222 Matrices
Lesson
Storing Data in Matrices
Chapter 4
4-1
BIG IDEA A variety of types of data, from numerical
information to coordinates of points, can be stored in matrices.
A rectangular arrangement of objects or numbers is called a matrix.
The plural of
matrix
is
matrices
. Each object in a matrix is called
an element of the matrix. Matrices are useful for storing data of all
kinds.
For example, the median salaries of collegiate head coaches for three
different sports, based on the highest degree an institution grants,
are shown in the matrix below. Entries are in dollars.
column 1
↓
column 2
↓
column 3
↓
column 4
↓
Doctoral Master’s Bachelor’s Associate’s
row 1
→
Football
185,000 72,070 62,499 31,314
row 2
→
Baseball
72,975 46,654 41,105 44,799
row 3
→
Basketball
157,500 63,347 51,092 45,982
Source: Chronicle of Higher Education, March 2006
Dimensions of a Matrix
The elements of the above matrix are enclosed by large
square brackets. (Sometimes large parentheses are used
in place of brackets.) This matrix has 3
rows
and 4
columns
.
Because of this, it is said to have the dimensions 3 by 4, written
3
×
4. In general, a matrix with
m
rows and
n
columns
has dimensions
m
×
n
. Each element of a matrix is
identifi ed fi rst by its row location, then by its column
location. For example, the element in the 3rd row and
2nd column of this matrix is 63,347. Headings are placed
outside the matrix, like the sports and degrees above.
A rectangular block of cells in a spreadsheet also constitutes a
matrix. Spreadsheets use the reverse order for identifying an
element—column fi rst (a letter) and row second (a number). Like
matrices, spreadsheets can have headings to identify their row(s)
and column(s).
Mental Math
Find an equation for a
line satisfying the
conditions.
a. slope 4 and y-intercept
2.5
b. unde ned slope and
passing through (
–
7, 2)
c. slope
1
__
3
and passing
through (0,
9
__
10
)
d. passing through
(17, 12) and (0.4, 12)
Mental Math
Find an equation for a
line satisfying the
conditions.
a. slope 4 and y-intercept
2.5
b. unde ned slope and
passing through (
–
7, 2)
c. slope
1
__
3
and passing
through (0,
9
__
10
)
d. passing through
(17, 12) and (0.4, 12)
Vocabulary
matrix
element
dimensions
equal matrices
point matrix
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Storing Data in Matrices 223
Lesson 4-1
Many calculators let you enter and manipulate matrices. Use the
Guided Example to see how to enter a matrix into a CAS and to store
a matrix as a variable.
Example
According to the Statistical Abstract of the United States, in 1980,
approximately 3.5 million males and 1.9 million females participated in
high school athletic programs. Ten years later, 3.4 million males and 1.9
million females participated. In 2000, 3.9 million males and 2.8 million
females participated.
a. Store the high school athletic participation information in a matrix M.
b. What are the dimensions of the matrix?
c. Enter the matrix from Part a into a CAS and store it as a variable.
Solution
a. You can write either of the two matrices below. Matrix M1 has the years
as rows, and matrix M2 has the years as columns. Either matrix is an
acceptable way to store the data.
Matrix
M
1:
Males Females
1980
??
1990
??
2000
??
Matrix
M
2:
1980 1990 2000
Males
???
Females
???
b. Matrix M1 has
?
rows and
?
columns.
The dimensions of M1 are
?
.
Matrix M2 has
?
rows and
?
columns.
The dimensions of M2 are
?
.
c. Use a CAS. Clear M1 or M2 before storing your matrix.
Although matrices
M
1 and
M
2 in the Example are both acceptable
ways to store and represent the data, the two matrices are not
considered equal. Matrices are equal matrices if and only if they
have the same dimensions
and
their corresponding elements
are equal.
GUIDEDGUIDED
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224 Matrices
Chapter 4
Matrices and Geometry
Points and polygons can also be represented by matrices.X
The ordered pair (
x
,
y
) is generally represented by the matrix
x
y
.
This 2
×
1 matrix is called a point matrix. Notice that the element in
the fi rst row is the
x
-coordinate and the element in the second row is
the
y
-coordinate. For instance, the point (5,
–
1) is represented by the
matrix
5
–
1
.
Similarly, polygons can be written as matrices. Each column of the
matrix contains the coordinates of a vertex of the polygon in the
order in which the polygon is named. The Activity illustrates this.
MATERIALS matrix polygon application
Step 1 Pentagon ABCDE with vertices A
=
(3,
–
5),
B
=
(6,
–
1), C
=
(4, 5), D
=
(
–
2.5, 4), and
E
=
(
–
5,
–
0.75), is shown at the right. Write a
matrix representing the coordinates of the
vertices of the pentagon, starting with point A.
Step 2 Write two other matrices representing the
pentagon. (Hint: Start with a different vertex.)
Step 3 Verify that your matrices from Step 2 are correct
by plotting the pentagon each matrix describes.
Each picture should be the same as pentagon
ABCDE. You can do this by using a matrix polygon
application supplied by your teacher.
Step 4 Plot
BA C D E
63 4
–
2.5
–
5
–
1
–
5
5 4 –0.75
.
Explain why BACDE does not describe a pentagon.
QY
ActivityActivity
QY
Are the four matrices in
the Activity equal? Explain
your answer.
y
x
2
246
6
᎑ 4᎑ 6 ᎑ 2
᎑ 6
4
᎑ 4
᎑ 2
B
=
(6, ᎑ 1)
C
=
(4, 5)
D
=
(᎑ 2.5, 4)
E
=
(᎑ 5, ᎑ 0.75)
A
=
(3, ᎑ 5)
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Storing Data in Matrices 225
Lesson 4-1
Questions
COVERING THE IDEAS
1.
