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8Matrices andDeterminants
8.1 Matrices and Systems of Equations
8.2 Operations with Matrices
8.3 The Inverse of a Square Matrix
8.4 The Determinant of a Square Matrix
8.5 Applications of Matrices and Determinants
In Mathematics
Matrices are used to model and solve a
variety of problems. For instance, you can
use matrices to solve systems of linear
equations.
In Real Life
Matrices are used to model inventory
levels, electrical networks, investment
portfolios, and other real-life situations.
For instance, you can use a matrix to
model the number of people in the United
States who participate in snowboarding.
(See Exercise 114, page 583.)
IN CAREERS
There are many careers that use matrices. Several are listed below.
Gra
ham
Hey
woo
d/is
tock
phot
o.co
m
• Bank Teller
Exercise 110, page 582
• Political Analyst
Exercise 70, page 597
• Small Business Owner
Exercises 69–72, pages 606 and 607
• Florist
Exercise 74, page 607
569
Matrices
In this section, you will study a streamlined technique for solving systems of linear
equations. This technique involves the use of a rectangular array of real numbers called
a matrix. The plural of matrix is matrices.
The entry in the th row and th column is denoted by the double subscript notation
For instance, refers to the entry in the second row, third column. A matrix having
rows and columns is said to be of order If the matrix is square of
order For a square matrix, the entries are the main
diagonal entries.
Order of Matrices
Determine the order of each matrix.
a. b.
c. d.
Solution
a. This matrix has one row and one column. The order of the matrix is
b. This matrix has one row and four columns. The order of the matrix is
c. This matrix has two rows and two columns. The order of the matrix is
d. This matrix has three rows and two columns. The order of the matrix is
Now try Exercise 9.
A matrix that has only one row is called a row matrix, and a matrix that has only
one column is called a column matrix.
3 2.
2 2.
1 4.
1 1.
�5
2
�7
0
�2
4��0
0
0
0�
�1 �3 012��2�
Example 1
a11, a22, a33, . . .m m �or n n�.m � n,m n.n
ma23
a ij.ji
570 Chapter 8 Matrices and Determinants
8.1 MATRICES AND SYSTEMS OF EQUATIONS
What you should learn
• Write matrices and identify theirorders.
• Perform elementary row operationson matrices.
• Use matrices and Gaussian elimination to solve systems of linear equations.
• Use matrices and Gauss-Jordan elimination to solve systems of linear equations.
Why you should learn it
You can use matrices to solve systems of linear equations in two or more variables. For instance, inExercise 113 on page 582, you will use a matrix to find a model for thepath of a ball thrown by a baseballplayer.
Foto
Age
ncy/
Phot
oLib
rary
Definition of Matrix
If and are positive integers, an (read “ by ”) matrix is a rectangular
array
Column 1 Column 2 Column 3 . . . Column n
in which each entry, of the matrix is a number. An matrix has rows
and columns. Matrices are usually denoted by capital letters.n
mm nai j,
�a11
a21
a31...
am1
a12
a22
a32...
am2
a13
a23
a33...
am3
. . .
. . .
. . .
. . .
a1n
a2n
a3n...
amn
�
nmm nnm
Row 1
Row 2
Row 3
Row m
.
.
.
Section 8.1 Matrices and Systems of Equations 571
A matrix derived from a system of linear equations (each written in standard form
with the constant term on the right) is the augmented matrix of the system. Moreover,
the matrix derived from the coefficients of the system (but not including the constant
terms) is the coefficient matrix of the system.
System:
Augmented
Matrix:
Coefficient
Matrix:
Note the use of 0 for the missing coefficient of the -variable in the third equation,
and also note the fourth column of constant terms in the augmented matrix.
When forming either the coefficient matrix or the augmented matrix of a system,
you should begin by vertically aligning the variables in the equations and using zeros
for the coefficients of the missing variables.
Writing an Augmented Matrix
Write the augmented matrix for the system of linear equations.
What is the order of the augmented matrix?
Solution
Begin by rewriting the linear system and aligning the variables.
Next, use the coefficients and constant terms as the matrix entries. Include zeros for the
coefficients of the missing variables.
The augmented matrix has four rows and five columns, so it is a matrix. The
notation is used to designate each row in the matrix. For example, Row 1 is
represented by
Now try Exercise 17.
R1.
Rn
4 5
R1
R2
R3
R4
�1
0
1
2
3
�1
0
4
0
4
�5
�3
�1
2
�6
0
...
...
...
...
9
�2
0
4�
�x � 3y � w � 9
�y � 4z � 2w � �2
x � 5z � 6w � 0
2x � 4y � 3z � 4
�x � 3y � w �
�y � 4z � 2w �
x � 5z � 6w �
2x � 4y � 3z �
9
�2
0
4
Example 2
y
�1
�1
2
�4
3
0
3
�1
�4�
�1
�1
2
�4
3
0
3
�1
�4
......
...
5
�3
6�
�x
�x
2x
� 4y
� 3y
� 3z �
� z �
� 4z �
5
�3
6The vertical dots in an
augmented matrix separate the
coefficients of the linear system
from the constant terms.
572 Chapter 8 Matrices and Determinants
Elementary Row Operations
In Section 7.3, you studied three operations that can be used on a system of linear
equations to produce an equivalent system.
1. Interchange two equations.
2. Multiply an equation by a nonzero constant.
3. Add a multiple of an equation to another equation.
In matrix terminology, these three operations correspond to elementary row
operations.An elementary row operation on an augmented matrix of a given system of
linear equations produces a new augmented matrix corresponding to a new (but
equivalent) system of linear equations. Two matrices are row-equivalent if one can be
obtained from the other by a sequence of elementary row operations.
Although elementary row operations are simple to perform, they involve a lot of
arithmetic. Because it is easy to make a mistake, you should get in the habit of noting
the elementary row operations performed in each step so that you can go back and
check your work.
Elementary Row Operations
a. Interchange the first and second rows of the original matrix.
Original Matrix New Row-Equivalent Matrix
b. Multiply the first row of the original matrix by
Original Matrix New Row-Equivalent Matrix
c. Add times the first row of the original matrix to the third row.
Original Matrix New Row-Equivalent Matrix
Note that the elementary row operation is written beside the row that is changed.
Now try Exercise 37.
�2R1 � R3→�1
0
0
2
3
�3
�4
�2
13
3
�1
�8��
1
0
2
2
3
1
�4
�2
5
3
�1
�2�
�2
12R1→ �
1
1
5
�2
3
�2
3
�3
1
�1
0
2��
2
1
5
�4
3
�2
6
�3
1
�2
0
2�
12.
R2
R1 ��1
0
2
2
1
�3
0
3
4
3
4
1��
0
�1
2
1
2
�3
3
0
4
4
3
1�
Example 3
Elementary Row Operations
1. Interchange two rows.
2. Multiply a row by a nonzero constant.
3. Add a multiple of a row to another row.
TECHNOLOGY
Most graphing utilities can perform elementary row operations on matrices. Consultthe user’s guide for your graphingutility for specific keystrokes.
After performing a row operation, the new row-equivalentmatrix that is displayed on yourgraphing utility is stored in theanswer variable. You should usethe answer variable and not theoriginal matrix for subsequentrow operations.
Section 8.1 Matrices and Systems of Equations 573
In Example 3 in Section 7.3, you used Gaussian elimination with back-substitution
to solve a system of linear equations. The next example demonstrates the matrix
version of Gaussian elimination. The two methods are essentially the same. The basic
difference is that with matrices you do not need to keep writing the variables.
Comparing Linear Systems and Matrix Operations
Linear System Associated Augmented Matrix
Add the first equation to the Add the first row to the
second equation. second row
Add times the first equation Add times the first row
to the third equation. to the third row
Add the second equation to the Add the second row to the
third equation. third row
Multiply the third equation by Multiply the third row by
At this point, you can use back-substitution to find x and y.
Substitute 2 for z.
Solve for y.
Substitute for y and 2 for z.
Solve for x.
The solution is and
Now try Exercise 39.
z � 2.y � �1,x � 1,
x � 1
�1x � 2��1� � 3�2� � 9
y � �1
y � 3�2� � 5
12R3→
�1
0
0
�2
1
0
3
3
1
.
..
.
..
.
..
9
5
2��
x � 2y � 3z � 9
y � 3z � 5
z � 2
�12R3�.1
212.
R2 � R3→�1
0
0
�2
1
0
3
3
2
.
..
.
..
.
..
9
5
4��
x � 2y � 3z �
y � 3z �
2z �
9
5
4
�R2 � R3�.
�2R1 � R3→�1
0
0
�2
1
�1
3
3
�1
.
..
.
..
.
..
9
5
�1��
x � 2y � 3z �
y � 3z �
�y � z �
9
5
�1
��2R1 � R3�.�2�2
R1 � R2→ �1
0
2
�2
1
�5
3
3
5
.
..
.
..
.
..
9
5
17��
x � 2y � 3z �
y � 3z �
2x � 5y � 5z �
9
5
17
�R1 � R2�.
�1
�1
2
�2
3
�5
3
0
5
.
..
.
..
.
..
9
�4
17��
x � 2y
�x � 3y
2x � 5y
� 3z
� 5z
�
�
�
9
�4
17
Example 4
Remember that you should
check a solution by substituting
the values of and into
each equation of the original
system. For example, you can
check the solution to Example 4
as follows.
Equation 1:
�
Equation 2:
�
Equation 3:
�2�1� � 5��1� � 5�2� � 17
�1 � 3��1� � �4
1 � 2��1� � 3�2� � 9
zy,x,
WARNING / CAUTION
Arithmetic errors are often
made when elementary row
operations are performed. Note
the operation you perform in
each step so that you can go
back and check your work.
The last matrix in Example 4 is said to be in row-echelon form. The term echelon
refers to the stair-step pattern formed by the nonzero elements of the matrix. To be in
this form, a matrix must have the following properties.
It is worth noting that the row-echelon form of a matrix is not unique. That is, two
different sequences of elementary row operations may yield different row-echelon
forms. However, the reduced row-echelon form of a given matrix is unique.
Row-Echelon Form
Determine whether each matrix is in row-echelon form. If it is, determine whether the
matrix is in reduced row-echelon form.
a. b.
c. d.
e. f.
Solution
The matrices in (a), (c), (d), and (f) are in row-echelon form. The matrices in
(d) and (f) are in reduced row-echelon form because every column that has a leading 1
has zeros in every position above and below its leading 1. The matrix in (b) is not in
row-echelon form because a row of all zeros does not occur at the bottom of the matrix.
The matrix in (e) is not in row-echelon form because the first nonzero entry in Row 2
is not a leading 1.
Now try Exercise 41.
Every matrix is row-equivalent to a matrix in row-echelon form. For instance, in
Example 5, you can change the matrix in part (e) to row-echelon form by multiplying its
second row by 12.
�0
0
0
1
0
0
0
1
0
5
3
0��
1
0
0
2
2
0
�3
1
1
4
�1
�3�
�1
0
0
0
0
1
0
0
0
0
1
0
�1
2
3
0��
1
0
0
0
�5
0
0
0
2
1
0
0
�1
3
1
0
3
�2
4
1�
�1
0
0
2
0
1
�1
0
2
2
0
�4��
1
0
0
2
1
0
�1
0
1
4
3
�2�
Example 5
574 Chapter 8 Matrices and Determinants
Row-Echelon Form and Reduced Row-Echelon Form
A matrix in row-echelon form has the following properties.
1. Any rows consisting entirely of zeros occur at the bottom of the matrix.
2. For each row that does not consist entirely of zeros, the first nonzero entry
is 1 (called a leading 1).
3. For two successive (nonzero) rows, the leading 1 in the higher row is farther
to the left than the leading 1 in the lower row.
A matrix in row-echelon form is in reduced row-echelon form if every column
that has a leading 1 has zeros in every position above and below its leading 1.
Section 8.1 Matrices and Systems of Equations 575
Gaussian Elimination with Back-Substitution
Gaussian elimination with back-substitution works well for solving systems of linear
equations by hand or with a computer. For this algorithm, the order in which the
elementary row operations are performed is important. You should operate from left
to right by columns, using elementary row operations to obtain zeros in all entries
directly below the leading 1’s.
Gaussian Elimination with Back-Substitution
Solve the system
Solution
The matrix is now in row-echelon form, and the corresponding system is
Using back-substitution, you can determine that the solution is
and
Now try Exercise 63.
w � 3.
z � 1,y � 2,x � �1,
�x � 2y � z
y � z
z
�
� 2w �
� w �
w �
2
�3
�2
3
.
Perform operations on R3
and R4 so third and fourth
columns have leading 1’s.
�113R4→
13R3→ �
1
0
0
0
2
1
0
0
�1
1
1
0
0
�2
�1
1
.
..
.
..
.
..
.
..
2
�3
�2
3�
Perform operations on R4
so second column has
zeros below its leading 1.
6R2 � R4→�1
0
0
0
2
1
0
0
�1
1
3
0
0
�2
�3
�13
.
..
.
..
.
..
.
..
2
�3
�6
�39�
Perform operations on R3
and R4 so first column has
zeros below its leading 1.�2R1 � R3→
�R1 � R4→�1
0
0
0
2
1
0
�6
�1
1
3
�6
0
�2
�3
�1
.
..
.
..
.
..
.
..
2
�3
�6
�21�
Interchange R1 and R2 so
first column has leading
1 in upper left corner.
R2
R1 �1
0
2
1
2
1
4
�4
�1
1
1
�7
0
�2
�3
�1
.
..
.
..
.
..
.
..
2
�3
�2
�19�
Write augmented matrix.�0
1
2
1
1
2
4
�4
1
�1
1
�7
�2
0
�3
�1
.
..
.
..
.
..
.
..
�3
2
�2
�19�
� x
2x
x
�
�
�
y � z
2y � z
4y � z
4y � 7z
� 2w �
�
� 3w �
� w �
�3
2
�2
�19
.
Example 6
The procedure for using Gaussian elimination with back-substitution is
summarized below.
When solving a system of linear equations, remember that it is possible
for the system to have no solution. If, in the elimination process, you obtain a
row of all zeros except for the last entry, it is unnecessary to continue the elimination
process. You can simply conclude that the system has no solution, or is inconsistent.
A System with No Solution
Solve the system
Solution
Write augmented matrix.
Perform row operations.
Perform row operations.
Note that the third row of this matrix consists entirely of zeros except for the last entry.
This means that the original system of linear equations is inconsistent. You can see why
this is true by converting back to a system of linear equations.
Because the third equation is not possible, the system has no solution.
Now try Exercise 81.
�x � y � 2z �
y � z �
0 �
5y � 7z �
4
2
�2
�11
R2 � R3→ �1
0
0
0
�1
1
0
5
2
�1
0
�7
.
..
.
..
.
..
.
..
4
2
�2
�11�
�R1 � R2→
�2R1 � R3→
�3R1 � R4→�1
0
0
0
�1
1
�1
5
2
�1
1
�7
.
..
.
..
.
..
.
..
4
2
�4
�11�
�1
1
2
3
�1
0
�3
2
2
1
5
�1
.
..
.
..
.
..
.
..
4
6
4
1�
�x
x
2x
3x
�
�
�
y � 2z � 4
� z � 6
3y � 5z � 4
2y � z � 1
.
Example 7
576 Chapter 8 Matrices and Determinants
Gaussian Elimination with Back-Substitution
1. Write the augmented matrix of the system of linear equations.
2. Use elementary row operations to rewrite the augmented matrix in
row-echelon form.
3. Write the system of linear equations corresponding to the matrix in
row-echelon form, and use back-substitution to find the solution.
Section 8.1 Matrices and Systems of Equations 577
Gauss-Jordan Elimination
With Gaussian elimination, elementary row operations are applied to a matrix to obtain
a (row-equivalent) row-echelon form of the matrix. A second method of elimination,
called Gauss-Jordan elimination, after Carl Friedrich Gauss and Wilhelm Jordan
(1842–1899), continues the reduction process until a reduced row-echelon form is
obtained. This procedure is demonstrated in Example 8.
Gauss-Jordan Elimination
Use Gauss-Jordan elimination to solve the system
Solution
In Example 4, Gaussian elimination was used to obtain the row-echelon form of the
linear system above.
Now, apply elementary row operations until you obtain zeros above each of the
leading 1’s, as follows.
The matrix is now in reduced row-echelon form. Converting back to a system of linear
equations, you have
Now you can simply read the solution, and which can be
written as the ordered triple
Now try Exercise 71.
The elimination procedures described in this section sometimes result in fractional
coefficients. For instance, in the elimination procedure for the system
you may be inclined to multiply the first row by to produce a leading 1, which will
result in working with fractional coefficients. You can sometimes avoid fractions by
judiciously choosing the order in which you apply elementary row operations.
12
�2x � 5y
3x � 2y
�3x � 3y
� 5z �
� 3z �
�
17
11
�6
�1,�1, 2�.z � 2,y � �1,x � 1,
�xyz
�
�
�
1
�1.
2
Perform operations on R1
and R2 so third column has
zeros above its leading 1.
�9R3 � R1→
�3R3 � R2→ �1
0
0
0
1
0
0
0
1
.
..
.
..
.
..
1
�1
2�
Perform operations on R1
so second column has a
zero above its leading 1.
2R2 � R1→ �1
0
0
0
1
0
9
3
1
.
..
.
..
.
..
19
5
2�
�1
0
0
�2
1
0
3
3
1
.
..
.
..
.
..
9
5
2�
�x � 2y
�x � 3y
2x � 5y
� 3z �
�
� 5z �
9
�4
17
.
Example 8
The advantage of using Gauss-
Jordan elimination to solve a
system of linear equations is
that the solution of the system
is easily found without using
back-substitution, as illustrated
in Example 8.
TECHNOLOGY
For a demonstration of a graphicalapproach to Gauss-Jordan elimination on a matrix,see the Visualizing RowOperations Program available for several models of graphingcalculators at the website for thistext at academic.cengage.com.
2 3
578 Chapter 8 Matrices and Determinants
Recall from Chapter 7 that when there are fewer equations than variables in a
system of equations, then the system has either no solution or infinitely many solutions.
A System with an Infinite Number of Solutions
Solve the system.
Solution
The corresponding system of equations is
Solving for and in terms of you have
and
To write a solution of the system that does not use any of the three variables of the
system, let represent any real number and let
Now substitute for in the equations for and
So, the solution set can be written as an ordered triple with the form
where a is any real number. Remember that a solution set of this form represents an
infinite number of solutions. Try substituting values for to obtain a few solutions.
Then check each solution in the original system of equations.
Now try Exercise 79.
a
��5a � 2, 3a � 1, a�
y � 3z � 1 � 3a � 1
x � �5z � 2 � �5a � 2
y.xza
z � a.
a
y � 3z � 1.x � �5z � 2
z,yx
�x � 5z � 2
y � 3z � �1.
�2R2 � R1→ �1
0
0
1
5
�3
.
..
.
..2
�1��R2→
�1
0
2
1
�1
�3
.
..
.
..0
�1��3R1 � R2→
�1
0
2
�1
�1
3
.
..
.
..0
1�
12R1→ �1
3
2
5
�1
0
.
..
.
..0
1�
�2
3
4
5
�2
0
.
..
.
..0
1�
�2x � 4y
3x � 5y
� 2z � 0
� 1
Example 9
In Example 9, and are
solved for in terms of the third
variable To write a solution
of the system that does not use
any of the three variables of the
system, let represent any real
number and let Then
solve for and The solution
can then be written in terms
of which is not one of the
variables of the system.
a,
y.x
z� a.
a
z.
yx
Section 8.1 Matrices and Systems of Equations 579
EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8.1
VOCABULARY: Fill in the blanks.
1. A rectangular array of real numbers that can be used to solve a system of linear equations is called a ________.
2. A matrix is ________ if the number of rows equals the number of columns.
3. For a square matrix, the entries are the ________ ________ entries.
4. A matrix with only one row is called a ________ matrix, and a matrix with only one column is called
a ________ matrix.
