Post on 10-Aug-2020
� � � � � � � �MATRICES ANDDETERMINANTSMATRICES ANDDETERMINANTS
c How can you organize data aboutmusic sales?
196
� � � � � � � �
APPLICATION: Music Sales
When you shop for music, your
purchases matter. Music sales by
people age 15 to 19 years
accounted for about 16% of the
$13.7 billion total music purchases
in 1998. The tables below give the
dollar value of the music sold by
age category in 1997 and 1998.
Think & Discuss
1. From 1997 to 1998, how much did sales in the15–19 age category increase? How much did salesin the 20–24 age category decrease?
2. Make a new table that gives the change in the salesin each age category from 1997 to 1998. (Usenegative numbers to indicate decreases.)
Learn More About It
You will find the total Hispanic music sales for 1996and 1997 in Exercise 40 on p. 204.
APPLICATION LINK Visit www.mcdougallittell.com
for more information about music sales.
INT
ERNET
C H A P T E R
4
197
c
1997 Music Sales
Age Sales(years) (billions)
10–14 108.58
15–19 204.96
20–24 168.36
25–29 142.74
30–34 134.2
35–39 141.52
40–44 107.36
45+ 201.3
1998 Music Sales
Age Sales(years) (billions)
10–14 124.67
15–19 216.46
20–24 167.14
25–29 156.18
30–34 156.18
35–39 172.52
40–44 113.71
45+ 247.97
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198 Chapter 4
What’s the chapter about?
Chapter 4 is about matrices and determinants. You can use matrices to organizenumerical data. In Chapter 4 you’ll learn
• how to add, subtract, and multiply matrices, and how to evaluate determinants.
• how to solve linear systems using Cramer’s rule and inverse matrices.
CHAPTER
4Study Guide
PREVIEW
Are you ready for the chapter?
SKILL REVIEW Do these exercises to review key skills that you’ll apply in thischapter. See the given reference page if there is something you don’t understand.
Perform the operation(s). (Skills Review, p. 905)
1. 4 + (º5) 2. º10 º 3 3. (º1)6 º 4(2) 4. 12(5) + 2(º10)
Identify the property shown. (Review Example 4, p. 5)
5. 3(º9) = º9(3) 6. 7 + y = y + 7 7. 4(x º 2) = 4x º 4(2)
Solve the system of linear equations using any algebraic method. (Review p. 149)
8. 2x = 30 9. x º y = 7 10. x º 2y = 24 11. 3x + y = º8x + 4y = 27 3x º 7y = 61 x + 3y = 20 8x º 3y = º10
PREPARE
Here’s a study strategy!
STUDY
STRATEGY
c Review
• multiplicative inverse, p. 5
• coefficient, p. 13
• constant, p. 13
• solution of a system of linearequations, p. 139
c New
• matrix, p. 199
• equal matrices, p. 199
• scalar, p. 200
• scalar multiplication, p. 200
• determinant, p. 214
• Cramer’s rule, p. 216
• identity matrix, p. 223
• inverse matrix, p. 223
KEY VOCABULARY
STUDENT HELP
Study Tip“Student Help” boxesthroughout the chaptergive you study tips andtell you where to look for extra help in this book and on the Internet.
Writing Out the StepsIn this chapter you will perform calculations withnumbers in matrices. When you work with matrices,point your fingers at the numbers you will add,subtract, or multiply. Before calculating, write thenumbers and the operation symbols in the correctlocation of the solution matrix. By doing this step, you are less likely to make a mistake. If you domake a mistake, you can easily trace backwards tofind it.
� � � � � � � �4.1
E X P L O R I N G D ATA
A N D S TAT I S T I C S
4.1 Matrix Operations 199
Matrix Operations
USING MATRIX OPERATIONS
A is a rectangular arrangement of numbers in rows and columns. Forinstance, matrix A below has two rows and three columns. The of this matrix are 2 ª 3 (read “2 by 3”). The numbers in a matrix are its In matrix A, the entry in the second row and third column is 5.
A = F G
Some matrices (the plural of matrix) have special names because of their dimensionsor entries.
NAME DESCRIPTION EXAMPLE
Row matrix A matrix with only 1 row f g
Column matrix A matrix with only 1 column F GSquare matrix A matrix with the same number F Gof rows and columns
Zero matrix A matrix whose entries are all zeros F GTwo matrices are if their dimensions are the same and the entries incorresponding positions are equal.
Comparing Matrices
a. The following matrices are equal because corresponding entries are equal.
F G = F Gb. The following matrices are not equal because corresponding entries in the second
row are not equal.
F G ≠ F G. . . . . . . . . .
To add or subtract matrices, you simply add or subtract corresponding entries. You can add or subtract matrices only if they have the same dimensions.
60
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6º3
º20
00.75
5º1
0
}
34
}
5
º}
44
}
E X A M P L E 1
equal
000
000
516
º10
º3
421
13
40º23
º15
20
6º2
entries.dimensions
matrix
GOAL 1
Add and subtract
matrices, multiply a matrix by
a scalar, and solve matrix
equations.
Use matrices in
real-life situations, such as
organizing data about health
care plans in Example 5.
. To organize real-life
data, such as the data for
Hispanic CD, cassette,
and music video sales
in Exs. 39–41.
Why you should learn it
GOAL 2
GOAL 1
What you should learn
REA
L LIFE
REA
L LIFE
2 rows
3 columns
� � � � �
200 Chapter 4 Matrices and Determinants
Adding and Subtracting Matrices
Perform the indicated operation, if possible.
a. F G + F G b. F G º F G c. F G + F GSOLUTION
a. Since the matrices have the same dimensions, you can add them.
F G + F G = F G = F Gb. Since the matrices have the same dimensions, you can subtract them.
F G º F G = F G = F Gc. Since the dimensions of F G are 2 ª 2 and the dimensions of F G are 2 ª 1,
you cannot add the matrices.
. . . . . . . . . .