What is a matrix?
In 2–4, refer to the matrix regarding coaching salaries at the
beginning of the lesson.
2. a. What is the element in row 2, column 3?
b. What does this element represent?
3. Write instructions that someone could use to enter the matrix on
your CAS.
4. What would be the dimensions of the matrix if
highest degrees
were rows, and
sports
were columns?
5. Refer to the Example. In 1985, there were 3,344,275 males and
1,807,121 females participating in high school athletic programs.
Construct a 2
×
4 matrix that incorporates this new information
with the old.
6. In the fall of 2008, mathematics classes at a local
community college had a total enrollment of 2850,
compared to 2241 in the fall of 2007. Additionally,
English classes had total enrollments of 2620 and
2051, biology classes had enrollments of 1160 and
1572, and psychology classes had enrollments of
740 and 784, all respectively.
a. Arrange the data into a matrix on a CAS or
graphing calculator, representing years as
rows.
b. Now arrange the data into a matrix representing
years as columns.
7. a. Write the ordered pair (
x
,
y
) as a matrix.
b. What is this matrix called?
8. Multiple Choice Which matrix represents the point
(
15
√
2 ,
–
7.3
)
?
A
15
√
2
–
7.3
B
–
7.3 15
√
2
C
–
7.3
15
√
2
D
15
√
2
–
7.3
9. Write
ABC
at the right as a matrix.
Many colleges campuses
have inner courtyards called
quadrangles.
Many colleges campuses
have inner courtyards called
quadrangles.
y
x
462
0
4
6
2
᎑ 4
᎑ 6
᎑ 2
᎑ 4᎑ 6
᎑ 2
A
B
C
y
x
462
0
4
6
2
᎑ 4
᎑ 6
᎑ 2
᎑ 4᎑ 6
᎑ 2
A
B
C
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226 Matrices
Chapter 4
10. Refer to the Activity.
a. The matrix at the right uses the same points as the Activity.
Use a matrix polygon application to draw (or plot by hand)
and connect the points in the matrix. Use the same window
as in the Activity. Why is the picture different from pentagon
ABCDE
?
b. How many matrices can represent the pentagon
ABCDE
?
11. The matrix at the right gives the numbers of professional
degrees earned in 2000 in four professions, separated by gender.
a. What are the dimensions of this matrix?
b. What does the sum of the elements in row 3 represent?
c. What does the sum of the elements in column 2 represent?
APPLYING THE MATHEMATICS
12.
Fill in the Blanks If
–
6 4.3
1
_
2
w
=
–
6 r
1
_
2
0.9
, then
w
=
?
and
r
=
?
.
13. Fill in the Blanks If
2
a
-
3
h
+
0.4
=
–
9
1
_
2
, then
a
=
?
and
h
=
?
.
14. Recall on your CAS the matrix
M
1 from the Guided Example.
If
M
1
=
xy
z
-
1
y
3
w
2.8
, fi nd
w
,
x
,
y
, and z.
15. In the English language, the vowels A, E, I, O, and U show up
with frequencies among all letters of about 8%, 13%, 7%, 8%, and
3%, respectively. In the board game SCRABBLE
®
, these letters
show up with frequencies 9%, 12%, 9%, 8%, and 4%, respectively.
a. Arrange this information into a 2
×
5 matrix.
b. Explain how to enter this matrix into a CAS.
Males Females
Medicine 8,759 6,527
Dentistry 2,546 1,704
Law 20,640 17,512
Theology 4,339 1,790
Males Females
Medicine 8,759 6,527
Dentistry 2,546 1,704
Law 20,640 17,512
Theology 4,339 1,790
ABC E D
364
–
5
–
2.5
–
5
–
15
–
0.75 4
Many colleges graduations use
different colored tassels or sashes
to denote individual majors.
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Storing Data in Matrices 227
Lesson 4-1
16. The endpoints of
___
PA
on line
m
are given by the matrix
1
_
2
2
17
_
2
13
.
The endpoints of
___
LN
on line
n
are defi ned by the matrix
4
–
1
8
–
7
.
Prove that lines
m
and
n
are parallel.
17. Use
a matrix polygon application to draw (or plot by hand) the
octagon
015 1 0
–
1
–
5
–
1
510
–
1
–
5
–
10 1
. Sketch a picture of the output,
and explain if the polygon is convex or nonconvex.
REVIEW
18.
Evaluate the following expressions. (Lesson 3-9)
a.
r
π
-
r
π
b.
n
10
-
n
10
c.
–
1
__
2
d.
–
3.6
-
–
3
e.
56.63
-
–
56.9
f.
5
-
5.02
g.
–
4.5
+
9.7
19. Shelby put $50 on her public transportation card. For every bus
or train ride she takes, $2 is deducted from her total. Express
the total left on her card as an explicit formula of an arithmetic
sequence dependent on the number of train or bus rides
taken.
(Lesson 3-8)
20. Fill in the Blank
Ax
+
By
=
C
is the
?
form of an equation
for a line.
(Lesson 3-2)
21. If the volume of cube
A
is 27 times the volume of cube
B
, how
do the lengths of their edges compare?
(Lesson 2-3)
22. a. What does the Commutative Property of Addition say?
b. Is subtraction commutative? If so, explain. If not, give a
counterexample. (Previous Course)
EXPLORATION
23.
What is a dot-matrix printer? How is it related to the matrices
discussed in this lesson?
QY ANSWER
No, corresponding elements
are not equal.
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