5. The matrix derived from a system of linear equations is called the ________ matrix of the system.
6. The matrix derived from the coefficients of a system of linear equations is called the ________ matrix of the system.
7. Two matrices are called ________ if one of the matrices can be obtained from the other by a sequence of
elementary row operations.
8. A matrix in row-echelon form is in ________ ________ ________ if every column that has a leading 1 has zeros
in every position above and below its leading 1.
SKILLS AND APPLICATIONS
a33, . . . , anna22,a11,
In Exercises 9–14, determine the order of the matrix.
9. 10.
11. 12.
13. 14.
In Exercises 15–20, write the augmented matrix for thesystem of linear equations.
15. 16.
17. 18.
19. 20.
In Exercises 21–26, write the system of linear equationsrepresented by the augmented matrix. (Use variables and if applicable.)
21. 22.
23.
24.
25.
26.
In Exercises 27–34, fill in the blank(s) using elementary rowoperations to form a row-equivalent matrix.
27. 28.
29. 30.
31. 32.
�1
0
0
0
1
0
6
0
1
1
��
��1
0
0
0
1
0
��2
1
�2
�7�
�1
0
0
0
�1
0
6
0
�1
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580 Chapter 8 Matrices and Determinants
33. 34.
In Exercises 35–38, identify the elementary row operation(s)being performed to obtain the new row-equivalent matrix.
Original Matrix New Row-Equivalent Matrix
35.
Original Matrix New Row-Equivalent Matrix
36.
Original Matrix New Row-Equivalent Matrix
37.
Original Matrix New Row-Equivalent Matrix
38.
39. Perform the sequence of row operations on the matrix.
What did the operations accomplish?
(a) Add times to
(b) Add times to
(c) Add times to
(d) Multiply by
(e) Add times to
40. Perform the sequence of row operations on the matrix.
What did the operations accomplish?
(a) Add to
(b) Interchange and
(c) Add 3 times to
(d) Add times to
(e) Multiply by
(f) Add the appropriate multiples of to and
In Exercises 41–44, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reducedrow-echelon form.
41. 42.
43. 44.
In Exercises 45–48, write the matrix in row-echelon form.(Remember that the row-echelon form of a matrix is notunique.)
45. 46.
47. 48.
In Exercises 49–54, use the matrix capabilities of a graphingutility to write the matrix in reduced row-echelon form.
49. 50.
51.
52.
53. 54.
In Exercises 55–58, write the system of linear equationsrepresented by the augmented matrix. Then use back-substitution to solve. (Use variables and if applicable.)
55. 56. �1
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Section 8.1 Matrices and Systems of Equations 581
57. 58.
In Exercises 59–62, an augmented matrix that represents a system of linear equations (in variables and if appli-cable) has been reduced using Gauss-Jordan elimination.Write the solution represented by the augmented matrix.
59. 60.
61. 62.
In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
63. 64.
65. 66.
67. 68.
69. 70.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
81. 82.
83.
84.
In Exercises 85–90, use the matrix capabilities of a graphingutility to reduce the augmented matrix corresponding to thesystem of equations, and solve the system.
85. 86.
87.
88.
89.
90.
In Exercises 91–94, determine whether the two systems of linear equations yield the same solution. If so, find thesolution using matrices.
91. (a) (b)
92. (a) (b)
93. (a) (b)
94. (a) (b)
In Exercises 95–98, use a system of equations to find thequadratic function that satisfies theequations. Solve the system using matrices.
95.
96. f �1� � 2, f �2� � 9, f �3� � 20
f �1� � 1, f �2� � �1, f �3� � �5
f x! ! ax21 bx 1 c
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y � 2z �
z �
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14
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y � 6z �
z �
19
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y � 5z � 42
z � 8�x � 4y � 5z �
y � 7z �
z �
27
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8
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y
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y � z �
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x � 3y � 3z � 2w �
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30
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582 Chapter 8 Matrices and Determinants
97.
98.
In Exercises 99–102, use a system of equations to find thecubic function that satisfies theequations. Solve the system using matrices.
99. 100.
101. 102.
103. Use the system
to write two different matrices in row-echelon form
that yield the same solution.
104. ELECTRICAL NETWORK The currents in an electrical
network are given by the solution of the system
where and are measured in amperes. Solve the
system of equations using matrices.
105. PARTIAL FRACTIONS Use a system of equations to
write the partial fraction decomposition of the rational
expression. Solve the system using matrices.
106. PARTIAL FRACTIONS Use a system of equations to
write the partial fraction decomposition of the rational
expression. Solve the system using matrices.
107. FINANCE A small shoe corporation borrowed
$1,500,000 to expand its line of shoes. Some of the
money was borrowed at 7%, some at 8%, and some
at 10%. Use a system of equations to determine
how much was borrowed at each rate if the annual
interest was $130,500 and the amount borrowed at
10% was 4 times the amount borrowed at 7%. Solve
the system using matrices.
108. FINANCE A small software corporation borrowed$500,000 to expand its software line. Some of themoney was borrowed at 9%, some at 10%, and someat 12%. Use a system of equations to determine howmuch was borrowed at each rate if the annual interestwas $52,000 and the amount borrowed at 10% was
times the amount borrowed at 9%. Solve the systemusing matrices.
109. TIPS A food server examines the amount of money
earned in tips after working an 8-hour shift. The server
has a total of $95 in denominations of $1, $5, $10, and
$20 bills. The total number of paper bills is 26. The
number of $5 bills is 4 times the number of $10 bills,
and the number of $1 bills is 1 less than twice the
number of $5 bills. Write a system of linear equations
to represent the situation. Then use matrices to find the
number of each denomination.
110. BANKING A bank teller is counting the total amount
of money in a cash drawer at the end of a shift. There
is a total of $2600 in denominations of $1, $5, $10,
and $20 bills The total number of paper bills is 235.
The number of $20 bills is twice the number of $1
bills, and the number of $5 bills is 10 more than the
number of $1 bills. Write a system of linear equations
to represent the situation. Then use matrices to find the
number of each denomination.
In Exercises 111 and 112, use a system of equations to findthe equation of the parabola that passesthrough the points. Solve the system using matrices. Use agraphing utility to verify your results.
111. 112.
113. MATHEMATICAL MODELING A video of the path
of a ball thrown by a baseball player was analyzed
with a grid covering the TV screen. The tape was
paused three times, and the position of the ball was
measured each time. The coordinates obtained are
shown in the table. ( and are measured in feet.)yx
−8
(2, 8)(3, 5)
(1, 9)
x
8
y
−4 8 12
12
−8
24
(1, 8)
(2, 13)
(3, 20)
x
y
−4 4 8 12
y ! ax 2" bx " c
212
8x2
�x � 1�2�x � 1� �A
x � 1�
B
x � 1�
C
�x � 1�2
4x 2
�x � 1�2�x � 1��
A
x � 1�
B
x � 1�
C
�x � 1�2
I3I1, I2,
�I1 � I2
3I1 � 4I2
I2
� I3 � 0
� 18
� 3I3 � 6
� x � 3y � z � 3
x � 5y � 5z � 1
2x � 6y � 3z � 8
f �2� � 7f �2� � �7
f �1� � 1f �1� � �4
f ��1� � �5f ��1� � 2
f ��2� � �17f ��2� � �7
f �3� � 44f �3� � 11
f �2� � 16f �2� � 1
f �1� � 4f �1� � �1
f ��1� � 4f ��1� � �5
f x! ! ax31 bx2
1 cx 1 d
f ��2� � �3, f �1� � �3, f �2� � �11
f ��2� � �15, f ��1� � 7, f �1� � �3
Horizontal distance, x Height, y
0
15
30
5.0
9.6
12.4
Section 8.1 Matrices and Systems of Equations 583
(a) Use a system of equations to find the equation of
the parabola that passes through
the three points. Solve the system using matrices.
(b) Use a graphing utility to graph the parabola.
(c) Graphically approximate the maximum height of the
ball and the point at which the ball struck the ground.
(d) Analytically find the maximum height of the ball
and the point at which the ball struck the ground.
(e) Compare your results from parts (c) and (d).
114. DATA ANALYSIS: SNOWBOARDERS The table
shows the numbers of people (in millions) in the
United States who participated in snowboarding in
selected years from 2003 to 2007. (Source: National
Sporting Goods Association)
(a) Use a system of equations to find the equation of
the parabola that passes through
the points. Let represent the year, with
corresponding to 2003. Solve the system using
matrices.
(b) Use a graphing utility to graph the parabola.
(c) Use the equation in part (a) to estimate the number
of people who participated in snowboarding in
2009. Does your answer seem reasonable? Explain.
(d) Do you believe that the equation can be used for
years far beyond 2007? Explain.
NETWORK ANALYSIS In Exercises 115 and 116, answerthe questions about the specified network. (In a network it isassumed that the total flow into each junction is equal to thetotal flow out of each junction.)
115. Water flowing through a network of pipes (in thousands
of cubic meters per hour) is shown in the figure.
(a) Solve this system using matrices for the water flow
represented by
(b) Find the network flow pattern when and
(c) Find the network flow pattern when and
116. The flow of traffic (in vehicles per hour) through a
network of streets is shown in the figure.
(a) Solve this system using matrices for the traffic
flow represented by
(b) Find the traffic flow when and
(c) Find the traffic flow when and
EXPLORATION
TRUE OR FALSE? In Exercises 117 and 118, determinewhether the statement is true or false. Justify your answer.
117. is a matrix.
118. The method of Gaussian elimination reduces a matrix
until a reduced row-echelon form is obtained.
119. THINK ABOUT IT The augmented matrix below
represents system of linear equations (in variables
and ) that has been reduced using Gauss-Jordan elim-
ination. Write a system of equations with nonzero
coefficients that is represented by the reduced matrix.
(There are many correct answers.)
120. THINK ABOUT IT
(a) Describe the row-echelon form of an augmented
matrix that corresponds to a system of linear
equations that is inconsistent.
(b) Describe the row-echelon form of an augmented
matrix that corresponds to a system of linear
equations that has an infinite number of solutions.
121. Describe the three elementary row operations that can
be performed on an augmented matrix.
123. What is the relationship between the three elementary
row operations performed on an augmented matrix and
the operations that lead to equivalent systems of
equations?
�100
0
1
0
3
4
0
�2
1
0�
z
y,x,
4 2� 5
�1
0
3
�2
�6
7
0�
x3 � 0.x2 � 150
x3 � 50.x2 � 200
xi, i � 1, 2, . . . , 5.
350200
x1
x2 x4x3
x5
300 150
x6 � 500.
x5 � 400
x 7� 0.
x6 � 0
xi, i � 1, 2, . . . , 7.
600 500
x1
x6 x7
x2
x3 x4 x5
600 500
t � 3t
y � at2 � bt � c
y
y � ax2 � bx � c
Year Number, y
2003
2005
2007
6.3
6.0
5.1
122. CAPSTONE In your own words, describe the
difference between a matrix in row-echelon form and
a matrix in reduced row-echelon form. Include an
example of each to support your explanation.
584 Chapter 8 Matrices and Determinants
8.2 OPERATIONS WITH MATRICES
What you should learn
• Decide whether two matrices areequal.
• Add and subtract matrices and multiply matrices by scalars.
• Multiply two matrices.
• Use matrix operations to model and solve real-life problems.
Why you should learn it
Matrix operations can be usedto model and solve real-life problems.For instance, in Exercise 76 on page598, matrix operations are used toanalyze annual health care costs.
© R
oyal
ty-F
ree/
Cor
bis
Equality of Matrices
In Section 8.1, you used matrices to solve systems of linear equations. There is a rich
mathematical theory of matrices, and its applications are numerous. This section and
the next two introduce some fundamentals of matrix theory. It is standard mathematical
convention to represent matrices in any of the following three ways.
Two matrices and are equal if they have the same order
and for and In other words, two matrices are equal if
their corresponding entries are equal.
Equality of Matrices
Solve for and in the following matrix equation.
Solution
Because two matrices are equal only if their corresponding entries are equal, you can
conclude that
and
Now try Exercise 7.
Be sure you see that for two matrices to be equal, they must have the same order
and their corresponding entries must be equal. For instance,
but �2
3
0
�1
4
0� � �2
3
�1
4�.� 2
4
�112� � �2
2
�1
0.5�
a22 � 0.a21 � �3,a12 � �1,a11 � 2,
�a11
a21
a12
a22� � � 2
�3
�1
0�a22a11, a12, a21,
Example 1
1 ! j ! n.1 ! i ! maij � bij
�m n�B � �bij �A � �aij�
Representation of Matrices
1. A matrix can be denoted by an uppercase letter such as A, B, or C.
2. A matrix can be denoted by a representative element enclosed in brackets,
such as or
3. A matrix can be denoted by a rectangular array of numbers such as
A � �aij � � �a11
a21
a31
...
am1
a12
a22
a32
...
am2
a13
a23
a33
...
am3
. . .
. . .
. . .
. . .
a1n
a2n
a3n
...
amn
�.
�cij�.�bij�,�aij �,
Section 8.2 Operations with Matrices 585
Definition of Matrix Addition
If and are matrices of order their sum is the
matrix given by
The sum of two matrices of different orders is undefined.
A � B � �aij � bij � .
m nm n,B � �bij�A � �aij�
Matrix Addition and Scalar Multiplication
In this section, three basic matrix operations will be covered. The first two are matrix
addition and scalar multiplication. With matrix addition, you can add two matrices (of
the same order) by adding their corresponding entries.
Addition of Matrices
a.
b.
c.
d. The sum of
and
is undefined because is of order and B is of order
Now try Exercise 13(a).
In operations with matrices, numbers are usually referred to as scalars. In this text,
scalars will always be real numbers. You can multiply a matrix by a scalar by
multiplying each entry in by c.A
cA
3 2.3 3A
B � �0
�1
2
1
3
4�
A � �2
4
3
1
0
�2
0
�1
2�
�1
�3
�2� � �
�1
3
2� � �
0
0
0�
�0
1
1
2
�2
3� � �0
0
0
0
0
0� � �0
1
1
2
�2
3�
� � 0
�1
5
3�
��1
0
2
1� � � 1
�1
3
2� � ��1 � 1
0 � ��1�2 � 3
1 � 2�
Example 2
HISTORICAL NOTE
Arthur Cayley (1821–1895), aBritish mathematician, invented
matrices around 1858. Cayleywas a Cambridge University
graduate and a lawyer by profession. His groundbreaking
work on matrices was begun as he studied the theory of
transformations. Cayley also wasinstrumental in the development
of determinants. Cayley and two American mathematicians,Benjamin Peirce (1809–1880)and his son Charles S. Peirce
(1839–1914), are credited withdeveloping “matrix algebra.”
The
Gra
nger
Col
lect
ion
Definition of Scalar Multiplication
If is an matrix and is a scalar, the scalar multiple of by is
the matrix given by
cA � �caij� .
m n
cAcm nA � �aij�
The symbol represents the negation of which is the scalar product
Moreover, if and are of the same order, then represents the sum of and
That is,
Subtraction of matrices
The order of operations for matrix expressions is similar to that for real
numbers. In particular, you perform scalar multiplication before matrix addition and
subtraction, as shown in Example 3(c).
Scalar Multiplication and Matrix Subtraction
For the following matrices, find (a) , (b) and (c)
and
Solution
a. Scalar multiplication
Multiply each entry by 3.
Simplify.
b. Definition of negation
Multiply each entry by
c. Matrix subtraction
Subtract corresponding entries.
Now try Exercise 13(b), (c), and (d).
It is often convenient to rewrite the scalar multiple by factoring out of every
entry in the matrix. For instance, in the following example, the scalar has been
factored out of the matrix.
�12
52
�32
12� � �
12�1�12�5�
12��3�
12�1�� � 1
2�1
5
�3
1�
12
ccA
� �4
�10
7
6
4
0
12
�6
4�
3A � B � �6
�9
6
6
0
3
12
�3
6� � �
2
1
�1
0
�4
3
0
3
2�
�1.� ��2
�1
1
0
4
�3
0
�3
�2�
�B � ��1��2
1
�1
0
�4
3
0
3
2�
� �6
�9
6
6
0
3
12
�3
6�
� �3�2�
3��3�3�2�
3�2�3�0�3�1�
3�4�3��1�
3�2��
3A � 3�2
�3
2
2
0
1
4
�1
2�
B � �2
1
�1
0
�4
3
0
3
2�A � �
2
�3
2
2
0
1
4
�1
2�
3A � B.�B,3A
Example 3
A � B � A � ��1�B.
��1�B.
AA � BBA
��1�A.A,�A
586 Chapter 8 Matrices and Determinants
Section 8.2 Operations with Matrices 587
The properties of matrix addition and scalar multiplication are similar to those of
addition and multiplication of real numbers.
Note that the Associative Property of Matrix Addition allows you to write
expressions such as without ambiguity because the same sum occurs no
matter how the matrices are grouped. This same reasoning applies to sums of four or
more matrices.
Addition of More than Two Matrices
By adding corresponding entries, you obtain the following sum of four matrices.
Now try Exercise 19.
Using the Distributive Property
Perform the indicated matrix operations.
Solution
Now try Exercise 21.
In Example 5, you could add the two matrices first and then multiply the matrix
by 3, as follows. Notice that you obtain the same result.
3���2
4
0
1� � �4
3
�2
7� � 3�2
7
�2
8� � � 6
21
�6
24�
� � 6
21
�6
24�
� ��6
12
0
3� � �12
9
�6
21�
3���2
4
0
1� � �4
3
�2
7� � 3��2
4
0
1� � 3�4
3
�2
7�
3���2
4
0
1� � �4
3
�2
7�
Example 5
�1
2
�3� � �
�1
�1
2� � �
0
1
4� � �
2
�3
�2� � �
2
�1
1�
Example 4
A � B � C
Properties of Matrix Addition and Scalar Multiplication
Let and be matrices and let and be scalars.
1. Commutative Property of Matrix Addition
2. Associative Property of Matrix Addition
3. Associative Property of Scalar Multiplication
4. Scalar Identity Property
5. Distributive Property
6. Distributive Property�c � d�A � cA � dA
c�A � B� � cA � cB
1A � A
�cd �A � c�dA)
A � �B � C� � �A � B� � C
A � B � B � A
dcm nCB,A,
You can review the properties
of addition and multiplication
of real numbers (and other
properties of real numbers) in
Appendix A.1.
TECHNOLOGY
Most graphing utilities have thecapability of performing matrixoperations. Consult the user’sguide for your graphing utility for specific keystrokes. Try using a graphing utility to find thesum of the matrices
and
B ! �#1
2
4
#5�.
A ! � 2
#1
#3
0�
588 Chapter 8 Matrices and Determinants
One important property of addition of real numbers is that the number 0 is the
additive identity. That is, for any real number For matrices, a similar
property holds. That is, if is an matrix and is the zero matrix
consisting entirely of zeros, then
In other words, is the additive identity for the set of all matrices. For
example, the following matrices are the additive identities for the sets of all and
matrices.
and
zero matrix zero matrix
The algebra of real numbers and the algebra of matrices have many similarities. For
example, compare the following solutions.
Real Numbers Matrices
(Solve for x.) (Solve for X.)
The algebra of real numbers and the algebra of matrices also have important
differences, which will be discussed later.
Solving a Matrix Equation
Solve for in the equation where
and
Solution
Begin by solving the matrix equation for to obtain
Now, using the matrices and you have
Substitute the matrices.
Subtract matrix from matrix
Multiply the matrix by
Now try Exercise 31.
13.� ��
43
23
2
�23�.
B.A�1
3��4
2
6
�2�
X �1
3���3
2
4
1� � �1
0
�2
3� B,A
X �1
3�B � A�.