In matrix algebra, a real number is often called a To multiply a matrix by a scalar, you multiply each entry in the matrix by the scalar. This process is called
Multiplying a Matrix by a Scalar
Perform the indicated operation(s), if possible.
a. 3F G b. º2F G + F GSOLUTION
a. 3F G = F G = F G
b. º2F G + F G = F G + F G= F G + F G= F G9
º14º4
º666
5º8
6
º46
º2
4º6
º10
º208
5º8
6
º46
º2
º2(º2)º2(3)º2(5)
º2(1)º2(0)
º2(º4)
5º8
6
º46
º2
º235
10
º4
0º21
º612
3(0)3(º7)
3(º2)3(4)
0º7
º24
5º8
6
º46
º2
º235
10
º4
0º7
º24
E X A M P L E 3
scalar multiplication.
scalar.
15
04
23
101
6º2
3 º (º7)0 º (º1)
8 º 24 º 6
º7º1
26
30
84
4º410
3 + 1º4 + 0
7 + 3
103
3º4
7
15
04
23
º7º1
26
30
84
103
3º4
7
E X A M P L E 2
STUDENT HELP
Study TipThe order of operationsfor matrix expressions issimilar to that for realnumbers. In particular,you perform scalarmultiplication beforematrix addition andsubtraction, as shown inpart (b) of Example 3.
� � � � �
4.1 Matrix Operations 201
You can use what you know about matrix operations and matrix equality to solve amatrix equation.
Solving a Matrix Equation
Solve the matrix equation for x and y: 2SF G + F GD= F GSOLUTION
Simplify the left side of the equation.
2SF G + F GD = F G2F G = F GF G = F G
Equate corresponding entries and solve the two resulting equations.
6x + 8 = 26 10 º 2y = 8
x = 3 y = 1
. . . . . . . . . .
In Example 4, you could have distributed the scalar 2 to each matrix inside theparentheses before adding the matrices.
2SF G + F GD = 2F G + 2F G= F G + F G= F G
This illustrates one of several properties of matrix operations stated below.
010 º 2y
6x + 812
2º2y
8º4
º210
6x16
1ºy
4º2
º15
3x8
1ºy
4º2
º15
3x8
08
26
120
10 º 2y
6x + 8
12
08
2612
05 º y
3x + 46
08
2612
1ºy
4º2
º15
3x8
08
2612
1ºy
4º2
º15
3x8
E X A M P L E 4
Let A, B, and C be matrices with the same dimensions and let c be a scalar.
When adding matrices, you can regroup them and change their order without
affecting the result.
ASSOCIATIVE PROPERTY OF ADDITION (A + B) + C = A + (B + C)
COMMUTATIVE PROPERTY OF ADDITION A + B = B + A
Multiplication of a sum or difference of matrices by a scalar obeys the
distributive property.
DISTRIBUTIVE PROPERTY OF ADDITION c(A + B) = cA + cB
DISTRIBUTIVE PROPERTY OF SUBTRACTION c(A º B) = cA º cB
PROPERTIES OF MATRIX OPERATIONS
Look Back For help with propertiesof real numbers, see p. 5.
STUDENT HELP
CONCEPT
SUMMARY
� � � � � � � �
202 Chapter 4 Matrices and Determinants
USING MATRICES IN REAL LIFE
Using Matrices to Organize Data
Use matrices to organize the following information about health care plans.
This Year For individuals, Comprehensive, HMO Standard, and HMO Plus cost$694.32, $451.80, and $489.48, respectively. For families, the Comprehensive, HMOStandard, and HMO Plus plans cost $1725.36, $1187.76, and $1248.12.
Next Year For individuals, Comprehensive, HMO Standard, and HMO Plus will cost$683.91, $463.10, and $499.27, respectively. For families, the Comprehensive, HMOStandard, and HMO Plus plans will cost $1699.48, $1217.45, and $1273.08.
SOLUTION
One way to organize the data is to use 3 ª 2 matrices, as shown.
THIS YEAR (A) NEXT YEAR (B)
Individual Family Individual Family
Comprehensive F G F GHMO Standard
HMO Plus
You can also organize the data using 2 ª 3 matrices where the row labels are levelsof coverage (individual and family) and the column labels are the types of plans(Comprehensive, HMO Standard, and HMO Plus).
Using Matrix Operations
HEALTH CARE A company offers the health care plans in Example 5 to its
employees. The employees receive monthly paychecks from which health care
payments are deducted. Use the matrices in Example 5 to write a matrix that shows
the monthly changes in health care payments from this year to next year.
SOLUTION
Begin by subtracting matrix A from matrix B to determine the yearly changes in
health care payments. Then multiply the result by }
112} and round answers to the
nearest cent to find the monthly changes.
}
112}(B º A) = }
112}SF G º F GD
= }
112}F G
≈ F Gc The monthly deductions for the Comprehensive plan will decrease, but the
monthly deductions for the other two plans will increase.
º$2.16$2.47$2.08
º$.87$.94$.82
º25.8829.6924.96
º10.4111.309.79
1725.361187.761248.12
694.32451.80489.48
1699.481217.451273.08
683.91463.10499.27
E X A M P L E 6
$1699.48$1217.45$1273.08
$683.91$463.10$499.27
$1725.36$1187.76$1248.12
$694.32$451.80$489.48
E X A M P L E 5
GOAL 2
RE
AL LIFE
RE
AL LIFE
Health Care
HEALTH SERVICES
MANAGER
Health services managers inhealth maintenanceorganizations (HMOs) plan
and organize the delivery of
health care.
CAREER LINK
www.mcdougallittell.com
INT
ERNET
RE
AL LIFE
RE
AL LIFE
FOCUS ON
CAREERS
� � � � � � � �
4.1 Matrix Operations 203
1. What is a matrix? Describe and give an example of a row matrix, a columnmatrix, and a square matrix.
2. Are the two matrices equal? Explain. 3 4 · F G3. To add or subtract two matrices, what must be true?
4. Use the matrices at the right to find º2(A + B). Is your answer the same as that for part (b) of Example 3? Explain.
5. Rework Example 5 by organizing the data using 2 ª 3 matrices.
Perform the indicated operation(s), if possible.
6. F G º F G 7. F G + F G8. º4F G 9. 6F G º 5F G
10. HEALTH CARE In Example 5, suppose the annual health care costs given inmatrix B increase by 4% the following year. Write a matrix that shows the newmonthly payment.
