to find x.
The matrices are equal if x = -6 and y = -2.
Check by substituting into the matrices.
More Examples
 4 -2 6 
Find A + B if A = 

1
3
-3


-1 2 5 
and B = 

-4
1
7


A + B =  4  (1) 2  2 6  5 
1  (4) 3  1 3  7 


=
 3 0 11
 3 4 4 


Find C – D if C =
C – D = C + -D =
=
5 2
 8 1  and D =


 -4 3 


 2 -1
2 5
 -3 4 


 6 -8 


3
5


 5 2   2 5
 8 1   3 4 



 4 3   6 8 

 

 2 1  3 5
 5  (2) 2  (5) 
 3 -3 
 83

 11 -3 
1

(

4)

 or 

 4  (6)
-10 11
38 




 2  (3) 1  (5) 
 -1 -6 
 1 3 4
If A = 
, find 2A.

 2 5 0 
 3 6 2 
 1 3 4
2  2 5 0  =
 3 6 2 
 2 6 8
 4 10 0 


 6 12 4 
 2(1) 2(3) 2(4) 
 2(2) 2(5) 2(0) 


 2(3) 2(6) 2(2) 
=
Multiply each element by 2.
2 4
Use matrices A =  0 1 


and C = -3 4 2 
1 5 0

,B=
 3 1 -2 
 4 0 -1


,
to find each product.

Find the product AB
AB =
 2 4   3 1 2 
 0 1    4 0 1

 

 2(3)  4(4) 2(1)  4(0) 2(2)  4(1) 
 22 2 -8
 0(3)  1(4) 0(1)  1(0) 0(2)  1(1)  =  4 0 -1




Find the product BC
B is a 2  3 matrix and C is a 2  3 matrix.
Since B does not have the same number of columns as
C has rows, the product BC does not exist.
BC is undefined.
Calculator Instructions
http://helmet.stetson.edu/~mhale/teach/ti83
.htm#matrices
http://dwb.unl.edu/calculators/activities/Matr
ices.html
http://occawlonline.pearsoned.com/bookbin
d/pubbooks/pirnot_awl/chapter1/medialib
/tech/ti83tutorial.pdf
Helpful Websites
MATRICES:
http://www.ping.be/~ping1339/matr.htm
http://www.purplemath.com/modules/matrices2.htm
http://www.purplemath.com/modules/mtrxadd.htm
http://www.purplemath.com/modules/mtrxmult.htm
http://www.purplemath.com/modules/mtrxmult2.htm
2-3 Self-Check Quiz:
http://www.glencoe.com/sec/math/studytools/cgibin/msgQuiz.php4?isbn=0-07-8608619&chapter=2&lesson=3&quizType=1&headerFile=4&state=