3X � B � A
X
B � ��3
2
4
1�.A � �1
0
�2
3�3X � A � B,X
Example 6
X � B � Ax � b � a
X � O � B � Ax � 0 � b � a
X � A � ��A� � B � ��A�x � a � ��a� � b � ��a�
X � A � Bx � a � b
m n
2 22 3
O � �0
0
0
0�O � �0
0
0
0
0
0�2 2
2 3
m nO
A � O � A.
m nOm nA
c.c � 0 � c
WARNING / CAUTION
Remember that matrices are
denoted by capital letters. So,
when you solve for you are
solving for a matrix that makes
the matrix equation true.
X,
Section 8.2 Operations with Matrices 589
Matrix Multiplication
Another basic matrix operation is matrix multiplication.At first glance, the definition
may seem unusual. You will see later, however, that this definition of the product of two
matrices has many practical applications.
The definition of matrix multiplication indicates a row-by-column multiplication,
where the entry in the th row and th column of the product is obtained by
multiplying the entries in the th row of by the corresponding entries in the th
column of and then adding the results. So for the product of two matrices to be
defined, the number of columns of the first matrix must equal the number of rows of
the second matrix. The general pattern for matrix multiplication is as follows.
Finding the Product of Two Matrices
Find the product using and
Solution
To find the entries of the product, multiply each row of by each column of
Now try Exercise 35.
� ��9
�4
�15
1
6
10�
� ���1���3� � ��3���4�
�4���3� � ��2���4��5���3� � ��0���4�
��1��2� � ��3��1��4��2� � ��2��1��5��2� � ��0��1��
AB � ��1
4
5
3
�2
0���3
�4
2
1�
B.A
B � ��3
�4
2
1�.A � ��1
4
5
3
�2
0�AB
Example 7
B
jAi
ABji
Definition of Matrix Multiplication
If is an matrix and is an matrix, the product
is an matrix
where ci j � ai1b1j � ai2b2 j � ai3b3j �. . . � ainbnj .
AB � �cij �
m p
ABn pB � �bij�m nA � �aij�
ai1b1j � ai2b2j � ai3b3j �. . . � ainbnj � cij
�c11
c21
.
..
ci1
.
..
cm1
c12
c22
.
..
ci2
.
..
cm2
. . .
. . .
. . .
. . .
c1j
c2j
.
..
cij
.
..
cmj
. . .
. . .
. . .
. . .
c1p
c2p
.
..
cip
.
..
cmp
��b11
b21
b31
.
..
bn1
b12
b22
b32
.
..
bn2
. . .
. . .
. . .
. . .
b1j
b2j
b3j
.
..
bnj
. . .
. . .
. . .
. . .
b1p
b2p
b3p
.
..
bnp
� ��a11
a21
a31...
ai1...
am1
a12
a22
a32...
ai2...
am2
a13
a23
a33...
ai3...
am3
. . .
. . .
. . .
. . .
. . .
a1n
a2n
a3n...
ain...
amn
�
In Example 7, the product is
defined because the number of
columns of is equal to the
number of rows of Also, note
that the product has order
3 2.
AB
B.
A
AB
Be sure you understand that for the product of two matrices to be defined, the number
of columns of the first matrix must equal the number of rows of the second matrix. That
is, the middle two indices must be the same. The outside two indices give the order of
the product, as shown below.
Finding the Product of Two Matrices
Find the product where
and
Solution
Note that the order of is and the order of is So, the product has
order
Now try Exercise 33.
Patterns in Matrix Multiplication
a.
b.
c. The product for the following matrices is not defined.
and
Now try Exercise 39.
3 43 2
B � ��2
0
2
3
1
�1
1
�1
0
4
2
1�A � �
�2
1
1
1
�3
4�
AB
3 13 13 3
�10
�5
�9���
1
2
�3��
6
3
1
2
�1
4
0
2
6�
2 22 22 2
� 3
�2
4
5���1
0
0
1�� 3
�2
4
5�
Example 9
� ��5
�3
7
6�
1�4� � 0�0� � 3�1�2�4� � ��1��0� � ��2��1��� � 1��2� � 0�1� � 3��1�
2��2� � ��1��1� � ��2���1�
AB � �1
2
0
�1
3
�2���2
1
�1
4
0
1�
2 2.
AB3 2.B2 3A
B � ��2
1
�1
4
0
1�.A � �1
2
0
�1
3
�2�
AB
Example 8
Equal
Order of AB
m pn pm n
AB�B A
590 Chapter 8 Matrices and Determinants
Section 8.2 Operations with Matrices 591
Patterns in Matrix Multiplication
a.
Now try Exercise 51.
In Example 10, note that the two products are different. Even if both and
are defined, matrix multiplication is not, in general, commutative. That is, for most
matrices, This is one way in which the algebra of real numbers and the
algebra of matrices differ.
AB � BA.
BAAB
1 13 11 3
� �1��2
�1
1��1 �2 �3�
Example 10
If A is an matrix, the identity matrix has the property that and
For example,
and
IA � A�1
0
0
0
1
0
0
0
1��
3
1
�1
�2
0
2
5
4
�3� � �
3
1
�1
�2
0
2
5
4
�3� .
AI � A�3
1
�1
�2
0
2
5
4
�3��
1
0
0
0
1
0
0
0
1� � �
3
1
�1
�2
0
2
5
4
�3�
InA � A.
AIn � An n
Properties of Matrix Multiplication
Let and be matrices and let be a scalar.
1. Associative Property of Matrix Multiplication
2. Distributive Property
3. Distributive Property
4. Associative Property of Scalar Multiplicationc�AB� � �cA�B � A�cB�
�A � B)C � AC � BC
A�B � C� � AB � AC
A�BC � � �AB�C
cCB,A,
Definition of Identity Matrix
The matrix that consists of 1’s on its main diagonal and 0’s elsewhere is
called the identity matrix of order and is denoted by
Identity matrix
Note that an identity matrix must be square. When the order is understood to be
you can denote simply by I.Inn n,
In � �1
0
0...
0
0
1
0...
0
0
0
1...
0
. . .
. . .
. . .
. . .
0
0
0...
1
�.
n n
n n
b.
3 31 33 1
� �2
�1
1
�4
2
�2
�6
3
�3��1 �2 �3��
2
�1
1�
Applications
Matrix multiplication can be used to represent a system of linear equations. Note how
the system
can be written as the matrix equation where is the coefficient matrix of the
system, and and are column matrices.
A X B
Solving a System of Linear Equations
Consider the following system of linear equations.
a. Write this system as a matrix equation,
b. Use Gauss-Jordan elimination on the augmented matrix to solve for the
matrix
Solution
a. In matrix form, the system can be written as follows.
b. The augmented matrix is formed by adjoining matrix to matrix
Using Gauss-Jordan elimination, you can rewrite this equation as
So, the solution of the system of linear equations is and
and the solution of the matrix equation is
Now try Exercise 61.
X � �x1x2x3� � �
�1
2
1�.
x3 � 1,x2 � 2,x1 � �1,
�I X� � �1
0
0
0
1
0
0
0
1
...
...
...
�1
2
1�.
�A B� � �1
0
2
�2
1
3
1
2
�2
..
.
..
.
..
.
�4
4
2�
A.B
�1
0
2
�2
1
3
1
2
�2� �
x1x2x3� � �
�4
4
2�
AX � B,
X.
�A B�AX � B.
�x1
2x1
� 2x2x2
� 3x2
� x3 �
� 2x3 �
� 2x3 �
�4
4
2
Example 11
�
�b1
b2
b3���
x1
x2
x3��
a11
a21
a31
a12
a22
a32
a13
a23
a33�
BX
AAX � B,
�a11x1 � a12x2 � a13x3 � b1
a21x1 � a22x2 � a23x3 � b2
a31x1 � a32x2 � a33x3 � b3
592 Chapter 8 Matrices and Determinants
The notation represents
the augmented matrix formed
when matrix is adjoined
to matrix The notation
represents the reduced
row-echelon form of the
augmented matrix that yields
the solution of the system.
�I ... X�A.
B
�A ... B�
The column matrix is also
called a constant matrix. Its
entries are the constant terms
in the system of equations.
B
Section 8.2 Operations with Matrices 593
Softball Team Expenses
Two softball teams submit equipment lists to their sponsors.
Women’s Team Men’s Team
Bats 12 15
Balls 45 38
Gloves 15 17
Each bat costs $80, each ball costs $6, and each glove costs $60. Use matrices to find
the total cost of equipment for each team.
Solution
The equipment lists and the costs per item can be written in matrix form as
and
The total cost of equipment for each team is given by the product
So, the total cost of equipment for the women’s team is $2130 and the total cost of
equipment for the men’s team is $2448.
Now try Exercise 69.
� �2130 2448�.
� �80�12� � 6�45� � 60�15� 80�15� � 6�38� � 60�17��
CE � �80 6 60��12
45
15
15
38
17�
C � �80 6 60� .
E � �12
45
15
15
38
17�
CE
Example 12
Problem Posing Write a matrix multiplication application problem that uses the matrix
Exchange problems with another student in your class. Form the matrices that represent the problem, and solve the problem. Interpret your solution in the contextof the problem. Check with the creator of the problem to see if you are correct.Discuss other ways to represent and/or approach the problem.
A ! [20
17
42
30
33
50] .
CLASSROOM DISCUSSION
Notice in Example 12 that you
cannot find the total cost using
the product because is
not defined. That is, the number
of columns of (2 columns)
does not equal the number of
rows of (1 row).C
E
ECEC
594 Chapter 8 Matrices and Determinants
EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8.2
VOCABULARY
In Exercises 1–4, fill in the blanks.
1. Two matrices are ________ if all of their corresponding entries are equal.
2. When performing matrix operations, real numbers are often referred to as ________.
3. A matrix consisting entirely of zeros is called a ________ matrix and is denoted by ________.
4. The matrix consisting of 1’s on its main diagonal and 0’s elsewhere is called the ________
matrix of order
In Exercises 5 and 6, match the matrix property with the correct form. and are matrices of order and and are scalars.
5. (a) (i) Distributive Property
(b) (ii) Commutative Property of Matrix Addition
(c) (iii) Scalar Identity Property
(d) (iv) Associative Property of Matrix Addition
(e) (v) Associative Property of Scalar Multiplication
6. (a) (i) Distributive Property
(b) (ii) Additive Identity of Matrix Addition
(c) (iii) Associative Property of Matrix Multiplication
(d) (iv) Associative Property of Scalar Multiplication
SKILLS AND APPLICATIONS
A�BC� � �AB�CA�B � C� � AB � AC
c�AB� � A�cB�A � O � A
A � B � B � A
�cd�A � c�dA��c � d�A � cA � dA
A � �B � C� � �A � B� � C
1A � A
dcm n,CB,A,
n n.
n n
In Exercises 7–10, find and
7. 8.
9.
10.
In Exercises 11–18, if possible, find (a) (b) (c ) and (d)
11.
12.
13.
14.
15.
16.
17.
18.
In Exercises 19–24, evaluate the expression.
19.
20.
21. 4���400
2
1
3� � �231
�6
�2
0� � 6
�1
8
0� � � 0
�3
5
�1� � ��112�7
�1���53
0
�6� � � 7
�2
1
�1� � ��1014�8
6�
B � ��4 6 2�A � �3
2
�1�,
B � �84�1
�3�A � � 6
�1
0
�4
3
0�,
B � ��3
2
10
3
0
5
�4
�9
2
1
1
�7
�1
�4
�2�A � �
�1
3
5
0
�4
4
�2
4
8
�1
0
2
�1
�6
0�,
B � � 1
�6
0
8
�1
2
1
�3
0
�7�
A � �415
2
�1
�2
3
�1
4
0�,
B � ��2�3
0
4
�5
�7�A � �10�1
6
3
9�,
B � �1
�1
1
6
�5
10�A � �
8
2
�4
�1
3
5�,B � ��34
�2
2�A � �122
1�,B � � 2
�1
�1
8�A � �12�1
�1�,3A ! 2B.3A,
A ! B,A 1 B,
�x � 2
1
7
8
2y
�2
�3
2x
y � 2� � �
2x � 6
1
7
8
18
�2
�3
�8
11�
�16
�3
0
4
13
2
5
15
4
4
6
0� � �
16
�3
0
4
13
2
2x � 1
15
3y � 5
4
3x
0�
��5yx
8� � ��51213
8�� x7�2
y � � ��47�2
22�y.x
Section 8.2 Operations with Matrices 595
22.
23.
24.
In Exercises 25–28, use the matrix capabilities of a graphingutility to evaluate the expression. Round your results to threedecimal places, if necessary.
25.
26.
27.
28.
In Exercises 29–32, solve for in the equation, given
and
29. 30.
31. 32.
In Exercises 33–40, if possible, find and state the order ofthe result.
33.
34.
35.
36.
37.
38.
39.
40.
In Exercises 41–46, use the matrix capabilities of a graphingutility to find if possible.
41.
42.
43.
44.
45.
46.
In Exercises 47–52, if possible, find (a) (b) and (c) (Note: )
47.
48.
49.
50.
51.
52. A � �3 2 1� , B � �2
3
0�
B � �1 1 2�A � �7
8
�1�,
B � � 1
�3
3
1�A � �11�1
1�,B � �13
�3
1�A � �31�1
3�,A � � 6
�2
3
�4�, B � ��220
4�
B � � 2
�1
�1
8�A � �142
2�,A2" AA.
A2.BA,AB,
A � �16
�4
�9
�18
13
21�, B � ��77
20
15
�1
26�
B � �5240�85
�35
27
60
45
82�
A � � 9
100
10
�50
�38
250
18
75�,
A � ��2
21
13
4
5
2
8
6
6�, B � �
2
�7
32
0.5
0
15
14
1.6�
B � �3
24
16
8
1
15
10
�4
6
14
21
10�A � �
�3
�12
5
8
15
�1
�6
9
1
8
6
5�,
A � �11
14
6
�12
10
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4
12
9�, B � �
12
�5
15
10
12
16�
A � �7
�2
10
5
5
�4
�4
1
�7�, B � �
2
8
�4
�2
1
2
3
4
�8�
AB,
B � �146
2�A � �160
13
3
8
�2
�17�,B � �6 �2 1 6�A � �1012�,
A � �0
0
0
0
0
0
5
�3
4�, B � �
6
8
0
�11
16
0
4
4
0�
A � �5
0
0
0
�8
0
0
0
7�, B � �
15
0
0
0
�18
0
0
012
�A � �
1
0
0
0
4
0
0
0
�2�, B � �
3
0
0
0
�1
0
0
0
5�
A � ��1
�4
0
6
5
3�, B � �20
3
9�
A � �0
6
7
�1
0
�1
2
3
8�, B � �
2
4
1
�1
�5
6�
A � �2
�3
1
1
4
6�, B � �
0
4
8
�1
0
�1
0
2
7�
AB
2A � 4B � �2X2X � 3A � B
2X � 2A � BX � 3A � 2B
B " [0
2
!4
3
0
!1].A " [
!2
1
3
!1
0
!4]
X
��10
�20
12
15
10
4� � 18��
�13
7
6
11
0
9� � �
�3
�3
�14
13
8
15�
��3.211
�1.004
0.055
6.829
4.914
�3.889� � �
�1.630
5.256
�9.768
�3.090
8.335
4.251�
55�� 14
�22
�11
19� � ��221320
6�
3
7�2
�1
5
�4� � 6��3
2
0
2�
��4
�2
9
11
�1
3� � 16��
�5
3
0
�1
4
13� � �
7
�9
6
5
�1
�1�
�3��07�3
2� � ��683
1� � 2�4
7
�4
�9�
12��5 �2 4 0� � �14 6 �18 9��
596 Chapter 8 Matrices and Determinants
In Exercises 53–56, evaluate the expression. Use the matrixcapabilities of a graphing utility to verify your answer.
53.
54.
55.
56.
In Exercises 57–64, (a) write the system of linear equations as a matrix equation, and (b) use Gauss-Jordanelimination on the augmented matrix to solve for thematrix
57. 58.
59. 60.
61.
62.
63.
64.
65. MANUFACTURING A corporation has three factories,
each of which manufactures acoustic guitars and
electric guitars. The number of units of guitars produced
at factory in one day is represented by in the matrix
Find the production levels if production is increased by
20%.
66. MANUFACTURING A corporation has four factories,
each of which manufactures sport utility vehicles and
pickup trucks. The number of units of vehicle produced
at factory in one day is represented by in the matrix
Find the production levels if production is increased by
10%.
67. AGRICULTURE A fruit grower raises two crops,
apples and peaches. Each of these crops is sent to three
different outlets for sale. These outlets are The Farmer’s
Market, The Fruit Stand, and The Fruit Farm. The
numbers of bushels of apples sent to the three outlets are
125, 100, and 75, respectively. The numbers of bushels
of peaches sent to the three outlets are 100, 175, and
125, respectively. The profit per bushel for apples is
$3.50 and the profit per bushel for peaches is $6.00.
(a) Write a matrix that represents the number of
bushels of each crop that are shipped to each outlet
State what each entry of the matrix represents.
(b) Write a matrix that represents the profit per
bushel of each fruit. State what each entry of the
matrix represents.
(c) Find the product and state what each entry of the
matrix represents.
68. REVENUE An electronics manufacturer produces three
models of LCD televisions, which are shipped to two
warehouses. The numbers of units of model that are
shipped to warehouse are represented by in the matrix
The prices per unit are represented by the matrix
Compute and interpret the result.
69. INVENTORY A company sells five models of computers
through three retail outlets. The inventories are repre-
sented by
Model
A B C D E
The wholesale and retail prices are represented by
Price
Wholesale Retail
Compute and interpret the result.ST
A
B
C
D
E
ModelT � �$840
$1200
$1450
$2650
$3050
$1100
$1350
$1650
$3000
$3200�
T.
1
2
3 OutletS � �3
0
4
2
2
2
2
3
1
3
4
3
0
3
2�
S.
BA
B � �$699.95 $899.95 $1099.95�.
A � �5,000
6,000
8,000
4,000
10,000
5,000�.
aijj
i
BA
bij
B
aijj.
i
A
A � �100
40
90
20
70
60
30
60�.
aijj
i
A � �70
35
50
100
25
70� .
aijj
�x1 �
x1 �
x2
3x2
�6x2
�
�
4x3 �
�
5x3 �
17
�11
40
�x1 �
�3x1 �
5x2
x2
�2x2
� 2x3 �
� x3 �
� 5x3 �
�20
8
�16
� x1
�x1
x1
�
�
�
x2
2x2
x2
�
�
3x3
x3
�
�
�
�1
1
2
�x1 � 2x2 � 3x3 �
�x1 � 3x2 � x3 �
2x1 � 5x2 � 5x3 �
9
�6
17
��4x1
x1
�
�
9x2
3x2
�
�
�13
12��2x1
6x1
�
�
3x2
x2
�
�
�4
�36
�2x1
x1
�
�
3x2
4x2
�
�
5
10� �x1
�2x1
�
�
x2
x2
�
�
4
0
X.[A B]
AX " B,
�3
�1
5
7���5 �6� � �7 �1� � ��8 9��
�0
4
2
1
�2
2���4
0
�1
0
�1
2� � �
�2
�3
0
3
5
�3�
�3��6
1
5
�2
�1
0��0
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4
3
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1�
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0
1
�2��1
�2
0
2��1
2
0
4�
Section 8.2 Operations with Matrices 597
70. VOTING PREFERENCES The matrix
From
R D I
is called a stochastic matrix. Each entry
represents the proportion of the voting population that
changes from party to party and represents the
proportion that remains loyal to the party from one
election to the next. Compute and interpret
71. VOTING PREFERENCES Use a graphing utility to
find and for the matrix given in
Exercise 70. Can you detect a pattern as is raised to
higher powers?
72. LABOR/WAGE REQUIREMENTS A company that
manufactures boats has the following labor-hour and
wage requirements.
Labor per boat
Department
Cutting AssemblyPackaging
Wages per hour
Plant
A B
Compute and interpret the result.