COMPARING MATRICES Tell whether the matrices are equal or not equal.
11. f g, F G 12. F G, F G
13. F G, F G 14. F G, 3 4ADDING AND SUBTRACTING MATRICES Perform the indicated operation,
if possible. If not possible, state the reason.
15. F G + F G 16. F G + F G17. F G º F G 18. F G + F G19. F G + F G 20. F G + F G21. F G º F G 22. F G º F G}
43
}
5
2
}
12
}
}
41
}
8
}
21
}
3
992
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7º211
6º6
8
1º1
2
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61
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42
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711
8.75.3
4.12.6
3.55.1
1.20.2
º79
2º4
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5º1
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40
52
3º5
º42
1º7
}
147}
0
}
23
}
º}
12
}
2
}
12
}
4.250
1.5º0.5
20.5
0º4
4º2
0º4
42
º81
00
18
º81
00
18
5º1
77º15
PRACTICE AND APPLICATIONS
0º3
º14
º10
º52
0º5
2º4
8º9
º512
º19
4º1
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º6º4
º11º10º6
20º22
9
0.5º5
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º643
}
21
}
º55
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43
GUIDED PRACTICE
Vocabulary Check ✓
Concept Check ✓
Skill Check ✓
A = F G B = F G5º8
6
º46
º2
º235
10
º4
Extra Practiceto help you masterskills is on p. 944.
STUDENT HELP
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 11–14Example 2: Exs. 15–22Example 3: Exs. 23–32Example 4: Exs. 33–36Examples 5, 6: Exs. 37–47
� � � � � � � �
204 Chapter 4 Matrices and Determinants
MULTIPLYING BY A SCALAR Perform the indicated operation.
23. º4F G 24. 5F G 25. 4F G26. º93 4 27. }
12
}F G 28. 2.53 4COMBINING MATRIX OPERATIONS Perform the indicated operations.
29. F G + 4F G 30. 2F G º F G31. 2F G + 4F G 32. 3F G º 2F GSOLVING MATRIX EQUATIONS Solve the matrix equation for x and y.
33. F G = F G 34. F G + F G = F G35. 2xF G = F G 36. F G + F G = F G
BASEBALL STATISTICS In Exercises 37 and 38, use the following
information about three Major League Baseball teams’ wins and losses in 1998
before and after the All-Star Game. c Source: CNN/SI
Before The Atlanta Braves had 59 wins and 29 losses, the Seattle Mariners had 37 wins and 51 losses, and the Chicago Cubs had 48 wins and 39 losses.
After The Atlanta Braves had 47 wins and 27 losses, the Seattle Mariners had 39 wins and 34 losses, and the Chicago Cubs had 42 wins and 34 losses.
37. Use matrices to organize the information.
38. Using your matrices from Exercise 37, write a matrix that shows the totalnumbers of wins and losses for the three teams in 1998.
HISPANIC MUSIC In Exercises 39–41, use the following information.
The figures below give the number (in millions) of Hispanic CD, cassette, and music video units shipped to all market channels and the dollar value (in millions) ofthose shipments (at suggested list prices). c Source: Recording Industry Association of America
1996 Number of units—CDs: 20,779; cassettes: 15,299; and music videos: 45. Dollar value—CDs: 268,441; cassettes: 122,329; and music videos: 916.
1997 Number of units—CDs: 26,277; cassettes: 17,799; and music videos: 70. Dollar value—CDs: 344,697; cassettes: 144,645; and music videos: 1,260.
39. Use matrices to organize the information.
40. Write a matrix that gives the total numbers of units shipped and total values forboth years.
41. Write a matrix that gives the change in units shipped and dollar value from 1996 to 1997.
º80
º7
º1y5
x7
º9
º5º7
4
º3º7
2
481
º16º20
12y
45
º3º11
º20
º16y
0º8
º4º7
º28
3xº1
yº9
6º10
º8º9
º2xº10
º75
º1º5
4º3
0º2
1º6
º78
º4º6
2º5
º73
7º1
1319
5º6
º1º3
2º4
º10º7
º64
0º2
5
º13
º4
º853
1200
3.4º5.1
4.4º8
º8.61.2
º4.810
4
º10
º2
}
161}
º2
}
14
}
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49
}
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}
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}
915
º11
35
º5
1º5º3
º61
º23
76
º30
º1º4
HOMEWORK HELP
Visit our Web sitewww.mcdougallittell.comfor help with Exs. 33–36.
INT
ERNET
STUDENT HELP
� � � � � � � �
COLLEGE COSTS
Since 1980, collegecosts have risen substan-tially. After adjusting forinflation, costs rose by 48%at public 4-year colleges and 71% at private 4-yearcolleges between 1980 and1995.
RE
AL LIFE
RE
AL LIFE
FOCUS ON
APPLICATIONSA
nnua
l cos
t ($1
000‘
s)
Cost of a 4-Year College
1965 1980 1995
Public Private
0
5
10
15
20
4.1 Matrix Operations 205
42. COLLEGE COSTS The matrices below show the average yearly cost (indollars) of tuition and room and board at colleges in the United States from 1995through 1997. Use matrix addition to write a matrix showing the totals of thesecosts. c Source: U.S. Department of Education
TUITION ROOM AND BOARD
1995 1996 1997 1995 1996 1997
3 4 3 4PSAT SCORES In Exercises 43 and 44, use the following information.
Eligibility for a National Merit Scholarship is based on a student’s PSAT score.Through 1996, this total score was found by doubling a student’s verbal score andadding this value to the student’s mathematics score. Let V represent the averageverbal scores and let M represent the average mathematics scores earned bysophomores and juniors at Central High for tests taken in 1993 through 1996.
VERBAL SCORES (V ) MATHEMATICS SCORES (M)
Sophomores Juniors Sophomores Juniors
3 4 3 443. Write an expression in terms of V and M that you could use to determine the
average total PSAT scores for sophomores and juniors at Central High from 1993 through 1996. Then evaluate the expression.
44. Use the matrix from Exercise 43 to determine the average total PSAT score forjuniors at Central High in 1996.
U.S. POPULATION In Exercises 45–47, use the following information.