73. PROFIT At a local dairy mart, the numbers of gallons
of skim milk, 2% milk, and whole milk sold over the
weekend are represented by
Skim 2% Wholemilk milk milk
The selling prices (in dollars per gallon) and the profits
(in dollars per gallon) for the three types of milk sold by
the dairy mart are represented by
Selling Profitprice
(a) Compute and interpret the result.
(b) Find the dairy mart’s total profit from milk sales for
the weekend.
74. PROFIT At a convenience store, the numbers of
gallons of 87-octane, 89-octane, and 93-octane gasoline
sold over the weekend are represented by
Octane
87 89 93
The selling prices (in dollars per gallon) and the profits
(in dollars per gallon) for the three grades of gasoline
sold by the convenience store are represented by
Selling Profitprice
(a) Compute and interpret the result.
(b) Find the convenience store’s profit from gasoline
sales for the weekend.
75. EXERCISE The numbers of calories burned by indi-
viduals of different body weights performing different
types of aerobic exercises for a 20-minute time period
are shown in matrix
Calories burned
120-lb 150-lbperson person
(a) A 120-pound person and a 150-pound person bicycled
for 40 minutes, jogged for 10 minutes, and walked for
60 minutes. Organize the time they spent exercising in
a matrix
(b) Compute and interpret the result.BA
B.
Bicycling
Jogging
Walking
A � �109
127
64
136
159
79�
A.
AB
87
89
93 OctaneB � �
$2.00
$2.10
$2.20
$0.08
$0.09
$0.10�
B.
Friday
Saturday
Sunday
A � �580
560
860
840
420
1020
320
160
540�
A.
AB
Skim milk
2% milk
Whole milk
B � �$3.45
$3.65
$3.85
$1.20
$1.30
$1.45�
B.
Friday
Saturday
Sunday
A � �40
60
76
64
82
96
52
76
84�
A.
ST
Cutting
Assembly
Packaging
DepartmentT � �$15
$12
$11
$13
$11
$10�
Small
Medium
Large
Boat sizeS � �1.0 h
1.6 h
2.5 h
0.5 h
1.0 h
2.0 h
0.2 h
0.2 h
1.4 h�
P
P8P7,P6,P5,P4,P3,
P2.
piij,i
pij �i � j�
R
D
I ToP � �
0.6
0.2
0.2
0.1
0.7
0.2
0.1
0.1
0.8�
598 Chapter 8 Matrices and Determinants
76. HEALTH CARE The health care plans offered this
year by a local manufacturing plant are as follows. For
individuals, the comprehensive plan costs $694.32, the
HMO standard plan costs $451.80, and the HMO Plus
plan costs $489.48. For families, the comprehensive
plan costs $1725.36, the HMO standard plan costs
$1187.76, and the HMO Plus plan costs $1248.12. The
plant expects the costs of the plans to change next year
as follows. For individuals, the costs for the compre-
hensive, HMO standard, and HMO Plus plans will be
$683.91, $463.10, and $499.27, respectively. For fami-
lies, the costs for the comprehensive, HMO standard,
and HMO Plus plans will be $1699.48, $1217.45, and
$1273.08, respectively.
(a) Organize the information using two matrices and
where represents the health care plan costs for
this year and represents the health care plan costs
for next year. State what each entry of each matrix
represents.
(b) Compute and interpret the result.
(c) The employees receive monthly paychecks from
which the health care plan costs are deducted. Use
the matrices from part (a) to write matrices that show
how much will be deducted from each employees’
paycheck this year and next year.
(d) Suppose instead that the costs of the health care
plans increase by 4% next year. Write a matrix that
shows the new monthly payments.
EXPLORATION
TRUE OR FALSE? In Exercises 77 and 78, determinewhether the statement is true or false. Justify your answer.
77. Two matrices can be added only if they have the same
order.
78. Matrix multiplication is commutative.
THINK ABOUT IT In Exercises 79–86, let matrices and be of orders and respectively.Determine whether the matrices are of proper order to perform the operation(s). If so, give the order of the answer.
79. 80.
81. 82.
83. 84.
85. 86.
87. Consider matrices and below. Perform the
indicated operations and compare the results.
(a) Find and
(b) Find then add to the resulting matrix.
Find then add to the resulting matrix.
(c) Find and then add the two resulting matrices.
Find then multiply the resulting matrix by 2.
88. Use the following matrices to find and
What do your results tell you about matrix
multiplication, commutativity, and associativity?
89. THINK ABOUT IT If and are real numbers such
that and then However, if
and are nonzero matrices such that then
is not necessarily equal to Illustrate this using the
following matrices.
90. THINK ABOUT IT If and are real numbers such
that then or However, if and
are matrices such that it is not necessarily true
that or Illustrate this using the following
matrices.
91. Let and be unequal diagonal matrices of the same
order. (A diagonal matrix is a square matrix in which
each entry not on the main diagonal is zero.) Determine
the products for several pairs of such matrices. Make
a conjecture about a quick rule for such products.
92. Let and let
and
(a) Find and Identify any similarities with
and
(b) Find and identify
93. Find two matrices and such that AB � BA.BA
B2.
i 4.i 3,
i 2,A4.A3,A2,
B � �0
i
�i
0�.A � � i00
i�i � �1
AB
BA
B � � 1
�1
�1
1�A � �3
4
3
4�,
B � O.A � O
AB � O,
BAb � 0.a � 0ab � 0,
ba
C � �2
2
3
3�B � �1
1
0
0�,A � �0
0
1
1�,
B.
AAC � BC,C
B,A,a � b.ac � bc,c � 0
cb,a,
C � �3
0
0
1�B � �0
2
1
3�,A � �1
3
2
4�,
A�BC�.�AB�C,BA,AB,
A � B,
2B,2A
AB � C,
CA � B,
B � A.A � B
C � �5
2
2
�6�B � ��2
8
0
1�,A � �3
4
�1
7�,
CB,A,
�BC � D�AD�A � 3B�CB � DBC � D
BCAB
B � 3CA � 2C
2 2,3 2,2 3,2 3,D
C,B,A,
A � B
B
A
B,A
94. CAPSTONE Let matrices and be of orders
and respectively. Answer the following
questions and explain your reasoning.
(a) Is it possible that
(b) Is defined?
(c) Is defined? If so, is it possible that AB � BA?AB
A � B
A � B?
2 2,3 2
BA
Section 8.3 The Inverse of a Square Matrix 599
8.3 THE INVERSE OF A SQUARE MATRIX
What you should learn
• Verify that two matrices are inverses of each other.
• Use Gauss-Jordan elimination to find the inverses of matrices.
• Use a formula to find the inverses of matrices.
• Use inverse matrices to solve systems of linear equations.
Why you should learn it
You can use inverse matrices to model and solve real-life problems.For instance, in Exercise 75 on page607, an inverse matrix is used to finda quadratic model for the enrollmentprojections for public universities inthe United States.
2 2
Alberto L. Pomares/istockphoto.com
The Inverse of a Matrix
This section further develops the algebra of matrices. To begin, consider the real number
equation To solve this equation for multiply each side of the equation by
(provided that ).
The number is called the multiplicative inverse of because The definition
of the multiplicative inverse of a matrix is similar.
The Inverse of a Matrix
Show that is the inverse of where
and
Solution
To show that is the inverse of show that as follows.
As you can see, This is an example of a square matrix that has an
inverse. Note that not all square matrices have inverses.
Now try Exercise 5.
Recall that it is not always true that even if both products are defined.
However, if and are both square matrices and it can be shown that
So, in Example 1, you need only to check that AB � I2.BA � In .
AB � In ,BA
AB � BA,
AB � I � BA.
BA � �1
1
�2
�1���1
�1
2
1� � ��1 � 2
�1 � 1
2 � 2
2 � 1� � �1
0
0
1�
AB � ��1
�1
2
1� �1
1
�2
�1� � ��1 � 2
�1 � 1
2 � 2
2 � 1� � �1
0
0
1�AB � I � BA,A,B
B � �1
1
�2
�1�.A � ��1
�1
2
1�A,B
Example 1
a�1a � 1.aa�1
x � a�1b
�1�x � a�1b
�a�1a�x � a�1b
ax � b
a � 0
a�1x,ax � b.
Definition of the Inverse of a Square Matrix
Let be an matrix and let be the identity matrix. If there exists a
matrix such that
then is called the inverse of The symbol is read “ inverse.”AA�1A.A�1
AA�1 � In � A�1A
A�1
n nInn nA
Finding Inverse Matrices
If a matrix has an inverse, is called invertible (or nonsingular); otherwise, is
called singular. A nonsquare matrix cannot have an inverse. To see this, note that if
is of order and is of order (where ), the products and are
of different orders and so cannot be equal to each other. Not all square matrices have
inverses (see the matrix at the bottom of page 602). If, however, a matrix does have an
inverse, that inverse is unique. Example 2 shows how to use a system of equations to
find the inverse of a matrix.
Finding the Inverse of a Matrix
Find the inverse of
Solution
To find the inverse of try to solve the matrix equation for
A X I
Equating corresponding entries, you obtain two systems of linear equations.
Linear system with two variables, and
Linear system with two variables, and
Solve the first system using elementary row operations to determine that and
From the second system you can determine that and
Therefore, the inverse of is
You can use matrix multiplication to check this result.
Check
�
�
Now try Exercise 15.
A�1A � ��3
1
�4
1��1
�1
4
�3� � �1
0
0
1�
AA�1 � � 1
�1
4
�3���3
1
�4
1� � �1
0
0
1�
� ��3
1
�4
1�.
X � A�1
A
x22 � 1.x12 � �4x21 � 1.
x11 � �3
x22.x12� x12 � 4x22 � 0
�x12 � 3x22 � 1
x21.x11� x11 � 4x21 � 1
�x11 � 3x21 � 0
� x11 � 4x21
�x11 � 3x21
x12 � 4x22
�x12 � 3x22� � �1
0
0
1�
� 1
�1
4
�3��x11
x21
x12
x22� � �1
0
0
1�
X.AX � IA,
A � � 1
�1
4
�3�.
Example 2
BAABm � nn mBm n
A
AAA
600 Chapter 8 Matrices and Determinants
Section 8.3 The Inverse of a Square Matrix 601
In Example 2, note that the two systems of linear equations have the same
coefficient matrix Rather than solve the two systems represented by
and
separately, you can solve them simultaneously by adjoining the identity matrix to the
coefficient matrix to obtain
A I
This “doubly augmented” matrix can be represented as By applying
Gauss-Jordan elimination to this matrix, you can solve both systems with a single
elimination process.
So, from the “doubly augmented” matrix you obtain the matrix
A I I
This procedure (or algorithm) works for any square matrix that has an inverse.
�1
0
0
1
.
..
.
..�3
1
�4
1�� 1
�1
4
�3
.
..
.
..1
0
0
1�A�1
�I A�1�.�A I�,
�4R2 � R1 → � 1
0
0
1
.
..
.
..�3
1
�4
1�R1 � R2 →
� 1
0
4
1
.
..
.
..1
1
0
1�
� 1
�1
4
�3
.
..
.
..1
0
0
1�
�A I�.
� 1
�1
4
�3
.
..
.
..1
0
0
1�.
� 1
�1
4
�3
.
..
.
..0
1�
� 1
�1
4
�3
.
..
.
..1
0�A.
Finding an Inverse Matrix
Let be a square matrix of order
1. Write the matrix that consists of the given matrix on the left and the
identity matrix on the right to obtain
2. If possible, row reduce to using elementary row operations on the entire
matrix The result will be the matrix If this is not possible,
is not invertible.
3. Check your work by multiplying to see that AA�1 � I � A�1A.
A
�I A�1�.�A I�.IA
�A I�.In n
An 2n
n.A
TECHNOLOGY
Most graphing utilities can findthe inverse of a square matrix.To do so, you may have to usethe inverse key .Consult theuser’s guide for your graphingutility for specific keystrokes.
x�1
Finding the Inverse of a Matrix
Find the inverse of
Solution
Begin by adjoining the identity matrix to to form the matrix
Use elementary row operations to obtain the form as follows.
So, the matrix is invertible and its inverse is
Confirm this result by multiplying and to obtain as follows.
Check
Now try Exercise 19.
The process shown in Example 3 applies to any matrix When using this
algorithm, if the matrix does not reduce to the identity matrix, then does not have
an inverse. For instance, the following matrix has no inverse.
To confirm that matrix above has no inverse, adjoin the identity matrix to to form
and perform elementary row operations on the matrix. After doing so, you will
see that it is impossible to obtain the identity matrix on the left. Therefore, is not
invertible.
AI
�A I�AA
A � �1
3
�2
2
�1
3
0
2
�2�
AA
A.n n
AA�1 � �1
1
6
�1
0
�2
0
�1
�3� ��2
�3
�2
�3
�3
�4
1
1
1� � �
1
0
0
0
1
0
0
0
1� � I
I,A�1A
A�1 � ��2
�3
�2
�3
�3
�4
1
1
1�.
A
R3 � R1 →
R3 � R2 → �1
0
0
0
1
0
0
0
1
.
..
.
..
.
..
�2
�3
�2
�3
�3
�4
1
1
1� � �I .
.. A�1�
R2 � R1 →
�4R2 � R3 →�1
0
0
0
1
0
�1
�1
1
.
..
.
..
.
..
0
�1
�2
1
1
�4
0
0
1�
�R1 � R2 →
�6R1 � R3 →�1
0
0
�1
1
4
0
�1
�3
.
..
.
..
.
..
1
�1
�6
0
1
0
0
0
1�
�I A�1�,
�A... I� � �
1
1
6
�1
0
�2
0
�1
�3
.
..
.
..
.
..
1
0
0
0
1
0
0
0
1�.
A
A � �1
1
6
�1
0
�2
0
�1
�3�.
Example 3
602 Chapter 8 Matrices and Determinants
WARNING / CAUTION
Be sure to check your solution
because it is easy to make
algebraic errors when using
elementary row operations.
Section 8.3 The Inverse of a Square Matrix 603
The Inverse of a 2 2 Matrix
Using Gauss-Jordan elimination to find the inverse of a matrix works well (even as a
computer technique) for matrices of order or greater. For matrices, however,
many people prefer to use a formula for the inverse rather than Gauss-Jordan
elimination. This simple formula, which works only for matrices, is explained as
follows. If is a matrix given by
then is invertible if and only if Moreover, if the inverse
is given by
Formula for inverse of matrix A
The denominator is called the determinant of the matrix You will
study determinants in the next section.
Finding the Inverse of a 2 2 Matrix
If possible, find the inverse of each matrix.
a.
b.
Solution
a. For the matrix apply the formula for the inverse of a matrix to obtain
Because this quantity is not zero, the inverse is formed by interchanging the entries
on the main diagonal, changing the signs of the other two entries, and multiplying
by the scalar as follows.
Substitute for and the determinant.
Multiply by the scalar
b. For the matrix you have
which means that is not invertible.
Now try Exercise 35.
B
� 0
ad � bc � �3��2� � ��1���6�
B,
14.� �
12
12
14
34�
d,c,b,a,A�1 �14�2
2
1
3�
14,
� 4.
ad � bc � �3��2� � ��1���2�
2 2A,
B � � 3
�6
�1
2�
A � � 3
�2
�1
2�
Example 4
A.2 2ad � bc
A�1 �1
ad � bc �d
�c
�b
a�.
ad � bc � 0,ad � bc � 0.A
A � �acb
d�2 2A
2 2
2 23 3
604 Chapter 8 Matrices and Determinants
Systems of Linear Equations
You know that a system of linear equations can have exactly one solution, infinitely
many solutions, or no solution. If the coefficient matrix of a square system (a system
that has the same number of equations as variables) is invertible, the system has a
unique solution, which is defined as follows.
Solving a System Using an Inverse Matrix
You are going to invest $10,000 in AAA-rated bonds, AA-rated bonds, and
B-rated bonds and want an annual return of $730. The average yields are 6% on AAA
bonds, 7.5% on AA bonds, and 9.5% on B bonds. You will invest twice as much in AAA
bonds as in B bonds. Your investment can be represented as
where and represent the amounts invested in AAA, AA, and B bonds, respectively.
Use an inverse matrix to solve the system.
Solution
Begin by writing the system in the matrix form
Then, use Gauss-Jordan elimination to find
Finally, multiply by on the left to obtain the solution.
The solution of the system is and So, you will invest
$4000 in AAA bonds, $4000 in AA bonds, and $2000 in B bonds.
Now try Exercise 65.
z � 2000.y � 4000,x � 4000,
� �4000
4000
2000�� �
15
�21.5
7.5
�200
300
�100
�2
3.5
�1.5� �
10,000
730
0�
X � A�1B
A�1B
A�1 � �15
�21.5
7.5
�200
300
�100
�2
3.5
�1.5�
A�1.
�x
y
z� � �
10,000
730
0��
1
0.06
1
1
0.075
0
1
0.095
�2�
AX � B.
zy,x,
� x
0.06x
x
�
�
y
0.075y
�
�
�
z �
0.095z �
2z �
10,000
730
0
Example 5
A
A System of Equations with a Unique Solution
If is an invertible matrix, the system of linear equations represented by
has a unique solution given by
X � A�1B.
AX � B
A
TECHNOLOGY
To solve a system of equationswith a graphing utility, enter thematrices and in the matrix editor. Then, using the inversekey, solve for
The screen will display thesolution, matrix X.
ENTERBA
X.
BA
x�1
Section 8.3 The Inverse of a Square Matrix 605
EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8.3VOCABULARY: Fill in the blanks.
1. In a ________ matrix, the number of rows equals the number of columns.
2. If there exists an matrix such that then is called the ________ of
3. If a matrix has an inverse, it is called invertible or ________; if it does not have an inverse, it is called ________.
4. If is an invertible matrix, the system of linear equations represented by has a unique solution given by
________.
SKILLS AND APPLICATIONS
X �
AX � BA
A
A.A�1AA�1 � In � A�1A,A�1n n
In Exercises 5–12, show that is the inverse of
5.
6.
7.
8.
9.
10.
11.
12.
In Exercises 13–24, find the inverse of the matrix (if it exists).
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
In Exercises 25–34, use the matrix capabilities of a graphingutility to find the inverse of the matrix (if it exists).
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
In Exercises 35–40, use the formula on page 603 to find theinverse of the matrix (if it exists).
35. 36.
37. 38. ��12
5
3
�2���4
2
�6
3�� 1
�3
�2
2�� 2
�1
3
5�2 2
�1
3
2
�1
�2
�5
�5
4
�1
�2
�2
4
�2
�3
�5
11��
�1
0
2
0
0
2
0
�1
1
0
�1
0
0
�1
0
1�
�0.6
0.7
1
0
�1
0
�0.3
0.2
�0.9��
0.1
�0.3
0.5
0.2
0.2
0.4
0.3
0.2
0.4�
��56
0
1
13
23
�12
116
2
�52
���12
1
0
34
0
�1
14
�3212
��
3
2
�4
2
2
4
2
2
3��
1
3
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1
1
0
2
0
3�
�10
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3
5
1
2
�7
4
�2��
1
3
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2
7
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0
0
0
3
2
0
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4
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0
0
6
1
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0
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0
0
1
0
0
0
0
4
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0
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3
2
0
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5
0
0
5��
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2
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0
0
5
0
0
7�
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3
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2
7
�4
2
9
�7��
1
3
3
1
5
6
1
4
5�
� 4
�3
�1
1��3
4
1
2���7
4
33
�19��1
2
�2
�3��1
3
2
7��2
0
0
3�
B �1
3��3
�3
0
�3
1
�1
1
�2
1
2
1
1
�3
�3
0
0�
A � ��1
1
�1
0
1
�1
1
�1
0
1
2
1
�1
0
0
1�,
A � ��2
1
0
2
�1
1
3
0
4�, B �
1
3��4
�4
1
�5
�8
2
3
3
0�
B � ��12
14
�14
1
�1
1
32
�114
74
�A � ��4
�1
0
1
2
�1
5
4
�1�,
A � �2
�1
0
�17
11
3
11
�7
�2�, B � �
1
2
3
1
4
6
2
�3
�5�
B � �35
�25
15
15�A � �1
2
�1
3�,
B � ��232
1
�12�A � �1
3
2
4�,
B � �2
1
1
1�A � � 1
�1
�1
2�,
B � � 3
�5
�1
2�A � �2
5
1
3�,
A.B
606 Chapter 8 Matrices and Determinants
39. 40.
In Exercises 41–44, use the inverse matrix found in Exercise15 to solve the system of linear equations.
41. 42.
43. 44.
In Exercises 45 and 46, use the inverse matrix found inExercise 19 to solve the system of linear equations.
45. 46.
In Exercises 47 and 48, use the inverse matrix found inExercise 34 to solve the system of linear equations.
47.
48.
In Exercises 49 and 50, use a graphing utility to solve the system of linear equations using an inverse matrix.
49.
50.
In Exercises 51–58, use an inverse matrix to solve (if possible)the system of linear equations.
51. 52.
53. 54.
55. 56.
57. 58.
In Exercises 59–62, use the matrix capabilities of a graphingutility to solve (if possible) the system of linear equations.
59. 60.
61.
62.
In Exercises 63 and 64, show that the matrix is invertible andfind its inverse.
63. 64.
INVESTMENT PORTFOLIO In Exercises 65–68, consider aperson who invests in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 6.5% on AAA bonds, 7%on A bonds, and 9% on B bonds. The person invests twice asmuch in B bonds as in A bonds. Let and represent theamounts invested in AAA, A, and B bonds, respectively.
Use the inverse of the coefficient matrix of this system to findthe amount invested in each type of bond.
Total Investment Annual Return
65. $10,000 $705
66. $10,000 $760
67. $12,000 $835
68. $500,000 $38,000
PRODUCTION In Exercises 69–72, a small home businesscreates muffins, bones, and cookies for dogs. In addition toother ingredients, each muffin requires 2 units of beef, 3 unitsof chicken, and 2 units of liver. Each bone requires 1 unit ofbeef, 1 unit of chicken, and 1 unit of liver. Each cookie requires2 units of beef, 1 unit of chicken, and 1.5 units of liver. Find thenumbers of muffins, bones, and cookies that the company cancreate with the given amounts of ingredients.
� x
0.065x
1
1
y 1
0.07y 1
2y !
z "
0.09z "
z "
!total investment"!annual return"0
zy,x,
A � �sec "
tan "
tan "
sec "�A � � sin "
�cos "
cos "
sin "�
��8x � 7y � 10z �
12x � 3y � 5z �
15x � 9y � 2z �
�151
86
187
�3x � 2y � z �
�4x � y � 3z �
x � 5y � z �
�29
37
�24
�2x � 3y � 5z � 4
3x � 5y � 9z � 7
5x � 9y � 17z � 13�
5x � 3y � 2z �
2x � 2y � 3z �
x � 7y � 8z �
2
3
�4
�4x � 2y � 3z �
2x � 2y � 5z �
8x � 5y � 2z �
�2
16
4�
4x � y � z �
2x � 2y � 3z �
5x � 2y � 6z �
�5
10
1
�56 x �43 x �
y � �2072 y � �51��
14 x �
38 y � �2
32 x �
34 y � �12
�0.2x �
�x �
0.6y �
1.4y �
2.4
�8.8��0.4x �
2x �
0.8y �
4y �
1.6
5
�18x � 12y � 13
30x � 24y � 23�3x � 4y �
5x � 3y �
�2
4
x1
2x1
x1
2x1
3x1
�
�
�
�
�
x2
x2
x2
x2
x2
�
�
�
�
�
x3
x3
x3
4x3
x3
�
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�
3x4
x4
2x4
x4
2x4
�
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�
x5
x5
x5
x5
x5
�
�
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�
3
4
3
�1
5
x1
x1
2x1
x1
2x1
�
�
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2x2
3x2
x2
x2
x2
�
�
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x3
x3
x3
2x3
x3
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3x4
2x4
3x4
x4
2x4
�
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�
x5
x5
x5
x5
x5
�
�
�
�
�
�3
�3
6
2
�3
�x1 � 2x2 � x3 � 2x4 �
3x1 � 5x2 � 2x3 � 3x4 �
2x1 � 5x2 � 2x3 � 5x4 �
�x1 � 4x2 � 4x3 � 11x4 �
1
�2
0
�3
�x1 � 2x2 � x3 � 2x4 �
3x1 � 5x2 � 2x3 � 3x4 �
2x1 � 5x2 � 2x3 � 5x4 �
�x1 � 4x2 � 4x3 � 11x4 �
0
1
�1
2
�x � y � z �
3x � 5y � 4z �
3x � 6y � 5z �
�1
2
0�x � y � z � 0
3x � 5y � 4z � 5
3x � 6y � 5z � 2
� x � 2y �
2x � 3y �
1
�2� x � 2y � 4
2x � 3y � 2
� x � 2y � 0
2x � 3y � 3� x � 2y � 5
2x � 3y � 10
��14
53
94
89��
72
15
�34
45�
Section 8.3 The Inverse of a Square Matrix 607
69. 700 units of beef 70. 525 units of beef
500 units of chicken 480 units of chicken
600 units of liver 500 units of liver
71. 800 units of beef 72. 1000 units of beef
750 units of chicken 950 units of chicken
725 units of liver 900 units of liver
73. COFFEE A coffee manufacturer sells a 10-pound
package that contains three flavors of coffee for $26.
French vanilla coffee costs $2 per pound, hazelnut fla-
vored coffee costs $2.50 per pound, and Swiss chocolate
flavored coffee costs $3 per pound. The package contains
the same amount of hazelnut as Swiss chocolate. Let
represent the number of pounds of French vanilla, rep-
resent the number of pounds of hazelnut, and represent
the number of pounds of Swiss chocolate.
(a) Write a system of linear equations that represents
the situation.
(b) Write a matrix equation that corresponds to your
system.
(c) Solve your system of linear equations using an
inverse matrix. Find the number of pounds of each
flavor of coffee in the 10-pound package.
74. FLOWERS A florist is creating 10 centerpieces for the
tables at a wedding reception. Roses cost $2.50 each,
lilies cost $4 each, and irises cost $2 each. The customer
has a budget of $300 allocated for the centerpieces and
wants each centerpiece to contain 12 flowers, with twice
as many roses as the number of irises and lilies combined.
(a) Write a system of linear equations that represents
the situation.
(b) Write a matrix equation that corresponds to your system.
(c) Solve your system of linear equations using an inverse
matrix. Find the number of flowers of each type that
the florist can use to create the 10 centerpieces.
75. ENROLLMENT The table shows the enrollment
projections (in millions) for public universities in the
United States for the years 2010 through 2012. (Source:
U.S. National Center for Education Statistics, Digest of
Education Statistics)
(a) The data can be modeled by the quadratic function
Create a system of linear equations
for the data. Let represent the year, with
corresponding to 2010.
(b) Use the matrix capabilities of a graphing utility to find
the inverse matrix to solve the system from part (a)
and find the least squares regression parabola
(c) Use the graphing utility to graph the parabola with the
data.
(d) Do you believe the model is a reasonable predictor of
future enrollments? Explain.
EXPLORATION
TRUE OR FALSE? In Exercises 77 and 78, determinewhether the statement is true or false. Justify your answer.
77. Multiplication of an invertible matrix and its inverse is
commutative.
78. If you multiply two square matrices and obtain the
identity matrix, you can assume that the matrices are
inverses of one another.
79. WRITING Explain how to determine whether the
inverse of a matrix exists. If so, explain how to
find the inverse.
80. WRITING Explain in your own words how to write a
system of three linear equations in three variables as a
matrix equation, as well as how to solve the
system using an inverse matrix.
81. Consider matrices of the form
(a) Write a matrix and a matrix in the form
of Find the inverse of each.
(b) Use the result of part (a) to make a conjecture about
the inverses of matrices in the form of
PROJECT: VIEWING TELEVISION To work an extended
application analyzing the average amounts of time spent
viewing television in the United States, visit this text’s
website at academic.cengage.com. (Data Source: The
Nielsen Company)
A.
A.
3 32 2
A � �a11
0
0
0
0
a22
0
0
0
0
a33
0
0
0
0
0
. . .
. . .
. . .
. . .
. . .
0
0
0
ann
�.
AX � B,
2 2
y � at2 � bt � c.
t � 10t
y � at2 � bt � c.
s
h
f
76. CAPSTONE If is a matrix
then is invertible if and only if If
verify that the inverse is
A�1 �1
ad � bc �d
�c
�b
a�.
ad � bc � 0,
ad � bc � 0.A
A � �acb
d�,2 2A
Year Enrollment projections
2010
2011
2012
13.89
14.04
14.20
608 Chapter 8 Matrices and Determinants
8.4 THE DETERMINANT OF A SQUARE MATRIX
What you should learn
• Find the determinants of matrices.
• Find minors and cofactors of square matrices.
• Find the determinants of squarematrices.
Why you should learn it
Determinants are often used in otherbranches of mathematics. For instance,Exercises 85–90 on page 615 showsome types of determinants that areuseful when changes in variables aremade in calculus.
2 2The Determinant of a 2 2 Matrix
Every square matrix can be associated with a real number called its determinant.
Determinants have many uses, and several will be discussed in this and the next section.
Historically, the use of determinants arose from special number patterns that occur
when systems of linear equations are solved. For instance, the system
has a solution
and
provided that Note that the denominators of the two fractions are the
same. This denominator is called the determinant of the coefficient matrix of the system.
Coefficient Matrix Determinant
The determinant of the matrix can also be denoted by vertical bars on both sides of
the matrix, as indicated in the following definition.
In this text, and are used interchangeably to represent the determinant of
Although vertical bars are also used to denote the absolute value of a real number, the
context will show which use is intended.
A convenient method for remembering the formula for the determinant of a
matrix is shown in the following diagram.
Note that the determinant is the difference of the products of the two diagonals of the
matrix.
det�A� � a1
a2
b1
b2 � a1b2 � a2b1
2 2
A.Adet�A�
A
det�A� � a1b2 � a2b1A � �a1
a2
b1
b2�
a1b2 � a2b1 � 0.
y �a1c2 � a2c1
a1b2 � a2b1
x �c1b2 � c2b1
a1b2 � a2b1
�a1x � b1y � c1
a2x � b2y � c2
Definition of the Determinant of a 2 2 Matrix
The determinant of the matrix
is given by
det�A� � A � a1
a2
b1
b2 � a1b2 � a2b1.
A � �a1
a2
b1
b2�
Section 8.4 The Determinant of a Square Matrix 609
The Determinant of a 2 2 Matrix
Find the determinant of each matrix.
a.
b.
c.
Solution
a.
b.
c.
Now try Exercise 9.
Notice in Example 1 that the determinant of a matrix can be positive, zero, or
negative.
The determinant of a matrix of order is defined simply as the entry of the
matrix. For instance, if then det�A� � �2.A � ��2�,1 1
� 0 � 3 � �3
� 0�4� � 2�32�
det�C� � 02 32
4� 4 � 4 � 0
� 2�2� � 4�1�
det�B� � 24 1
2� 7� 4 � 3
� 2�2� � 1��3�
det�A� � 21 �3
2
C � �0
2
32
4�B � �2
4
1
2�
A � �2
1
�3
2�
Example 1
TECHNOLOGY
Most graphing utilities can evaluate the determinant of a matrix. For instance, youcan evaluate the determinant of
by entering the matrix as and then choosing the determinant feature. The resultshould be 7, as in Example 1(a). Try evaluating the determinants of other matrices.Consult the user’s guide for your graphing utility for specific keystrokes.
[A]A " [2
1
!3
2]
610 Chapter 8 Matrices and Determinants
Minors and Cofactors
To define the determinant of a square matrix of order or higher, it is convenient
to introduce the concepts of minors and cofactors.
In the sign pattern for cofactors at the left, notice that odd positions (where is
odd) have negative signs and even positions (where is even) have positive signs.
Finding the Minors and Cofactors of a Matrix
Find all the minors and cofactors of
Solution
To find the minor delete the first row and first column of and evaluate the
determinant of the resulting matrix.
Similarly, to find delete the first row and second column.
Continuing this pattern, you obtain the minors.
Now, to find the cofactors, combine these minors with the checkerboard pattern of signs
for a matrix shown at the upper left.
Now try Exercise 29.
C33 � �6C32 � 3C31 � 5
C23 � 8C22 � �4C21 � �2
C13 � 4C12 � 5C11 � �1
3 3
M33 � �6M32 � �3M31 � 5
M23 � �8M22 � �4M21 � 2
M13 � 4M12 � �5M11 � �1
M12 � 34 2
1 � 3�1� � 4�2� � �5�0
3
4
2
�1
0
1
2
1�,
M12,
M11 � �1
0
2
1 � �1�1� � 0�2� � �1�0
3
4
2
�1
0
1
2
1�,
AM11,
A � �0
3
4
2
�1
0
1
2
1�.
Example 2
i � j
i � j
3 3
Minors and Cofactors of a Square Matrix
If is a square matrix, the minor of the entry is the determinant of the
matrix obtained by deleting the th row and th column of The cofactor
of the entry is
Ci j � ��1�i� jMi j.
ai j
Ci jA.ji
ai jMi jA
Sign Pattern for Cofactors
matrix
matrix
matrixn n
��
�
�
�
�
.
..
�
�
�
�
�
.
..
�
�
�
�
�
.
..
�
�
�
�
�
.
..
�
�
�
�
�
.
..
. . .
. . .
. . .
. . .
. . .�4 4
��
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�3 3
��
�
�
�
�
�
�
�
��
Section 8.4 The Determinant of a Square Matrix 611
The Determinant of a Square Matrix
The definition below is called inductive because it uses determinants of matrices of
order to define determinants of matrices of order
Try checking that for a matrix
this definition of the determinant yields as previously defined.
The Determinant of a Matrix of Order 3 3
Find the determinant of
Solution
Note that this is the same matrix that was in Example 2. There you found the cofactors
of the entries in the first row to be
and
So, by the definition of a determinant, you have
First-row expansion
Now try Exercise 39.
In Example 3, the determinant was found by expanding by the cofactors in the first
row. You could have used any row or column. For instance, you could have expanded
along the second row to obtain
Second-row expansion
� 14.
� 3��2� � ��1���4� � 2�8�A � a21C21 � a22C22 � a23C23
� 14.
� 0��1� � 2�5� � 1�4�A � a11C11 � a12C12 � a13C13
C13 � 4.C12 � 5,C11 � �1,
A � �0
3
4
2
�1
0
1
2
1�.
Example 3
A � a1b2 � a2b1,
A � �a1
a2
b1
b2�
2 2
n.n � 1
Determinant of a Square Matrix
If is a square matrix (of order or greater), the determinant of is the
sum of the entries in any row (or column) of multiplied by their respective
cofactors. For instance, expanding along the first row yields
Applying this definition to find a determinant is called expanding by cofactors.
A � a11C11 � a12C12 �. . . � a1nC1n.
A
A2 2A
When expanding by cofactors, you do not need to find cofactors of zero entries,
because zero times its cofactor is zero.
So, the row (or column) containing the most zeros is usually the best choice for expansion
by cofactors. This is demonstrated in the next example.
The Determinant of a Matrix of Order 4 4
Find the determinant of
Solution
After inspecting this matrix, you can see that three of the entries in the third column are
zeros. So, you can eliminate some of the work in the expansion by using the third
column.
Because and have zero coefficients, you need only find the cofactor
To do this, delete the first row and third column of and evaluate the determinant of
the resulting matrix.
Delete 1st row and 3rd column.
Simplify.
Expanding by cofactors in the second row yields
So, you obtain
Now try Exercise 49.
Try using a graphing utility to confirm the result of Example 4.
� 15.
� 3�5�A � 3C13
� 5.
� 0 � 2�1���8� � 3��1���7�
C13 � 0��1�314 2
2 � 2��1�4�1
3
2
2 � 3��1�5�1
3
1
4
� �1
0
3
1
2
4
2
3
2C13 � ��1�1�3�1
0
3
1
2
4
2
3
2A
C13.C43C23, C33,
A � 3�C13� � 0�C23� � 0�C33� � 0�C43�
A � �1
�1
0
3
�2
1
2
4
3
0
0
0
0
2
3
2�.
Example 4
aijCij � �0�Cij � 0
612 Chapter 8 Matrices and Determinants
Section 8.4 The Determinant of a Square Matrix 613
EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8.4VOCABULARY: Fill in the blanks.
1. Both and represent the ________ of the matrix
2. The ________ of the entry is the determinant of the matrix obtained by deleting the th row and th
column of the square matrix
3. The ________ of the entry of the square matrix is given by
4. The method of finding the determinant of a matrix of order or greater is called ________ by ________.
SKILLS AND APPLICATIONS
2 2
��1�i� jMij.AaijCij
A.
jiaijMij
A.Adet�A�
In Exercises 5–20, find the determinant of the matrix.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–24, use the matrix capabilities of a graphingutility to find the determinant of the matrix.
21. 22.
23. 24.
In Exercises 25–32, find all (a) minors and (b) cofactors of thematrix.
25. 26.
27. 28.
29. 30.
31. 32.
In Exercises 33–38, find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column.
33. 34.
(a) Row 1 (a) Row 2
(b) Column 2 (b) Column 3
35. 36.
(a) Row 2 (a) Row 3
(b) Column 2 (b) Column 1
37. 38.
(a) Row 2 (a) Row 3
(b) Column 2 (b) Column 1
In Exercises 39–54, find the determinant of the matrix.Expand by cofactors on the row or column that appears tomake the computations easiest.
39. 40.
41. 42.
43. 44. �1
�1
4
0
�1
11
0
0
5��
�1
0
0
8
3
0
�3
�6
3�
�1
3
�2
1
1
0
2
0
3��
6
0
4
3
0
�6
�7
0
3�
��2
1
0
2
�1
1
3
0
4��
2
4
4
�1
2
2
0
1
1�
�10
4
0
1
8
0
3
0
3
5
2
�3
�7
�6
7
2��
6
4
�1
8
0
13
0
6
�3
6
7
0
5
�8
4
2�
�10
30
0
�5
0
10
5
10
1��
5
0
1
0
12
6
�3
4
3�
��3
6
4
4
3
�7
2
1
�8��
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4
2
2
5
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1
6
1�
��2
7
6
9
�6
7
4
0
�6��
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7
1
6
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0
3
8
�5�
�1
3
4
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2
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0
5
4��
4
�3
1
0
2
�1
2
1
1�
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7
5
�2�� 3
�2
1
�4��0
3
10
�4��4
3
5
�6�
�0.1
7.5
0.3
0.1
6.2
0.6
�4.3
0.7
�1.2��
0.9
�0.1
�2.2
0.7
0.3
4.2
0
1.3
6.1�
�0.1
�0.3
0.5
0.2
0.2
0.4
0.3
0.2
0.4��
0.3
0.2
�0.4
0.2
0.2
0.4
0.2
0.2
0.3�
�23
�1
43
�13���
12
�6
1313�
� 0
�3
6
2���712
6
3�� 4
�2
7
5���3
�6
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�1�� 2
�6
�3
9��2
0
6
3��4
0
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0���7
3
0
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4
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2
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6
0
2��8
2
4
3���10��4�
614 Chapter 8 Matrices and Determinants
45. 46.
47. 48.
49. 50.
51. 52.
53.
54.
In Exercises 55–62, use the matrix capabilities of a graphingutility to evaluate the determinant.
55. 56.
57. 58.
59. 60.
61.
62.
In Exercises 63–70, find (a) (b) (c) and (d)
63.
64.
65.
66.
67.
68.
69.
70.
In Exercises 71–76, evaluate the determinant(s) to verify theequation.
71. 72.
73.
74.
75.
76.
In Exercises 77–84, solve for
77. 78.
79. 80.
81. 82.