The matrices show the number of people (in thousands) who lived in each region ofthe United States in 1991 and the number of people (in thousands) projected to live ineach region in 2010. The regional populations are separated into three age categories.
1991 2010
0–17 18–65 Over 65 0–17 18–65 Over 65
3 4 3 445. The total population in 1991 was 252,177,000 and the projected total population
in 2010 is 297,716,000. Rewrite the matrices to give the information as percentsof the total population. (Hint: Multiply each matrix by the reciprocal of the totalpopulation (in thousands), and then multiply by 100.)
46. Write a matrix that gives the projected change in the percent of the population ineach region and age group from 1991 to 2010.
47. Based on the result of Exercise 46, which region(s) and age group(s) areprojected to show relative growth from 1991 to 2010?
7,3778,980
14,8322,7075,511
33,82241,09567,33712,42031,125
12,49315,84025,4285,094
13,655
7,0437,857
10,9421,5804,331
31,79136,55453,4718,461
25,001
12,14215,81422,5043,993
10,693
Northeast
Midwest
South
Mountain
Pacific
50.450.050.850.9
49.048.349.449.8
49.048.948.848.6
48.948.948.748.2
1993
1994
1995
1996
3,1284,3454,6995,555
2,9784,1664,4695,368
2,9443,9904,2565,121
1,2832,9867,190
12,920
1,2392,8487,094
12,243
1,1922,6816,914
11,481
Public 2-year college
Public 4-year college
Private 2-year college
Private 4-year college
DATA UPDATE of U.S. Bureau of the Census data at www.mcdougallittell.com INT
ERNET
! " # $ % & '
206 Chapter 4 Matrices and Determinants
48. MULTI-STEP PROBLEM The matrices show the number of hardcover volumessold and the average price per volume (in dollars) for different subject areas.
c Source: The Bowker Annual
1995 (A) 1996 (B )
Volumes Average price Volumes Average pricesold per volume sold per volume
3 4 3 4a. Calculate B º A. How many more (or fewer) law volumes were sold in 1996
than in 1995? How much more (or less) did the average music book cost in1996 than in 1995?
b. Calculate B + A. Does the “volumes sold” column in B + A give youmeaningful information? Does the “average price per volume” column in B + A give you meaningful information? Explain.
c. Writing What conclusions can you make about the number of volumes soldand the average price per volume of these books from 1995 to 1996?
49. A triangle has vertices (2, 2), (8, 2), and (5, 6). Assigna letter to each vertex and organize the triangle’s vertices in a matrix. When youmultiply the matrix by 4, what does the “new” triangle look like? How are thetwo triangles related? Use a graph to help you.
TRANSFORMING FIGURES Draw the figure
produced by each transformation of the figure
shown. (Skills Review, p. 921)
50. a 270° counterclockwise rotation about the origin
51. a translation by 2 units right and 4 units down
52. a reflection over the y-axis
MULTIPLYING REAL NUMBERS Find the product. (Skills Review, p. 905)
53. º4(º5) 54. 8(º2) 55. º7(º1)
56. }
12
}(º7) 57. }
56
} • }
37
} 58. 3.2(2.4 + 8.1)
CHECKING SOLUTIONS Check whether the ordered pairs are solutions of the
inequality. (Review 2.6)
59. x + 2y ≤ º3; (0, 3), (º5, 1) 60. 5x º y > 2; (º5, 0), (5, 23)
61. º8x º 3y < 5; (º1, 1), (3, º9) 62. 21x º 10y > 4; (2, 3), (º1, 0)
FINDING A SOLUTION Give an ordered pair that is a solution of the system.
(Lesson 3.3)
63. x + y < 10 64. x º y ≥ 3 65. 3x > yy > 1 y < 12 x ≤ 15
MIXED REVIEW
GEOMETRY CONNECTION
53.4088.5139.2133.92
1,070,000827,000253,000179,000
41.2373.0943.2738.30
1,116,000716,000251,000199,000
Art
Law
Music
Travel
★★Challenge
TestPreparation
()
*( (4, 5) (6, 5)
(2, 2) (8, 2)
+ , - . / 0 1 2
4.1 Technology Activity 207
Using Matrix OperationsYou can use a graphing calculator to perform matrix operations.
c EXAMPLE
The matrices show music sales (in thousands) for a chain of stores during a two-month period. The music formats are CDs and cassette tapes. The categories of music are Rock (R), Country (C), Jazz (J), and Easy Listening (E).
a. Find the total sales for each format and category for May and June.
b. Estimate the sales for July if they are expected to decrease 3% from the June sales.
c SOLUTION
a. To find the total sales in each format and category for May and June, let matrix Arepresent May sales and let matrix B represent June sales. Use a graphingcalculator to enter each matrix. Then find the sum of matrix A and matrix B.
For each matrix, enter the Then enter the data row by To find the sum, select each
dimensions of the matrix. row until all 8 entries have matrix from the matrix menu
been filled for each matrix. and add.
b. To estimate the sales for July, multiply matrix Bby the scalar 0.97. The calculator shows, forinstance, that July sales for rock CDs areexpected to be 23,280.
c EXERCISES
Use a graphing calculator to perform the indicated operation(s).
1. F G + F G 2. F G º F G3. 1.043F G 4. 5.08F G º 0.07F G5. MUSIC Write a matrix that gives the change in sales from May to June for
the chain of stores discussed in the example. For which categories of CDs andcassettes did sales increase from May to June? For which did sales decrease?
0.05234.15
1.00911.2
1.250.01
0.063.32
1.67
6.222.1
359184299
512428735
418452146
418452146
º9.24.22
11.56.03
3.1º2.5
º4.99.3
[A]+[B][[56 31 6 15] [53 22 9 30]]
MATRIX [A] 2X4[32 16 3 8][28 12 5 15]
2,4=15
MATRIX [A] 2X4[0 0 0 0][0 0 0 0]
Using Technology
Graphing Calculator Activity for use with Lesson 4.1ACTIVITY 4.1
.97[B][[23.28 14.5... [24.25 9.7 ...
KEYSTROKE HELP
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R C J E R C J E
F G F G715
34
1510
2425
815
35
1612
3228
CDs
Tapes
+ , - . / 0 1 2
208 Chapter 4 Matrices and Determinants
Multiplying Matrices
MULTIPLYING TWO MATRICES
The product of two matrices A and B is defined provided the number of columns in Ais equal to the number of rows in B.