83. 84. x � 4
7
�2
x � 5 � 0x � 3
1
2
x � 2 � 0
x � 2
�3
�1
x � 0x � 1
3
2
x � 2 � 0
x � 1
�1
2
x � 4x2 1
x � 2 � �1
x
�1
4
x � 20x1 2
x � 2
x.
a � b
a
a
a
a � b
a
a
a
a � b � b2�3a � b�
111 x
y
z
x 2
y 2
z 2 � �y � x��z � x��z � y�
wcw x
cx � 0
wy x
z � wy x � cw
z � cy wy cx
cz � cwy x
zwy x
z � � yw z
x
B � �2
0
3
�1
1
�2
4
3
1�A � �2
1
3
0
�1
1
1
2
0�,
B � ��1
0
0
0
2
0
0
0
3�A � ��1
1
0
2
0
1
1
1
0�,
B � ��3
0
�2
0
2
�1
1
�1
1�A � �
3
�1
�2
2
�3
0
0
4
1�,
B � �3
1
3
�2
�1
1
0
2
1�A � �
0
�3
0
1
�2
4
2
1
1�,
B � �0
1
6
�2�A � �5
3
4
�1�,
B � ��1
�2
1
2�A � �4
3
0
�2�,
B � �1
0
2
�1�A � ��2
4
1
�2�,
B � �2
0
0
�1�A � ��1
0
0
3�,
AB.AB,B,A,
�2
0
0
0
0
0
3
0
0
0
0
0
�1
0
0
0
0
0
2
0
0
0
0
0
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�1
5
4
1
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0
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7
2
4
2
0
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3
3
1
3
0
0
1
0
2
0
2 0
8
�4
�7
�3
1
6
0
8
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0
0
2
6
9
141220 �1
6
0
2
8
0
2
8
4
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6
0 3
�2
12
0
5
5
0
0
7 7
�2
�6
0
5
2
�14
4
12 5
9
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7
7
0
4
1308 8
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1
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4
6
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0
0
0
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2
1
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0
0
0
4
2
3
0
0
3
6
4
0
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2
3
1
2
��
3
�2
1
6
3
2
0
0
0
0
4
1
0
2
5
�1
3
4
�1
1
5
2
0
0
0
��
1
�5
0
3
4
6
0
�2
3
2
0
1
2
1
0
5��
5
4
0
0
3
6
2
1
0
4
�3
�2
6
12
4
2�
�3
�2
1
0
6
0
1
3
�5
6
2
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4
0
2
�1��
2
2
1
3
6
7
5
7
6
3
0
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2
6
1
7�
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7
1
0
11
2
0
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2��
2
0
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4
3
0
6
1
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1
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1
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4
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3�
Section 8.4 The Determinant of a Square Matrix 615
In Exercises 85–90, evaluate the determinant in which theentries are functions. Determinants of this type occur whenchanges of variables are made in calculus.
85. 86.
87. 88.
89. 90.
EXPLORATION
TRUE OR FALSE? In Exercises 91 and 92, determinewhether the statement is true or false. Justify your answer.
91. If a square matrix has an entire row of zeros, the
determinant will always be zero.
92. If two columns of a square matrix are the same, the
determinant of the matrix will be zero.
93. Find square matrices and to demonstrate that
94. Consider square matrices in which the entries are
consecutive integers. An example of such a matrix is
(a) Use a graphing utility to evaluate the determinants
of four matrices of this type. Make a conjecture
based on the results.
(b) Verify your conjecture.
95. WRITING Write a brief paragraph explaining the
difference between a square matrix and its determinant.
96. THINK ABOUT IT If is a matrix of order such
that is it possible to find Explain.
PROPERTIES OF DETERMINANTS In Exercises 97–99, a property of determinants is given ( and are square matrices). State how the property has been applied to thegiven determinants and use a graphing utility to verify theresults.
97. If is obtained from by interchanging two rows of
or interchanging two columns of then
(a)
(b)
98. If is obtained from by adding a multiple of a row
of to another row of or by adding a multiple of a
column of to another column of then
(a)
(b)
99. If is obtained from by multiplying a row by a
nonzero constant or by multiplying a column by a
nonzero constant then
(a)
(b)
In Exercises 101–104, evaluate the determinant.
101. 102.
103. 104.
105. CONJECTURE A triangular matrix is a square
matrix with all zero entries either below or above its
main diagonal. A square matrix is upper triangular if
it has all zero entries below its main diagonal and is
lower triangular if it has all zero entries above its
main diagonal. A matrix that is both upper and lower
triangular is called diagonal. That is, a diagonal
matrix is a square matrix in which all entries above
and below the main diagonal are zero. In Exercises
101–104, you evaluated the determinants of triangular
matrices. Make a conjecture based on your results.
106. Use the matrix capabilities of a graphing utility to find the
determinant of What message appears on the screen?
Why does the graphing utility display this message?
A � �1
�1
3
2
0
�2�
A.
1
�4
5
0
�1
1
0
0
5�1
0
0
2
3
0
�5
4
3�2
0
0
0
0
2
0
0
0
0
1
0
0
0
0
3100 0
5
0
0
0
2
137 8
�12
4
�3
6
9 � 12137 2
�3
1
�1
2
352 10
�3 � 512 2
�3B � cA.c,
c
AB
527 4
�3
6
2
4
3 � 127 10
�3
6
�6
4
315 �3
2 � 10 �3
17B � A.A,A
AA
AB
1
�2
1
3
2
6
4
0
2 � � 1
�2
1
6
2
3
2
0
4 1
�7
6
3
2
1
4
�5
2 � � 1
�7
6
4
�5
2
3
2
1B��A.A,
AAB
BA
2A?A � 5,
3 3A
�4
7
10
5
8
11
6
9
12�.
A � B � A � B.BA
x1 x ln x
1 � ln x x1 ln x
1�x e�x�e�x
xe�x
�1 � x�e�x e2x
2e2x
e3x
3e3x3x 2
1
�3y 2
1 4u
�1
�1
2v
100. CAPSTONE If is an matrix, explain how to
find the following.
(a) The minor of the entry
(b) The cofactor of the entry
(c) The determinant of A
aijCij
aijMij
n nA
Cramer’s Rule
So far, you have studied three methods for solving a system of linear equations: substitution,
elimination with equations, and elimination with matrices. In this section, you will
study one more method, Cramer’s Rule, named after Gabriel Cramer (1704–1752).
This rule uses determinants to write the solution of a system of linear equations. To see
how Cramer’s Rule works, take another look at the solution described at the beginning
of Section 8.4. There, it was pointed out that the system
has a solution
and
provided that Each numerator and denominator in this solution can
be expressed as a determinant, as follows.
Relative to the original system, the denominator for and is simply the determinant
of the coefficient matrix of the system. This determinant is denoted by The numerators
for and are denoted by and respectively. They are formed by using the
column of constants as replacements for the coefficients of and as follows.
Coefficient
Matrix D
For example, given the system
the coefficient matrix, D, and are as follows.
Coefficient
Matrix D
2
�4
3
838 �5
3 2
�4
�5
3� 2
�4
�5
3�DyDx
DyDx,
� 2x � 5y � 3
�4x � 3y � 8
a1
a2
c1
c2c1
c2
b1
b2a1
a2
b1
b2�a1
a2
b1
b2�
DyDx
y,x
Dy,Dxyx
D.
yx
y �a1c2 � a2c1
a1b2 � a2b1
�a1
a2
c1
c2a1
a2
b1
b2x �
c1b2 � c2b1
a1b2 � a2b1
�c1
c2
b1
b2a1
a2
b1
b2
a1b2 � a2b1 � 0.
y �a1c2 � a2c1
a1b2 � a2b1
x �c1b2 � c2b1
a1b2 � a2b1
�a1x � b1y � c1
a2x � b2y � c2
616 Chapter 8 Matrices and Determinants
8.5 APPLICATIONS OF MATRICES AND DETERMINANTS
What you should learn
• Use Cramer’s Rule to solve systems of linear equations.
• Use determinants to find the areas of triangles.
• Use a determinant to test forcollinear points and find an equation of a line passing through two points.
• Use matrices to encode and decode messages.
Why you should learn it
You can use Cramer’s Rule to solvereal-life problems. For instance, inExercise 69 on page 627, Cramer’sRule is used to find a quadratic modelfor the per capita consumption of bottled water in the United States.
MAFORD/istockphoto.com
Section 8.5 Applications of Matrices and Determinants 617
Cramer’s Rule generalizes easily to systems of equations in variables. The
value of each variable is given as the quotient of two determinants. The denominator is
the determinant of the coefficient matrix, and the numerator is the determinant of the
matrix formed by replacing the column corresponding to the variable (being solved for)
with the column representing the constants. For instance, the solution for in the
following system is shown.
Using Cramer’s Rule for a 2 2 System
Use Cramer’s Rule to solve the system of linear equations.
Solution
To begin, find the determinant of the coefficient matrix.
Because this determinant is not zero, you can apply Cramer’s Rule.
So, the solution is and Check this in the original system.
Now try Exercise 7.
y � �1.x � 2
y �Dy
D�
43 10
11�14
�44 � 30
�14�
14
�14� �1
x �Dx
D�
10
11
�2
�5�14
��50 � ��22�
�14��28
�14� 2
D � 43 �2
�5 � �20 � ��6� � �14
�4x � 2y � 10
3x � 5y � 11
Example 1
x3 �A3A �
a11
a21
a31
a12
a22
a32
b1
b2
b3a11
a21
a31
a12
a22
a32
a13
a23
a33�a11x1 � a12x2 � a13x3 � b1
a21x1 � a22x2 � a23x3 � b2
a31x1 � a32x2 � a33x3 � b3
x3
nn
Cramer’s Rule
If a system of linear equations in variables has a coefficient matrix with a
nonzero determinant the solution of the system is
where the th column of is the column of constants in the system of equations.
If the determinant of the coefficient matrix is zero, the system has either no
solution or infinitely many solutions.
Aii
xn �AnA. . . ,x2 �
A2A ,x1 �
A1A ,
A,Ann
Using Cramer’s Rule for a 3 3 System
Use Cramer’s Rule to solve the system of linear equations.
Solution
To find the determinant of the coefficient matrix
expand along the second row, as follows.
Because this determinant is not zero, you can apply Cramer’s Rule.
The solution is Check this in the original system as follows.
Check
Now try Exercise 13.
Remember that Cramer’s Rule does not apply when the determinant of the coefficient
matrix is zero. This would create division by zero, which is undefined.
��45�
�45
2�45�85
3�45�
125
�
�
�
�
2��32�3
4��32�6
�
�
�
�
�
�
3��85�
245
��85�85
4��85�
325
�?
�
�?
�
�?
�
1
1
0
0
2
2
� 45, �
32, �
85�.
z �Dz
D�
�1
2
3
2
0
�4
1
0
210
��16
10� �
8
5
y �Dy
D�
�1
2
3
1
0
2
�3
1
410
��15
10� �
3
2
x �Dx
D�
102 2
0
�4
�3
1
410
�8
10�
4
5
� 10
� �2��4� � 0 � 1��2�
D � 2��1�3 2
�4
�3
4 � 0��1�4�1
3
�3
4 � 1��1�5�1
3
2
�4
��1
2
3
2
0
�4
�3
1
4�
��x
2x
3x
� 2y
� 4y
� 3z � 1
� z � 0
� 4z � 2
Example 2
618 Chapter 8 Matrices and Determinants
Substitute into Equation 1.
Equation 1 checks. �
Substitute into Equation 2.
Equation 2 checks. �
Substitute into Equation 3.
Equation 3 checks. �
Section 8.5 Applications of Matrices and Determinants 619
Area of a Triangle
Another application of matrices and determinants is finding the area of a triangle whose
vertices are given as points in a coordinate plane.
Finding the Area of a Triangle
Find the area of a triangle whose vertices are and as shown in
Figure 8.1.
Solution
Let and Then, to find the area of the
triangle, evaluate the determinant.
Using this value, you can conclude that the area of the triangle is
Choose so that the area is positive.
Now try Exercise 25.
�3
2 square units.
� �1
2��3�
��� Area � �1
2 124 0
2
3
1
1
1� �3.
� 1��1� � 0 � 1��2�
� 1��1�223 1
1 � 0��1�324 1
1 � 1��1�424 2
3x1
x2
x3
y1
y2
y3
1
1
1 � 124 0
2
3
1
1
1�x3, y3� � �4, 3�.�x1, y1� � �1, 0�, �x2, y2� � �2, 2�,
�4, 3�,�1, 0�, �2, 2�,
Example 3
Area of a Triangle
The area of a triangle with vertices and is
where the symbol indicates that the appropriate sign should be chosen to yield
a positive area.
±
Area � ±1
2 x1
x2
x3
y1
y2
y3
1
1
1�x3, y3��x2, y2�,�x1, y1�,
x
1 2 3 4
1
2
3
(1, 0)
(2, 2)
(4, 3)
y
FIGURE 8.1
620 Chapter 8 Matrices and Determinants
Lines in a Plane
What if the three points in Example 3 had been on the same line? What would have
happened had the area formula been applied to three such points? The answer is that the
determinant would have been zero. Consider, for instance, the three collinear points
and as shown in Figure 8.2. The area of the “triangle” that has these
three points as vertices is
The result is generalized as follows.
Testing for Collinear Points
Determine whether the points and are collinear. (See Figure 8.3.)
Solution
Letting and you have
Because the value of this determinant is not zero, you can conclude that the three points
do not lie on the same line. Moreover, the area of the triangle with vertices at these
points is square units.
Now try Exercise 39.
��12���6� � 3
� �6.
� �2��4� � 2��6� � 1��2�
� �2��1�215 1
1� ��2���1�317 1
1� 1��1�417 1
5x1
x2
x3
y1
y2
y3
1
1
1 � �2
1
7
�2
1
5
1
1
1�x3, y3� � �7, 5�,�x1, y1� � ��2, �2�, �x2, y2� � �1, 1�,
�7, 5���2, �2�, �1, 1�,
Example 4
� 0.
�1
2�0 � 1��2� � 1��2��
1
2024 1
2
3
1
1
1 � 1
2�0��1�223 1
1� 1��1�324 1
1� 1��1�424 2
3��4, 3�,�0, 1�, �2, 2�,
Test for Collinear Points
Three points and are collinear (lie on the same line) if
and only if
x1
x2
x3
y1
y2
y3
1
1
1 � 0.
�x3, y3��x1, y1�, �x2, y2�,
x
(7, 5)
(1, 1)
(−2, −2)
−1 2
1
2
3
4
5
6
7
y
3 4 5 6 71
FIGURE 8.3
x
1 2 3 4
1
2
3
(0, 1)
(2, 2)
(4, 3)
y
FIGURE 8.2
Section 8.5 Applications of Matrices and Determinants 621
The test for collinear points can be adapted to another use. That is, if you are given
two points on a rectangular coordinate system, you can find an equation of the line
passing through the two points, as follows.
Finding an Equation of a Line
Find an equation of the line passing through the two points and as shown
in Figure 8.4.
Solution
Let and Applying the determinant formula for the
equation of a line produces
To evaluate this determinant, you can expand by cofactors along the first row to obtain
the following.
So, an equation of the line is
Now try Exercise 47.
Note that this method of finding the equation of a line works for all lines, including
horizontal and vertical lines. For instance, the equation of the vertical line through
and is
x � 2.
4 � 2x � 0
x22 y
0
2
1
1
1 � 0
�2, 2��2, 0�
x � 3y � 10 � 0.
x � 3y � 10 � 0
x�1��1� � y��1��3� � �1��1��10� � 0
x��1�243 1
1� y��1�3 2
�1
1
1� 1��1�4 2
�1
4
3 � 0
x
2
�1
y
4
3
1
1
1 � 0.
�x2, y2� � ��1, 3�.�x1, y1� � �2, 4�
��1, 3�,�2, 4�
Example 5
Two-Point Form of the Equation of a Line
An equation of the line passing through the distinct points and is
given by
xx1
x2
y
y1
y2
1
1
1 � 0.
�x2, y2��x1, y1�
x
(2, 4)
(−1, 3)
−1 1 2 3 4
1
2
4
5
y
FIGURE 8.4
Cryptography
A cryptogram is a message written according to a secret code. (The Greek word kryptos
means “hidden.”) Matrix multiplication can be used to encode and decode messages. To
begin, you need to assign a number to each letter in the alphabet (with 0 assigned to a
blank space), as follows.
Then the message is converted to numbers and partitioned into uncoded row matrices,
each having entries, as demonstrated in Example 6.
Forming Uncoded Row Matrices
Write the uncoded row matrices of order for the message
MEET ME MONDAY.
Solution
Partitioning the message (including blank spaces, but ignoring punctuation) into groups
of three produces the following uncoded row matrices.
Note that a blank space is used to fill out the last uncoded row matrix.
Now try Exercise 55(a).
To encode a message, use the techniques demonstrated in Section 8.3 to choose an
invertible matrix such as
and multiply the uncoded row matrices by (on the right) to obtain coded row matrices.
Here is an example.
Uncoded Matrix Encoding Matrix A Coded Matrix
�13 �26 21���1
�1
1
�2
1
�1
2
3
�4��13 5 5�
A
A � �1
�1
1
�2
1
�1
2
3
�4�
n n
M E E T M E M O N D A Y
�13 5 5� �20 0 13� �5 0 13� �15 14 4� �1 25 0�
1 3
Example 6
n
26 � Z17 � Q8 � H
25 � Y16 � P7 � G
24 � X15 � O6 � F
23 � W14 � N5 � E
22 � V13 � M4 � D
21 � U12 � L3 � C
20 � T11 � K2 � B
19 � S10 � J1 � A
18 � R19 � I0 � _
622 Chapter 8 Matrices and Determinants
Section 8.5 Applications of Matrices and Determinants 623
Encoding a Message
Use the following invertible matrix to encode the message MEET ME MONDAY.
Solution
The coded row matrices are obtained by multiplying each of the uncoded row matrices
found in Example 6 by the matrix as follows.
Uncoded Matrix Encoding Matrix A Coded Matrix
So, the sequence of coded row matrices is
Finally, removing the matrix notation produces the following cryptogram.
Now try Exercise 55(b).
For those who do not know the encoding matrix decoding the cryptogram found
in Example 7 is difficult. But for an authorized receiver who knows the encoding matrix
decoding is simple. The receiver just needs to multiply the coded row matrices by
(on the right) to retrieve the uncoded row matrices. Here is an example.
A�1
�13 �26 21���1
�1
0
�10
�6
�1
�8
�5
�1� � �13 5 5�
A�1
A,
A,
13 �26 21 33 �53 �12 18 �23 �42 5 �20 56 �24 23 77
��24 23 77�.�5 �20 56��18 �23 �42��33 �53 �12��13 �26 21�
�1 25 0� �1
�1
1
�2
1
�1
2
3
�4� � ��24 23 77�
�15 14 4� �1
�1
1
�2
1
�1
2
3
�4� � �5 �20 56�
�5 0 13� �1
�1
1
�2
1
�1
2
3
�4� � �18 �23 �42�
�20 0 13� �1
�1
1
�2
1
�1
2
3
�4� � �33 �53 �12�
�13 5 5� �1
�1
1
�2
1
�1
2
3
�4� � �13 �26 21�
A,
A � �1
�1
1
�2
1
�1
2
3
�4�
Example 7
Coded Uncoded
624 Chapter 8 Matrices and Determinants
Decoding a Message
Use the inverse of the matrix
to decode the cryptogram
Solution
First find by using the techniques demonstrated in Section 8.3. is the decoding
matrix. Then partition the message into groups of three to form the coded row matrices.
Finally, multiply each coded row matrix by (on the right).
Coded Matrix Decoding Matrix Decoded Matrix
So, the message is as follows.
Now try Exercise 63.
M E E T M E M O N D A Y
�13 5 5� �20 0 13� �5 0 13� �15 14 4� �1 25 0�
��24 23 77� ��1
�1
0
�10
�6
�1
�8
�5
�1� � �1 25 0�
�5 �20 56� ��1
�1
0
�10
�6
�1
�8
�5
�1� � �15 14 4�
�18 �23 �42� ��1
�1
0
�10
�6
�1
�8
�5
�1� � �5 0 13�
�33 �53 �12� ��1
�1
0
�10
�6
�1
�8
�5
�1� � �20 0 13�
�13 �26 21� ��1
�1
0
�10
�6
�1
�8
�5
�1� � �13 5 5�
A�1
A�1
A�1A�1
13 �26 21 33 �53 �12 18 �23 �42 5 �20 56 �24 23 77.
A � �1
�1
1
�2
1
�1
2
3
�4�
Example 8
Cryptography Use your school’s library, the Internet, or some other referencesource to research information about another type of cryptography. Write a shortparagraph describing how mathematics is used to code and decode messages.
CLASSROOM DISCUSSION
HISTORICAL NOTE
During World War II, Navajo soldiers created a code usingtheir native language to sendmessages between battalions.Native words were assigned
to represent characters in theEnglish alphabet, and they
created a number of expressionsfor important military terms,
such as iron-fish to mean submarine. Without the NavajoCode Talkers, the Second World
War might have had a very different outcome.
Bettmann/Corbis
Section 8.5 Applications of Matrices and Determinants 625
EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8.5
VOCABULARY: Fill in the blanks.
1. The method of using determinants to solve a system of linear equations is called ________ ________.
2. Three points are ________ if the points lie on the same line.
3. The area of a triangle with vertices and is given by ________.
4. A message written according to a secret code is called a ________.
5. To encode a message, choose an invertible matrix and multiply the ________ row matrices by
(on the right) to obtain ________ row matrices.
6. If a message is encoded using an invertible matrix then the message can be decoded by multiplying
the coded row matrices by ________ (on the right).
SKILLS AND APPLICATIONS
A,
AA
�x3, y3��x2, y2�,�x1, y1�,A
In Exercises 7–16, use Cramer’s Rule to solve (if possible) thesystem of equations.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
In Exercises 17–20, use a graphing utility and Cramer’s Ruleto solve (if possible) the system of equations.
17. 18.
19. 20.
In Exercises 21–32, use a determinant and the given verticesof a triangle to find the area of the triangle.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
In Exercises 33 and 34, find a value of such that the trianglewith the given vertices has an area of 4 square units.
33. 34.
In Exercises 35 and 36, find a value of such that the trianglewith the given vertices has an area of 6 square units.
35.
36.
37. AREA OF A REGION A large region of forest has
been infested with gypsy moths. The region is roughly
triangular, as shown in the figure on the next page. From
the northernmost vertex of the region, the distances to
the other vertices are 25 miles south and 10 miles east
(for vertex ), and 20 miles south and 28 miles east (for
vertex ). Use a graphing utility to approximate the
number of square miles in this region.
C
B
A
�1, 0�, �5, �3�, ��3, y���2, �3�, �1,�1�, ��8, y�
y
��4, 2�, ��3, 5�, ��1, y���5, 1�, �0, 2�, ��2, y�
y
�92, 0�, �2, 6�, �0, �
32���4, 2�, �0,
72�, �3, �
12�
��2, 4�, �1, 5�, �3, �2���3, 5�, �2, 6�, �3, �5��0, �2�, ��1, 4�, �3, 5���2, 4�, �2, 3�, ��1, 5�
(6, −1)
(6, 10)
(−4, −5)
−8
4
8
y
x
(4, 3)
( )
1
1
2
3
4
1
20,
( )5
2, 0
x
y
2 3 4
(1, 6)
(−2, 1)
4
2
6
y
x
(3, −1)
−2
(0, 4)
(2, −3)(−2, −3)
−4 2−2
4
y
x
4
(4, 5)
(5, −2)
(0, 0)
−1 4
−2
1
2
3
4
5
y
x
1 6x
(0, 0)
(1, 5)
(3, 1)
5
1
2
3
4
5
y
4321
�3x
2x
x
�
�
�
y
y
y
�
�
�
3z
2z
z
�
�
�
1
�4
5�2x
x
3x
�
�
�
y
2y
y
�
�
�
z
z
z
�
�
�
5
1
4
�x � 2y � z �
2x � 2y � 2z �
�x � 3y � 4z �
�7
�8
8�
3x � 3y � 5z � 1
3x � 5y � 9z � 2
5x � 9y � 17z � 4
�5x� 4y� z�
�x� 2y� 2z�
3x� y� z�
�14
10
1�
x� 2y� 3z�
�2x� y� z�
3x� 3y� 2z�
�3
6
�11
�4x � 2y � 3z �
2x � 2y � 5z �
8x � 5y � 2z �
�2
16
4�
4x � y � z �
2x � 2y � 3z �
5x � 2y � 6z �
�5
10
1
� 2.4x � 1.3y �
�4.6x � 0.5y �
14.63
�11.51��0.4x � 0.8y � 1.6
0.2x � 0.3y � 2.2
� 6x � 5y �
�13x � 3y �
17
�76�3x � 2y �
6x � 4y �
�2
4
�4x6x
�
�
3y
9y
�
�
�10
12��7x
3x
�
�
11y
9y
�
�
�1
9
626 Chapter 8 Matrices and Determinants
FIGURE FOR 37
38. AREA OF A REGION You own a triangular tract of
land, as shown in the figure. To estimate the number
of square feet in the tract, you start at one vertex, walk
65 feet east and 50 feet north to the second vertex, and
then walk 85 feet west and 30 feet north to the third
vertex. Use a graphing utility to determine how many
square feet there are in the tract of land.
In Exercises 39–44, use a determinant to determine whetherthe points are collinear.
39. 40.
41. 42.
43. 44.
In Exercises 45 and 46, find such that the points arecollinear.
45. 46.
In Exercises 47–52, use a determinant to find an equation ofthe line passing through the points.
47. 48.
49. 50.
51. 52.
In Exercises 53 and 54, (a) write the uncoded rowmatrices for the message. (b) Then encode the message usingthe encoding matrix.
Message Encoding Matrix
53. COME HOME SOON
54. HELP IS ON THE WAY
In Exercises 55 and 56, (a) write the uncoded row matrices for the message. (b) Then encode the message usingthe encoding matrix.
Message Encoding Matrix
55. CALL ME TOMORROW
56. PLEASE SEND MONEY
In Exercises 57–60, write a cryptogram for the message usingthe matrix
57. LANDING SUCCESSFUL
58. ICEBERG DEAD AHEAD
59. HAPPY BIRTHDAY
60. OPERATION OVERLOAD
In Exercises 61–64, use to decode the cryptogram.
61.
11 21 64 112 25 50 29 53 23 46
40 75 55 92
62.
85 120 6 8 10 15 84 117 42 56 90
125 60 80 30 45 19 26
63.
9 38 28
25 41 21 31 9 �4�5�64�80
�19�9�19�19�9�1
A � �1
1
�6
�1
0
2
0
�1
3�
A � �2
3
3
4�
A � �1
3
2
5�A 1
A ! [1
3
1
2
7
4
2
9
7]
A.
�4
�3
3
2
�3
2
1
�1
1�
�1
1
�6
�1
0
2
0
�1
3�
1 " 3
��2
�1
3
1��1
3
2
5�
1 " 2
� 23, 4�, �6, 12���1
2, 3�, � 52, 1�
�10, 7�, ��2, �7���4, 3�, �2, 1��0, 0�, ��2, 2��0, 0�, �5, 3�
��6, 2�, ��5, y�, ��3, 5��2, �5�, �4, y�, �5, �2�
y
�2, 3�, �3, 3.5�, ��1, 2��0, 2�, �1, 2.4�, ��1, 1.6��0,
12�, �2, �1�, ��4,
72��2, �
12�, ��4, 4�, �6, �3�
�3, �5�, �6, 1�, �4, 2��3, �1�, �0, �3�, �12, 5�
65
85
50
30
N
S
EW
A
C
B
25
28
10
20
N
S
EW
64.
112 83 19 13 72 61 95
71 20 21 38 35 36 42 32
In Exercises 65 and 66, decode the cryptogram by using theinverse of the matrix
65. 20 17 1
62 143 181
66. 13 61 112 106 11
24 29 65 144 172
67. The following cryptogram was encoded with a
matrix.
8 21 5 10 5 25 5 19
6 20 40 1 16
The last word of the message is _RON. What is the
message?
68. The following cryptogram was encoded with a
matrix.
5 2 25 11 32 14 8
38 19 37 16
The last word of the message is _SUE. What is the
message?
69. DATA ANALYSIS: BOTTLED WATER The table
shows the per capita consumption of bottled water
(in gallons) in the United States from 2000 through
2007. (Source: Economic Research Service, U.S.
Department of Agriculture)
(a) Use the technique demonstrated in Exercises 77–80
in Section 7.3 to create a system of linear equations
for the data. Let represent the year, with
corresponding to 2000.
(b) Use Cramer’s Rule to solve the system from part (a)
and find the least squares regression parabola
(c) Use a graphing utility to graph the parabola from
part (b).
(d) Use the graph from part (c) to estimate when the
per capita consumption of bottled water will exceed
35 gallons.
70. HAIR PRODUCTS A hair product company sells
three types of hair products for $30, $20, and $10 per
unit. In one year, the total revenue for the three products
was $800,000, which corresponded to the sale of 40,000
units. The company sold half as many units of the $30
product as units of the $20 product. Use Cramer’s Rule
to solve a system of linear equations to find how many
units of each product were sold.
EXPLORATION
TRUE OR FALSE? In Exercises 71–74, determine whetherthe statement is true or false. Justify your answer.
71. In Cramer’s Rule, the numerator is the determinant of
the coefficient matrix.
72. You cannot use Cramer’s Rule when solving a system of
linear equations if the determinant of the coefficient
matrix is zero.
73. In a system of linear equations, if the determinant of the
coefficient matrix is zero, the system has no solution.
74. The points and are collinear.
75. WRITING Use your school’s library, the Internet, or
some other reference source to research a few current
real-life uses of cryptography. Write a short summary of
these uses. Include a description of how messages are
encoded and decoded in each case.
77. Use determinants to find the area of a triangle with
vertices and Confirm your
answer by plotting the points in a coordinate plane and
using the formula
Area �12�base��height�.
�7, 5�.�7, �1�,�3, �1�,
�3, 11��0, 2�,��5, �13�,
y � at2 � bt � c.
t � 0t
y
�19�19�13
��15�15�7�2
2 2
�18�18�1
�13�13�10�15
2 2
�131�73�17�59�9
�65�25�104�56�12�15
A ! [1
3
1
2
7
4
2
9
7]
A.
�48�23�118
�76�25�140
A � �3
0
4
�4
2
�5
2
1
3�
Section 8.5 Applications of Matrices and Determinants 627
Year Consumption, y
2000
2001
2002
2003
2004
2005
2006
2007
16.7
18.2
20.1
21.6
23.2
25.5
27.7
29.1
76. CAPSTONE
(a) State Cramer’s Rule for solving a system of linear
equations.
(b) At this point in the text, you have learned several
methods for solving systems of linear equations.
Briefly describe which method(s) you find easiest
to use and which method(s) you find most
difficult to use.
628 Chapter 8 Matrices and Determinants
CHAPTER SUMMARY8
What Did You Learn? Explanation/Examples ReviewExercises
Sec
tio
n 8
.2Sec
tio
n 8
.3Sec
tio
n 8
.1
Write matrices and identify their
orders (p. 570).
Perform elementary row
operations on matrices (p. 572).
Use matrices and Gaussian
elimination to solve systems of
linear equations (p. 575).
Use matrices and Gauss-Jordan
elimination to solve systems of
linear equations (p. 577).
Decide whether two matrices are
equal (p. 584).
Add and subtract matrices and
multiply matrices by scalars
(p. 585).
Multiply two matrices (p. 589).
Use matrix operations to model and
solve real-life problems (p. 592).
Verify that two matrices are
inverses of each other (p. 599).
Elementary Row Operations
1. Interchange two rows.
2. Multiply a row by a nonzero constant.
3. Add a multiple of a row to another row.
Gaussian Elimination with Back-Substitution
1. Write the augmented matrix of the system of linear
equations.
2. Use elementary row operations to rewrite the augmented
matrix in row-echelon form.
3. Write the system of linear equations corresponding to
the matrix in row-echelon form, and use back-substitution
to find the solution.
Gauss-Jordan elimination continues the reduction process
on a matrix in row-echelon form until a reduced row-echelon
form is obtained. (See Example 8.)
Two matrices are equal if their corresponding entries are
equal.
Definition of Matrix Addition
If and are matrices of order their
sum is the matrix given by
Definition of Scalar Multiplication
If is an matrix and is a scalar, the scalar
multiple of by is the matrix given by
Matrix Multiplication
If is an matrix and is an
matrix, the product is an matrix
where
Matrix operations can be used to find the total cost of
equipment for two softball teams. (See Example 12.)
Inverse of a Square Matrix
Let be an matrix and let be the identity
matrix. If there exists a matrix such that
then is the inverse of A.A�1
AA�1 � In � A�1A
A�1
n nInn nA
cij � ai1b1j � ai2b2j � ai3b3j �. . . � ainbnj.
AB � �cij�m pAB
n pB � �bij�m nA � �aij�
cA � �cij�m ncA
cm nA � �aij�
A � B � �aij � bij�.m n
m n,B � �bij�A � �aij�
2 13 21 32 2
��1
4
1
7� ��2 3 0� �4
5
�2
�3
0
1� � 8
�8�1–8
9, 10
11–28
29–36
37–40
41–54
55–68
69–72
73–76
Chapter Summary 629
What Did You Learn? Explanation/Examples ReviewExercises
Sec
tio
n 8
.5Sec
tio
n 8
.4Sec
tio
n 8
.3
Use Gauss-Jordan elimination
to find the inverses of matrices
(p. 600).
Use a formula to find the inverses
of matrices (p. 603).
Use inverse matrices to solve
systems of linear equations
(p. 604).
Find the determinants of
matrices (p. 608).
Find minors and cofactors of
square matrices (p. 610).
Find the determinants of square
matrices (p. 611).
Use Cramer’s Rule to solve systems
of linear equations (p. 616).
Use determinants to find the areas
of triangles (p. 619).
Use a determinant to test for
collinear points and find an
equation of a line passing
through two points (p. 620).
Use matrices to encode and
decode messages (p. 622).
2 2
2 2
Finding an Inverse Matrix
Let be a square matrix of order
1. Write the matrix that consists of the given matrix
on the left and the identity matrix on the right to
obtain
2. If possible, row reduce to using elementary row
operations on the entire matrix The result will be the
matrix If this is not possible, is not invertible.
3. Check your work to see that
If and then
If is an invertible matrix, the system of linear equations
represented by has a unique solution given by
The determinant of the matrix is given by
If is a square matrix, the minor of the entry is the
determinant of the matrix obtained by deleting the th row
and th column of The cofactor of the entry is
If is a square matrix (of order or greater), the
determinant of is the sum of the entries in any row (or
column) of multiplied by their respective cofactors.
Cramer’s Rule uses determinants to write the solution of a
system of linear equations.
The area of a triangle with vertices and
is
where the symbol indicates that the appropriate sign
should be chosen to yield a positive area.
Three points and are collinear (lie
on the same line) if and only if
The inverse of a matrix can be used to decode a cryptogram.
(See Example 8.)
x1
x2
x3
y1
y2
y3
1
1
1 � 0.
�x3, y3��x2, y2�,�x1, y1�,
±
Area � ±1
2x1
x2
x3
y1
y2
y3
1
1
1�x3, y3�
�x2, y2�,�x1, y1�,
A
A
2 2A
Cij � ��1�i� jMij.
aijCijA.j
i
aijMijA
det�A� � A � a1
a2
b1
b2 � a1b2 � a2b1.
A � �a1
a2
b1
b2�
X � A�1B.
AX � B
A
A�1 �1
ad � bc�d
�c
�b
a�.ad � bc � 0,A � �ac
b
d�
AA�1 � I � A�1A.
A�I A�1�.�A I�.IA
�A I�.In n
An 2n
n.A
77–84
85–92
93–110
111–114
115–118
119–128
129–132
133–136
137–142
143–146
In Exercises 1–4, determine the order of the matrix.
1. 2.
3. 4.
In Exercises 5 and 6, write the augmented matrix for thesystem of linear equations.
5. 6.
In Exercises 7 and 8, write the system of linear equations represented by the augmented matrix. (Use variables and if applicable.)
7.
8.
In Exercises 9 and 10, write the matrix in row-echelon form.(Remember that the row-echelon form of a matrix is notunique.)
9. 10.
In Exercises 11–14, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve the system. (Use variables and )
11.
12.
13.
14.
In Exercises 15–28, use matrices and Gaussian elimination withback-substitution to solve the system of equations (if possible).
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27.
28.
In Exercises 29–34, use matrices and Gauss-Jordan eliminationto solve the system of equations.
29. 30.
31.
32.
33.
34.
��3x � y � 7z �
5x � 2y � z �
�x � y � 4z �
�20
34
�8
�2x � y � 9z �
�x � 3y � 4z �
5x � 2y � z �
�8
�15
17
�4x � 4y � 4z � 5
4x � 2y � 8z � 1
5x � 3y � 8z � 6
��x � y � 2z �
2x � 3y � z �
5x � 4y � 2z �
1
�2
4
� x
3x
�x
�
�
�
3y
y
y
�
�
�
z
z
3z
�
�
�
2
�6
�2� xx
2x
�
�
�
2y
y
y
�
�
�
z
z
3z
�
�
�
3
�3
10
�x
4x
2x
�
�
2y �
�3y �
4y �
�
3z
z
z
w � 3
� 0
� 2w � 0
� 3
�2x
3x
x
�
�
y
�2y
3y
�
�
�
�
z
3z
2z
z
�
� w �
� 2w �
� 3w �
6
9
�11
14
�2x � 3y � 3z � 3
6x � 6y � 12z � 13
12x � 9y � z � 2�
2x � 3y � z �
2x � 3y � 3z �
4x � 2y � 3z �
10
22
�2
�x � 2y � 6z �
2x � 5y � 15z �
3x � y � 3z �
1
4
�6�
2x � y
2x � 2y
2x � y
� 2z � 4
� 5
� 6z � 2
� x
2x
�x
�
�
�
2y
y
3y
�
�
�
z
2z
2z
�
�
�
4
�24
20� x
2x
�x
�
�
�
2y
y
3y
�
�
�
z
2z
2z
�
�
�
7
�4
�3
��x2x
�
�
2y
4y
�
�
3
�6��x2x
�
�
2y
4y
�
�
3
6
�0.2x � 0.1y �
0.4x � 0.5y �
0.07
�0.01�0.3x � 0.1y � �0.13
0.2x � 0.3y � �0.25
�2x � 5y � 2
3x � 7y � 1� 5x � 4y �
�x � y �
2
�22
�1
0
0
�8
1
0
0
�1
1
�2
�7
1�
�1
0
0
�5
1
0
4
2
1
1
3
4�
�1
0
0
3
1
0
�9
�1
1
4
10
�2�
�1
0
0
2
1
0
3
�2
1
9
2
0�
z.y,x,
�4
3
�2
8
�1
10
16
2
12��
0
1
2
1
2
2
1
3
2�
�13
1
4
16
21
10
7
8
�4
3
5
3
2
12
�1�
�5
4
9
1
2
4
7
0
2
�9
10
3�
w,z,y,x,
�8x � 7y � 4z � 12
3x � 5y � 2z � 20
5x � 3y � 3z � 26
�3x � 10y � 15
5x � 4y � 22
�6 2 �5 8 0��3�
� 3
�2
�1
7
0
1
6
4���4
0
5�
8.1
630 Chapter 8 Matrices and Determinants
REVIEW EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8
Review Exercises 631
In Exercises 35 and 36, use the matrix capabilities of a graphingutility to reduce the augmented matrix corresponding to thesystem of equations, and solve the system.