If A is an m ª n matrix and B is ann ª p matrix, then the product ABis an m ª p matrix.
Describing Matrix Products
State whether the product AB is defined. If so, give the dimensions of AB.
a. A: 2 ª 3, B: 3 ª 4 b. A: 3 ª 2, B: 3 ª 4
SOLUTION
a. Because A is a 2 ª 3 matrix and B is a 3 ª 4 matrix, the product AB is definedand is a 2 ª 4 matrix.
b. Because the number of columns in A (two) does not equal the number of rowsin B (three), the product AB is not defined.
Finding the Product of Two Matrices
Find AB if A = F G and B = F G.
SOLUTION
Because A is a 3 ª 2 matrix and B is a 2 ª 2 matrix, the product AB is defined and is a 3 ª 2 matrix. To write the entry in the first row and first column of AB, multiplycorresponding entries in the first row of A and the first column of B. Then add. Use asimilar procedure to write the other entries of the product.
AB = F G F G
AB = F GAB = F G6
º1318
º47
º6
(º2)(3) + (3)(4)(1)(3) + (º4)(4)(6)(3) + (0)(4)
(º2)(º1) + (3)(º2)(1)(º1) + (º4)(º2)(6)(º1) + (0)(º2)
34
º1
º2
3
º40
º2
16
34
º1º2
3º4
0
º216
E X A M P L E 2
E X A M P L E 1
GOAL 1
Multiply two
matrices.
Use matrix
multiplication in real-life
situations, such as finding the
number of calories burned in
Ex. 40.
. To solve real-life
problems, such as
calculating the cost of
softball equipment in
Example 5.
Why you should learn it
GOAL 2
GOAL 1
What you should learn
4.2R
E
AL LIFE
RE
AL LIFE
• =
equal
dimensions of AB
ABm ª p
Bn ª p
Am ª n
3 4 5 6 7 8 9 :
4.2 Multiplying Matrices 209
Finding the Product of Two Matrices
If A = F G and B = F G, find each product.
a. AB b. BA
SOLUTION
a. AB = F G F G = F Gb. BA = F G F G = F G
. . . . . . . . . .
Notice in Example 3 that AB ≠ BA. Matrix multiplication is not, in general,commutative.
Using Matrix Operations
If A = F G, B = F G, and C = F G, simplify each expression.
a. A(B + C) b. AB + AC
SOLUTION
a. A(B + C) = F G 1F G + F G2A(B + C) = F G F G = F G
b. AB + AC = F G F G + F G F GA(B + C) = F G + F G = F G
. . . . . . . . . .
Notice in Example 4 that A(B + C) = AB + AC, which is true in general. This andother properties of matrix multiplication are summarized below.
611
522
45
58
26
014
12
13
13
2º1
02
º24
13
2º1
611
522
14
º17
13
2º1
12
13
02
º24
13
2º1
12
13
02
º24
13
2º1
E X A M P L E 4
24
75
20
3º1
º41
12
º104
7º1
º41
12
20
3º1
º41
12
20
3º1
E X A M P L E 3
Let A, B, and C be matrices and let c be a scalar.
ASSOCIATIVE PROPERTY OF MATRIX MULTIPLICATION A(BC ) = (AB )C
LEFT DISTRIBUTIVE PROPERTY A(B + C ) = AB + AC
RIGHT DISTRIBUTIVE PROPERTY (A + B )C = AC + BC
ASSOCIATIVE PROPERTY OF SCALAR MULTIPLICATION c(AB) = (cA)B = A(cB )
PROPERTIES OF MATRIX MULTIPLICATION
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CONCEPT
SUMMARY
3 4 5 6 7 8 9 :
210 Chapter 4 Matrices and Determinants
USING MATRIX MULTIPLICATION IN REAL LIFE
Matrix multiplication is useful in business applications because an inventory matrix,when multiplied by a cost per item matrix, results in a total cost matrix.
F G • F G = F Gm ª n n ª p m ª p
For the total cost matrix to be meaningful, the column labels for the inventory matrixmust match the row labels for the cost per item matrix.
Using Matrices to Calculate the Total Cost
SPORTS Two softball teams submit equipment lists for the season.
Each bat costs $21, each ball costs $4, and each uniform costs $30. Use matrixmultiplication to find the total cost of equipment for each team.
SOLUTION
To begin, write the equipment lists and the costs per item in matrix form. Becauseyou want to use matrix multiplication to find the total cost, set up the matrices so thatthe columns of the equipment matrix match the rows of the cost matrix.
The total cost of equipment for each team can now be obtained by multiplying theequipment matrix by the cost per item matrix. The equipment matrix is 2 3 3 and thecost per item matrix is 3 3 1, so their product is a 2 3 1 matrix.
F G F G = F G = F GThe labels for the product matrix are as follows.
TOTAL COST
Dollars
F Gc The total cost of equipment for the women’s team is $882, and the total
cost of equipment for the men’s team is $977.
882977
Women’s team
Men’s team
882977
12(21) + 45(4) + 15(30)15(21) + 38(4) + 17(30)
214
30
1517
4538
1215
E X A M P L E 5
Total costmatrix
Cost per itemmatrix
Inventorymatrix
GOAL 2
EQUIPMENT COST
Bats Balls Uniforms Dollars
F G F G214
30
Bats
Balls
Uniforms
1517
4538
1215
Women’s team
Men’s team
DOT RICHARDSON
helped lead theUnited States to the firstwomen’s softball gold medalin the 1996 Olympics byplaying shortstop.
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Women’s team
12 bats
45 balls
15 uniforms
Men’s team
15 bats
38 balls
17 uniforms
; < = > ? @ A B
4.2 Multiplying Matrices 211
1. Complete this statement: The product of matrices A and B is defined providedthe number of
ooo
? in A is equal to the number of ooo
? in B.
2. Matrix A is 6 ª 1. Matrix B is 1 ª 2. Which of the products is defined, AB orBA? Explain.
3. Tell whether the matrix equation is true or false. Explain.
F G F G = F G F GState whether the product AB is defined. If so, give the dimensions of AB.