35.
36.
In Exercises 37– 40, find and
37.
38.
39.
40.
In Exercises 41– 44, if possible, find (a) (b) (c) and (d)
41.
42.
43.
44.
In Exercises 45–48, perform the matrix operations. If it is notpossible, explain why.
45.
46.
47.
48.
In Exercises 49 and 50, use the matrix capabilities of a graphingutility to evaluate the expression.
49.
50.
In Exercises 51–54, solve for in the equation, given
and
51. 52.
53. 54.
In Exercises 55–58, find if possible.
55.
56.
57.
58.
In Exercises 59– 66, perform the matrix operations, if possible.If it is not possible, explain why.
59.
60.
61.
62. �1
0
0
3
2
0
2
�4
3��
4
0
0
�3
3
0
2
�1
2�
�1
2
5
�4
6
0��6
�2
8
4
0
0�
�1
2
5
�4
6
0� �6
4
�2
0
8
0�
�1
5
6
2
�4
0��6
4
�2
0
8
0�
B � ��1
4
8�A � �6 �5 7�,
B � � 4
20
12
40�A � �5
�7
11
4
2
2�,
B � �4
20
15
12
40
30�A � �
5
�7
11
4
2
2�,B � ��3
12
10
8�A � �2
3
�2
5�,AB,
2A � 5B � 3X3X � 2A � B
6X � 4A � 3BX � 2A � 3B
B ! �1
2
4
2
1
4�.A ! �
4
1
3
0
5
2�
X
�5�2
7
8
0
�2
2� � 4�
4
6
�1
�2
11
3�
3 �8
1
�2
3
5
�1� � 6 �4
2
�2
7
�3
6�
��8
�2
0
�1
4
�6
8
12
0� � 5�
�2
3
6
0
�1
12
�4
1
�8�
�2�1
5
6
2
�4
0� � 8�
7
1
1
1
2
4�
��11
�7
16
�2
19
1� � �6
8
�2
0
�4
10�
� 7
�1
3
5� � �10
14
�20
�3�
B � ��1
4
8�A � �6 �5 7�,
B � �0
4
20
3
12
40�A � �
5
�7
11
4
2
2�,
B � �3
15
20
11
25
29�A � �
4
�6
10
3
1
1�,B � ��3
12
10
8�A � �2
3
�2
5�,A # 3B.4A,
A B,A # B,
��9
012x
4
�3
�1
x � 10
7
1
�5
2y
0���
�9
0
6
4
�3
�1
2
7
1
�5
�4
0�
�x � 3
0
�2
�4
�3
y � 5
4y
2
6x� � �
5x � 1
0
�2
�4
�3
16
44
2
6�
��1
x
�4
0
5
y� � �
�1
8
�4
0
5
0�
��1
y
x
9� � ��1
�7
12
9�y.x8.2
�4x � 12y � 2z �
x � 6y � 4z �
x � 6y � z �
�2x � 10y � 2z �
20
12
8
�10
�3x � y � 5z � 2w �
x � 6y � 4z � w �
5x � y � z � 3w �
4y � z � 8w �
�44
1
�15
58
632 Chapter 8 Matrices and Determinants
63.
64.
65.
66.
In Exercises 67 and 68, use the matrix capabilities of a graphingutility to find the product.
67.
68.
69. MANUFACTURING A tire corporation has three
factories, each of which manufactures two models of
tires. The number of units of model produced at factory
in one day is represented by in the matrix
Find the production levels if production is decreased by 5%.
70. MANUFACTURING A power tool company has four
manufacturing plants, each of which produces three
types of cordless power tools. The number of units of
cordless power tool produced at plant in one day is
represented by in the matrix
Find the production levels if production is increased by 20%.
71. MANUFACTURING An electronics manufacturing
company produces three different models of headphones
that are shipped to two warehouses. The number of
units of model that are shipped to warehouse is
represented by in the matrix
The price per unit is represented by the matrix
Compute and interpret the result.
72. CELL PHONE CHARGES The pay-as-you-go charges
(in dollars per minute) of two cellular telephone companies
for calls inside the coverage area, regional roaming calls,
and calls outside the coverage area are represented by
Company
A B
Coverage area
Each month, you plan to use 120 minutes on calls inside
the coverage area, 80 minutes on regional roaming calls,
and 20 minutes on calls outside the coverage area.
(a) Write a matrix that represents the times spent on
the phone for each type of call.
(b) Compute and interpret the result.
In Exercises 73–76, show that is the inverse of
73.
74.
75.
76.
In Exercises 77–80, find the inverse of the matrix (if it exists).
77. 78.
79. 80.
In Exercises 81–84, use the matrix capabilities of a graphingutility to find the inverse of the matrix (if it exists).
81. 82.
83. 84. �8
4
1
�1
0
�2
2
4
2
0
1
1
8
�2
4
1��
1
4
3
�1
3
4
4
2
1
2
1
�1
6
6
2
�2�
�1
2
�1
4
�3
18
6
1
16��
�1
3
1
�2
7
4
�2
9
7�
�0
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7
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�2
3
1
�3
4��
2
�1
2
0
1
�2
3
1
1�
��3
2
�5
3���6
�5
5
4�
B � ��2
�3
2
1
1
�2
12
12
�12
�A � �1
�1
8
�1
0
�4
0
�1
2�,B � �
�2
3
2
�3
3
4
1
�1
�1�A � �
1
1
6
1
0
2
0
1
3�,B � � �2
�11
1
5�A � � 5
11
�1
�2�,B � ��2
7
�1
4�A � ��4
7
�1
2�,A.B8.3
TC
T
Inside
Regional roaming
Outside
C � �0.07
0.10
0.28
0.095
0.08
0.25�
C.
BA
B � �$79.99 $109.95 $189.99�.
A � �8200
6500
5400
7400
9800
4800�.
aij
ji
A � �80
50
90
70
30
60
90
80
100
40
20
50�.
aij
ji
A � �80
40
120
100
140
80�.
aijj
i
��2
4
3
�2
10
2��1
�5
3
1
2
2�
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11
12
1
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3��3
2
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6
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�3�1
4
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2���0
1
3
2� �1
5
0
�3� �2
6
1
0��� 4
�3
2
1� � ��2
0
4
4�
�4 �2 6� ��2
0
2
1
�3
0�
�1
0
1
2
4
1
�1
�2
3��1 �1 2�
Review Exercises 633
In Exercises 85–92, use the formula below to find the inverseof the matrix, if it exists.
85. 86.
87. 88.
89. 90.
91. 92.
In Exercises 93–104, use an inverse matrix to solve (ifpossible) the system of linear equations.
93. 94.
95. 96.
97. 98.
99. 100.
101.
102.
103.
104.
In Exercises 105–110, use the matrix capabilities of a graphingutility to solve (if possible) the system of linear equations.
105. 106.
107. 108.
109.
110.
In Exercises 111–114, find the determinant of the matrix.
111. 112.
113. 114.
In Exercises 115–118, find all (a) minors and (b) cofactors ofthe matrix.
115. 116.
117. 118.
In Exercises 119–128, find the determinant of the matrix.Expand by cofactors on the row or column that appears tomake the computations easiest.
119. 120.
121. 122.
123. 124.
125. 126.
127. 128.
In Exercises 129–132, use Cramer’s Rule to solve (ifpossible) the system of equations.
129. 130.
131.
��2x � 3y � 5z �
4x � y � z �
�x � 4y � 6z �
�11
�3
15
�3x � 8y �
9x � 5y �
�7
37� 5x � 2y �
�11x � 3y �
6
�23
8.5
��5
0
�3
1
6
1
4
6
0
�1
�5
0
0
2
1
3��
3
0
6
0
0
8
1
3
�4
1
8
�4
0
2
2
1�
�1
4
2
0
�2
1
3
�2
1
4
3
�4
2
1
0
2��
1
1
2
2
2
2
�4
0
�1
�4
�3
0
0
1
1
0�
�1
�4
0
1
1
1
4
2
�1��
�2
�6
5
4
0
3
1
2
4�
��1
2
�5
�2
3
�1
1
0
3��
4
2
1
1
3
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�1
2
0�
�0
0
�1
1
1
�1
�2
2
3��
�2
2
�1
0
�1
1
0
0
�3�
�8
6
�4
3
5
1
4
�9
2��
3
�2
1
2
5
8
�1
0
6�
�3
5
6
�4��2
7
�1
4�
�14
12
�24
�15��50
10
�30
5���9
7
11
�4��8
2
5
�4�8.4
�x � 3y � 2z �
�2x � 7y � 3z �
x � y � 3z �
8
�19
3
��3x � 3y � 4z �
y � z �
4x � 3y � 4z �
2
�1
�1
�5x
2x
�
�
10y
y
�
�
7
�98�65x
�125 x
�
�
47y
127 y
�
�
65
�175
� x � 3y �
�6x � 2y �
23
�18� x � 2y � �1
3x � 4y � �5
�3x � y � 5z �
�x � y � 6z �
�8x � 4y � z �
�14
8
44
��2x � y � 2z �
�x � 4y � z �
�y � z �
�13
�11
0
��x � 4y � 2z �
2x � 9y � 5z �
�x � 5y � 4z �
12
�25
10
�3x � 2y � z �
x � y � 2z �
5x � y � z �
6
�1
7
�3.5x
2.5x
�
�
4.5y
7.5y
�
�
8
25�0.3x
0.4x
�
�
0.7y
0.6y
�
�
10.2
7.6
��56x
4x
�
�
38y
3y
�
�
�2
0�12x
�3x
�
�
13y
2y
�
�
2
0
� 4x � 2y �
�19x � 9y �
�10
47��3x � 10y �
5x � 17y �
8
�13
� 5x � y �
�9x � 2y �
13
�24��x � 4y �
2x � 7y �
8
�5
�1.6
1.2
�3.2
�2.4�� 0.5
�0.2
0.1
�0.4���
34
�45
52
�83���
12
310
20
�6���18
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10
6
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7
4
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2
2�
A 1!
1
ad bc[ d
c
b
a]
634 Chapter 8 Matrices and Determinants
132.
In Exercises 133–136, use a determinant and the given vertices of a triangle to find the area of the triangle.
133. 134.
135. 136.
In Exercises 137 and 138, use a determinant to determinewhether the points are collinear.
137.
138.
In Exercises 139–142, use a determinant to find an equationof the line passing through the points.
139. 140.
141. 142.
In Exercises 143 and 144, (a) write the uncoded rowmatrices for the message, and (b) encode the message usingthe encoding matrix.
Message Encoding Matrix
143. LOOK OUT BELOW
144. HEAD DUE WEST
In Exercises 145 and 146, decode the cryptogram by using theinverse of the matrix
145. 5 11 2 370 265 225 57 48 33 32
15 20 245 171 147
146. 145 105 92 264 188 160 23 16 15
129 84 78 9 8 5 159 118 100 219
152 133 370 265 225 105 84 63
EXPLORATION
TRUE OR FALSE? In Exercises 147 and 148, determinewhether the statement is true or false. Justify your answer.
147. It is possible to find the determinant of a matrix.
148.
149. Use the matrix capabilities of a graphing utility to find
the inverse of the matrix
What message appears on the screen? Why does the
graphing utility display this message?
150. Under what conditions does a matrix have an inverse?
151. WRITING What is meant by the cofactor of an entry
of a matrix? How are cofactors used to find the
determinant of the matrix?
152. Three people were asked to solve a system of
equations using an augmented matrix. Each person
reduced the matrix to row-echelon form. The reduced
matrices were
and
Can all three be right? Explain.
153. THINK ABOUT IT Describe the row-echelon form of
an augmented matrix that corresponds to a system of
linear equations that has a unique solution.
154. Solve the equation for
2 � !3
5
�8 � ! � 0
!.
�1
0
2
0
3
0�.
�1
0
0
1
1
1�,�1
0
2
1
3
1�,
A � � 1
�2
�3
6�.
a11
a21
a31
a12
a22
a32
a13
a23
a33 � a11
a21
c1
a12
a22
c2
a13
a23
c3 a11
a21
a31 � c1
a12
a22
a32 � c2
a13
a23
a33 � c3 �4 5
����
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A ! [ 5
10
8
4
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Chapter Test 635
CHAPTER TEST See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8
Take this test as you would take a test in class. When you are finished, check your workagainst the answers given in the back of the book.
In Exercises 1 and 2, write the matrix in reduced row-echelon form.
1. 2.
3. Write the augmented matrix corresponding to the system of equations and solve the
system.
4. Find (a) (b) 3 (c) and (d) (if possible).
In Exercises 5 and 6, find the inverse of the matrix (if it exists).
5. 6.
7. Use the result of Exercise 5 to solve the system.
In Exercises 8–10, evaluate the determinant of the matrix.
8. 9. 10.
In Exercises 11 and 12, use Cramer’s Rule to solve (if possible) the system of equations.
11. 12.
13. Use a determinant to find the area of the triangle in the figure.
14. Find the uncoded row matrices for the message KNOCK ON WOOD. Then
encode the message using the matrix below.
15. One hundred liters of a 50% solution is obtained by mixing a 60% solution with a
20% solution. How many liters of each solution must be used to obtain the desired
mixture?
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6
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5
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14
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8
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FIGURE FOR 13
636
PROOFS IN MATHEMATICS
Proofs without words are pictures or diagrams that give a visual understanding of why
a theorem or statement is true. They can also provide a starting point for writing a
formal proof. The following proof shows that a determinant is the area of a
parallelogram.
The following is a color-coded version of the proof along with a brief explanation
of why this proof works.
Area of Area of orange Area of yellow Area of blue
Area of pink Area of white quadrilateral
Area of Area of orange Area of pink Area of green
quadrilateral
Area of Area of white quadrilateral Area of blue Area of yellow
Area of green quadrilateral
Area of Area of
From “Proof Without Words” by Solomon W. Golomb, Mathematics Magazine, March 1985, Vol. 58, No. 2,
pg. 107. Reprinted with permission.
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# �# ��$ �
# �# �! �
# �
# �# �# �" �
ac b
d � ad � bc � !" ! � ! &! � !$!
(0, d)
(0, 0)
(a, d)
(a, b)
(a, 0)
(a + c, d)
(a + c, b + d)(a, b + d)
ac b
d � ad � bc � !" ! � ! &! � !$!
(0, d)
(0, 0)
(a, d)
(a, b)
(a, 0)
(a + c, d)
(a + c, b + d)(a, b + d)
2 2
637
1. The columns of matrix show the coordinates of the
vertices of a triangle. Matrix is a transformation matrix.
(a) Find and Then sketch the original triangle
and the two transformed triangles. What transformation
does represent?
(b) Given the triangle determined by describe the
transformation process that produces the triangle
determined by and then the triangle determined by
2. The matrices show the number of people (in thousands)
who lived in each region of the United States in 2000 and
the number of people (in thousands) projected to live
in each region in 2015. The regional populations are
separated into three age categories. (Source: U.S.
Census Bureau)
2000
0–17 18–64 65 +
2015
0–17 18–64 65 +
(a) The total population in 2000 was approximately
281,422,000 and the projected total population in
2015 is 322,366,000. Rewrite the matrices to give the
information as percents of the total population.
(b) Write a matrix that gives the projected change in the
percent of the population in each region and age
group from 2000 to 2015.
(c) Based on the result of part (b), which region(s) and
age group(s) are projected to show relative growth
from 2000 to 2015?
3. Determine whether the matrix is idempotent. A square
matrix is idempotent if
(a) (b)
(c) (d)
4. Let
(a) Show that where is the identity
matrix of order 2.
(b) Show that
(c) Show in general that for any square matrix satisfying
the inverse of is given by
5. Two competing companies offer satellite television to a
city with 100,000 households. Gold Satellite System has
25,000 subscribers and Galaxy Satellite Network has
30,000 subscribers. (The other 45,000 households do not
subscribe.) The percent changes in satellite subscriptions
each year are shown in the matrix below.
Percent Changes
From From From Non-
Gold Galaxy subscriber
(a) Find the number of subscribers each company will
have in 1 year using matrix multiplication. Explain
how you obtained your answer.
(b) Find the number of subscribers each company will
have in 2 years using matrix multiplication. Explain
how you obtained your answer.
(c) Find the number of subscribers each company will
have in 3 years using matrix multiplication. Explain
how you obtained your answer.
(d) What is happening to the number of subscribers to
each company? What is happening to the number of
nonsubscribers?
6. Find such that the matrix is equal to its own inverse.
7. Find such that the matrix is singular.
8. Find an example of a singular matrix satisfying
A2 � A.
2 2
A � � 4
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x
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x
A � � 3
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x
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x
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0.20
0.10
0.15
0.80
0.05
0.15
0.15
0.70�
To Gold
To Galaxy
To Nonsubscriber
PercentChanges
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A
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16,363
29,372
6,016
12,826
35,288
42,249
73,495
14,231
33,294
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16,648
25,567
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2,030
4,892�
T.AT
AAT,
A
AAT.AT
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2
4
3
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T
This collection of thought-provoking and challenging exercises further explores andexpands upon concepts learned in this chapter.
PROBLEM SOLVING
Northeast
Midwest
South
Mountain
Pacific
Northeast
Midwest
South
Mountain
Pacific
638
9. Verify the following equation.
10. Verify the following equation.
11. Verify the following equation.
12. Use the equation given in Exercise 11 as a model to find
a determinant that is equal to
13. The atomic masses of three compounds are shown in the
table. Use a linear system and Cramer’s Rule to find the
atomic masses of sulfur (S), nitrogen (N), and fluorine (F).
14. A walkway lighting package includes a transformer, a
certain length of wire, and a certain number of lights on
the wire. The price of each lighting package depends on
the length of wire and the number of lights on the wire.
Use the following information to find the cost of a
transformer, the cost per foot of wire, and the cost of a
light. Assume that the cost of each item is the same in
each lighting package.
• A package that contains a transformer, 25 feet of
wire, and 5 lights costs $20.
• A package that contains a transformer, 50 feet of
wire, and 15 lights costs $35.
• A package that contains a transformer, 100 feet of
wire, and 20 lights costs $50.
15. The transpose of a matrix, denoted is formed by
writing its columns as rows. Find the transpose of each
matrix and verify that
16. Use the inverse of matrix to decode the cryptogram.
23 13 31 63 25 61
24 14 41 20 40
38 116 13 1 22
41 85 28 16
17. A code breaker intercepted the encoded message below.
45 38 18 35 81
42 75 2 22 15
Let
(a) You know that and
that where is the
inverse of the encoding matrix Write and solve
two systems of equations to find and
(b) Decode the message.
18. Let
Use a graphing utility to find Compare with
Make a conjecture about the determinant of the
inverse of a matrix.
19. Let be an matrix each of whose rows adds up to
zero. Find
20. Consider matrices of the form
(a) Write a matrix and a matrix in the form
of
(b) Use a graphing utility to raise each of the matrices
to higher powers. Describe the result.
(c) Use the result of part (b) to make a conjecture about
powers of if is a matrix. Use a graphing
utility to test your conjecture.
(d) Use the results of parts (b) and (c) to make a
conjecture about powers of if is an
matrix.
n nAA
4 4AA
A.
3 32 2
A � �0
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a12
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a13
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a2n
a3n
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A.A�1A�1.
A � �6
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z.y,x,w,
A.
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A�1 � �wyx
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B � ��3
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�AB�T � BTAT.
AT,
ax3 � bx2 � cx � d.
x
�1
0
0
x
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c
b
a � ax2 � bx � c
1
a
a3
1
b
b3
1
c
c3 � �a � b��b � c��c � a��a � b � c�
1
a
a2
1
b
b2
1
c
c2 � �a � b��b � c��c � a�
Compound Formula Atomic Mass
Tetrasulfur
tetranitride
Sulfur
hexafluoride
Dinitrogen
tetrafluoride
N2F4
SF6
S4N4 184
146
104