4. A: 3 ª 2, B: 2 ª 3 5. A: 3 ª 3, B: 3 ª 3 6. A: 3 ª 2, B: 3 ª 2
Find the product.
7. F G F G 8. f g F G 9. F G F G10. SOFTBALL EQUIPMENT Use matrix multiplication to find the total cost
of equipment in Example 5 if the women’s team needs 16 bats, 42 balls, and 16 uniforms and the men’s team needs 14 bats, 43 balls, and 15 uniforms.
MATRIX PRODUCTS State whether the product AB is defined. If so, give the
dimensions of AB.
11. A: 1 ª 3, B: 3 ª 2 12. A: 2 ª 4, B: 4 ª 3 13. A: 4 ª 2, B: 3 ª 5
14. A: 5 ª 5, B: 5 ª 4 15. A: 3 ª 4, B: 4 ª 1 16. A: 3 ª 3, B: 2 ª 4
FINDING MATRIX PRODUCTS Find the product. If it is not defined, state the
reason.
17. 3 4 F G 18. F G f g
19. F G F G 20. F G F G
21. F G F G 22. F G F G
23. F G F G 24. F G F G25. F G F G 26. F G F G46
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PRACTICE AND APPLICATIONS
0º1
1º2
3º2º1
º330
º2º3
4403
21
0º1
1º2
35
5º3
01
20
01
20
35
5º3
GUIDED PRACTICE
Vocabulary Check ✓
Concept Check ✓
Skill Check ✓
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Example 1: Exs. 11–16Examples 2, 3: Exs. 17–26Example 4: Exs. 27–32Example 5: Exs. 35–40
; < = > ? @ A B
212 Chapter 4 Matrices and Determinants
SIMPLIFYING EXPRESSIONS Using the given matrices, simplify the
expression.
A = F G, B = F G, C = F G, D = F G, E = F G27. 2AB 28. AB + AC 29. D(D + E)
30. (E + D)E 31. º3(AC) 32. 0.5(AB) + 2AC
SOLVING MATRIX EQUATIONS Solve for x and y.
33. F G F G = F G 34. F G F G = F GAGRICULTURE In Exercises 35 and 36, use the following information.
The percents of the total 1997 world production of wheat, rice, and maize are shown in the matrix for the four countries that grow the most grain: China, India, the Commonwealth of Independent States (formerly the Soviet Union), and the United States. The total 1997 world production (in thousands of metric tons) of wheat, rice, and maize is 608,846, 570,906, and 586,923, respectively.
c Source: Food and Agriculture Organization of the United Nations
35. Rewrite the matrix to give the percents as decimals.
36. Show how matrix multiplication can be used to determine how many metric tonsof all three grains were produced in each of the four countries.
CLASS DEBATE In Exercises 37º39, use the
following information.
Three teams participated in a debating competition. The final score for each team is based on how many students ranked first, second, and third in a debate. The results of 12 debates are shown in matrix A.
37. Teams earn 6 points for each first place, 5 points for each second place, and 4 points for each third place. Organize this information into a matrix B.
38. Find the product AB.
39. LOGICAL REASONING Which team won the competition? How many pointsdid the winning team score?
40. EXERCISE The numbers of calories burned by people of different weights doing different activities for 20 minutes are shown in the matrix. Show how matrix multiplication can be used to write the total number of calories burned by a 120 pound person and a 150 pound person who each bicycled for 40 minutes, jogged for 10 minutes, and then walked for 60 minutes.
c Source: Medicine and Science in Sports and Exercise
511
yº13
º214
92
º1
31
1x
4º2
619
y
1x3
244
12
º2
º230
62
º4
541
º2º1
3
143
º22
º3
3º1º2
31
º1º2
04
1º2
º2º1
46
MATRIX A
1st 2nd 3rd
F G452
526
354
Team 1
Team 2
Team 3
GRAIN PRODUCTION
Wheat Rice Maize
3 418%
1.7%0.5%
40.5%
34.8%21.5%0.1%1.4%
20.1%22%
7.3%11.3%
China
India
C.I.S.
U.S.
EXERCISE
A 120 pound personwalking at a moderate paceof 3 mi/h would burn about64 Cal in 20 min. At a briskpace of 4.5 mi/h, the personwould burn about 82 Cal in20 min.
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120 lb 150 lb
person person
F G13615979
10912764
Bicycling
Jogging
Walking
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C D E F G H I J41. Writing Describe the process you use when multiplying any two matrices.
42. MULTIPLE CHOICE What is the product of F G and F G?
¡A F G ¡B F G ¡C F G ¡D F G43. MULTIPLE CHOICE If A is a 2 ª 3 matrix and B is a 3 ª 2 matrix, what are the
dimensions of BA?
¡A 2 ª 2 ¡B 3 ª 3 ¡C 3 ª 2 ¡D 2 ª 3 ¡E BA not defined
44. ROTATIONAL MATRIX Matrix A is a 90° rotational matrix. Matrix B containsthe coordinates of the triangle’s vertices shown in the graph.
A = F G B = F Ga. Calculate AB. Graph the coordinates of the
vertices given by AB. What rotation does AB represent in the graph?
b. Find the 180° and 270° rotations of the original triangle by using repeated multiplication of the 90° rotational matrix. What are the coordinates of the vertices of the rotated triangles?
CALCULATING AREA Find the area of the figure. (Skills Review, p. 914)
45. 46. 47.
WRITING EQUATIONS Write an equation of the line with the given properties.
(Review 2.4)
48. slope: }12
}, passes through (1, 8) 49. slope: º}14
}, passes through (0, 4)
50. passes through (3, º6) and (2, 10) 51. passes through (1, 5) and (4, 14)
52. x-intercept: º7, y-intercept: º5 53. x-intercept: 4, y-intercept: º6
SOLVING SYSTEMS Solve the system of linear equations using any algebraic
method. (Review 3.2 for 4.3)
54. 4x + y = 6 55. 2x + y = º9 56. º9x + 5y = 1º3x º 2y = 8 3x + 5y = 4 3x º 2y = 2
57. 2x º 2y = 8 58. º3x + 4y = º1 59. 5x + 2y = º10x º y = 1 6x + 2y = 7 º3x º 8y = 40
60. 7x + 3y = 11 61. 5x º 4y = º1 62. ºx + 7y = º49º2x + 5y = 32 2x º 9y = 10 12x + y = º24
3 ft
6 m
4 m
15 m
12 m
MIXED REVIEW
º42
º48
º74
º10
01
08
1º30
08
1º26
06
8º30
0º24
2º26
º20
7º1
º1º2
0º4
4.2 Multiplying Matrices 213
TestPreparation
★★Challenge
2 K K LM
(24, 8)
(24, 2)
(27, 4)
EXTRA CHALLENGE
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C D E F G H I JDeterminants
and Cramer’s Rule
EVALUATING DETERMINANTS
Associated with each square matrix is a real number called its The determinant of a matrix A is denoted by det A or by |A|.
Evaluating Determinants
Evaluate the determinant of the matrix.
a. F G b. F GSOLUTION
a. * * = 1(5) º 2(3) = 5 º 6 = º1
b. * * = [0 + (º1) + (º12)] º (0 + 4 + 8) = º13 º 12
= º25
. . . . . . . . . .
You can use a determinant to find the area of a triangle whose vertices are points in acoordinate plane.
º102
2º2
1
314
º102
2º2
1
35
12
314
º102
2º2
1
35
12
E X A M P L E 1
determinant.
GOAL 1
214 Chapter 4 Matrices and Determinants
Evaluate
determinants of 2 ª 2 and
3 ª 3 matrices.
Use Cramer’s rule
to solve systems of linear
equations, as applied in
Example 5.
. To solve real-lifeproblems, such as finding the
area of the Golden Triangle of
India in Ex. 58.
Why you should learn it
GOAL 2
GOAL 1
What you should learn
4.3R
EAL LIFE
REA
L LIFE
DETERMINANT OF A 2 ª 2 MATRIX
det F G = * * = ad º cb
DETERMINANT OF A 3 ª 3 MATRIX
det F G = * * = (aei + bfg + cdh) º (gec + hfa + idb)
b
e
h
a
d
g
c
f
i
b
e
h
a
d
g
c
f
i
b
e
h
a
d
g
b
d
a
c
b
d
a
c
THE DETERMINANT OF A MATRIX
Subtract the sum of the products in red
from the sum of the products in blue.
2
The determinant of a 2 ª 2 matrix is the difference
of the products of the entries on the diagonals.
Repeat the first two columns
to the right of the
determinant.
1
N O P Q R S T U
The Area of a Triangle
The area of the triangle shown is:
Area = ±}
12}* *
= ±}
12}[(0 + 12 + 8) º (0 + 2 + 8)] = 5
The Area of a Triangular Region
BERMUDA TRIANGLE The Bermuda Triangle is a large triangular region in the Atlantic Ocean. Many ships and airplanes have been lost in this region. The triangle is formed by imaginary lines connecting Bermuda, Puerto Rico, and Miami, Florida. Use a determinant to estimate the area of the Bermuda Triangle.
SOLUTION
The approximate coordinates of the Bermuda Triangle’s three vertices are (938, 454), (900, º518), and (0, 0). So, the area of the region is as follows:
Area = ±}
12}* *
= ±}
12} [(º485,884 + 0 + 0) º (0 + 0 + 408,600)]
= 447,242
c The area of the Bermuda Triangle is about 447,000 square miles.
111
454º518
0
938900
0
E X A M P L E 3
111
202
146
E X A M P L E 2
4.3 Determinants and Cramer's Rule 215
VV W X(1, 2) (6, 2)
(4, 0)
Bermuda(938, 454)
Puerto Rico(900, 2518)
Miami (0, 0)
Each tick markrepresents 200 mi.
YZ [\
The area of a triangle with vertices
(x1, y1), (x2, y2), and (x3, y3) is given by
Area = ±}
12
}* *where the symbol ± indicates that the
appropriate sign should be chosen to
yield a positive value.
1
1
1
y1
y2
y3
x1
x2
x3
AREA OF A TRIANGLE
XW
(x1, y1)
(x2, y2)
(x3, y3)
BERMUDA
TRIANGLE
The U.S.S. Cyclops, shownabove, disappeared in theBermuda Triangle in March,1918.
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* *5º4
2º10
216 Chapter 4 Matrices and Determinants
USING CRAMER’S RULE
You can use determinants to solve a system of linear equations. The method, called and named after the Swiss mathematician Gabriel Cramer(1704º1752), uses the of the linear system.
LINEAR SYSTEM COEFFICIENT MATRIX
ax + by = ecx + dy = f
In Cramer’s rule, notice that the denominator for x and y is the determinant of thecoefficient matrix of the system. The numerators for x and y are the determinants of the matrices formed by using the column of constants as replacements for thecoefficients of x and y, respectively.
Using Cramer’s Rule for a 232 System
Use Cramer’s rule to solve this system:
SOLUTION
Evaluate the determinant of the coefficient matrix.
* * = º32 º 10 = º42
Apply Cramer’s rule since the determinant is not 0.
x = }º42} = }º8 º
º4(2º50)} = }º
4422} = º1
y = }º42} = }º8º04º2
4} = }ºº
84
42} = 2
c The solution is (º1, 2).
✓CHECK Check this solution 8(º1) + 5(2) · 2 2(º1) º 4(2) · º10in the original equations. 2 = 2 ✓ º10 = º10 ✓
5º4
82
8x + 5y = 22x º 4y = º10
E X A M P L E 4
coefficient matrix
Cramer’s rule
GOAL 2
F Gbd
ac
Let A be the coefficient matrix of this linear system:
ax + by = e
cx + dy = f
If det A ≠ 0, then the system has exactly one solution. The solution is:
x = }det A
} and y = }det A
}
CRAMER’S RULE FOR A 2 X 2 SYSTEM
* *b
d
e
f * *e
f
a
c
* *2º10
82
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] ^ _ ` a b c d
CHEMIST
Chemists researchand put to practical useknowledge about chemicals.Research on the chemistry ofliving things sparks advancesin medicine, agriculture, andother fields.
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4.3 Determinants and Cramer's Rule 217
Using Cramer’s Rule for a 3 ª3 System
SCIENCE CONNECTION The atomicweights of three compounds are shown.Use a linear system and Cramer’s rule tofind the atomic weights of carbon (C),hydrogen (H), and oxygen (O).
SOLUTION
Write a linear system using the formula for each compound. Let C, H, and Orepresent the atomic weights of carbon, hydrogen, and oxygen.
C + 4H = 163C + 8H + 3O = 92
2H + O = 18
Evaluate the determinant of the coefficient matrix.
* * = (8 + 0 + 0) º (0 + 6 + 12) = º10
Apply Cramer’s rule since the determinant is not 0.
C = }º10} = }ºº11200
} = 12 Atomic weight of carbon
H = }º10} = }ºº1100} = 1 Atomic weight of hydrogen
O = }º10} = }ºº11600
} = 16 Atomic weight of oxygen
c The weights of carbon, hydrogen, and oxygen are 12, 1, and 16, respectively.
031
482
130
E X A M P L E 5
Let A be the coefficient matrix of this linear system:
ax + by + cz = j
dx + ey + fz = k
gx + hy + iz = l
If det A ≠ 0, then the system has exactly one solution. The solution is:
x = }det A
}, y = }det A
}, and z = }det A
}
CRAMER’S RULE FOR A 3 X 3 SYSTEM
Compound Formula Atomic weight
Methane CH4 16
Glycerol C3H8O3 92
Water H2O 18
* *c
f
i
b
e
h
j
k
l * *c
f
i
j
k
l
a
d
g * *j
k
l
b
e
h
a
d
g
* *031
169218
130
* *169218
482
130
* *031
482
169218
e f g h i j k l1. Explain Cramer’s rule and how it is used.
2. Can two different matrices have the same determinant? If so, give an example.
3. ERROR ANALYSIS Find the error in each calculation.a. b.
4. To use Cramer’s rule to solve a linear system, what must be true of thedeterminant of the coefficient matrix?
Evaluate the determinant of the matrix.
5. F G 6. F G 7. F GUse Cramer’s rule to solve the linear system.
8. 6x º 8y = 4 9. 2x + 7y = º3 10. 12x º 2y = 24x º 5y = º4 3x º 8y = º23 º14x + 11y = 51
11. SCHOOL SPIRIT You are making a large pennant for your school football team. A diagram of the pennant is shown at the right. The coordinates given are measured in inches. How many square inches of material will you need to make the pennant?
2 ª 2 DETERMINANTS Evaluate the determinant of the matrix.
12. F G 13. F G 14. F G15. F G 16. F G 17. F G18. F G 19. F G 20. F G3 ª 3 DETERMINANTS Evaluate the determinant of the matrix.
21. F G 22. F G 23. F G24. F G 25. F G 26. F G27. F G 28. F G 29. F Gº10
6º14
402
15º10º8
201812
29
15
º3º10
11
º10
º2
1296
310
º5
994
23
13
8123
º192
085
º400
º22
º4
1641
1207
2º4º1
5134
010
º5
411
º921
540
º121
438
12º2
5
28
12º5
º310
08
59
º6º3
89
15
04
4º3
112
º7º7
31
9º2
03
8º1
2º2
º45
PRACTICE AND APPLICATIONS
º24
8º2
4º1
º15
12
06
GUIDED PRACTICE
218 Chapter 4 Matrices and Determinants
Vocabulary Check ✓Concept Check ✓
Skill Check ✓
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 12–29Example 2: Exs. 30–35Example 3: Exs. 54–58Example 4: Exs. 36–44,
59Example 5: Exs. 45–53,
60
Extra Practiceto help you masterskills is on p. 945.
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* * = 6 º 5 = 1 * * = º2 º 12 = º1423
4º1
º12
35
m no n GO TEAM
pq(0, 50)
(70, 20)
(0, 0)
e f g h i j k l
SAILING Theedges of a sail are
called the luff, leech, andfoot. The luff length of the jibis usually 80% to 90% of thedistance from the deck tothe head of the jib.
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APPLICATIONS
leech luff
foot mainsail jib
head
4.3 Determinants and Cramer's Rule 219
AREA OF A TRIANGLE Find the area of the triangle with the given vertices.
30. A(0, 1), B(2, 7), C(5, 5) 31. A(3, 6), B(3, 0), C(1, 3)
32. A(6, º1), B(2, 2), C(4, 8) 33. A(º4, 2), B(3, º1), C(º2, º2)
34. A(2, º6), B(º1, º4), C(0, 2) 35. A(1, 3), B(º2, 6), C(º1, 1)
USING CRAMER’S RULE Use Cramer’s rule to solve the linear system.
36. 2x + y = 3 37. 7x º 5y = 11 38. 9x + 2y = 75x + 6y = 4 3x + 10y = º56 4x º 3y = 42
39. x + 7y = º3 40. ºx º 12y = 44 41. 4x º 3y = 183x º 5y = 17 12x º 15y = º51 8x º 7y = 34
42. 4x º 5y = 13 43. 8x º 9y = 32 44. 3x + 10y = 502x º 7y = 24 º5x + 7y = 40 12x + 15y = 64
SOLVING SYSTEMS Use Cramer’s rule to solve the linear system.
45. x + 2y º 3z = º2 46. x + 3y º z = 1 47. 3x + 2y º 5z = º10x º y + z = º1 º2x º 6y + z = º3 6x º z = 83x + 4y º 4z = 4 3x + 5y º 2z = 4 ºy + 3z = º2
48. x + 2y + z = 9 49. 4x + y + 6z = 7 50. x + 4y º z = º7x + y + z = 3 3x + 3y + 2z = 17 2x º y + 2z = 155x º 2z = º1 ºx º y + z = º9 º3x + y º 3z = º22
51. 2x + y + z = 5 52. ºx + 2y + 7z = 13 53. º3x + y + 2z = º14x + 4y º 2z = 9 2x º y º 2z = º2 9x º y + 2z = º86x + 5y = 16 3x + 5y + 2z = º14 8x + 5y º 4z = 6
54. BIRDS Black-necked stilts are birds that live throughoutFlorida and surrounding areas but breed mostly in the triangularregion shown on the map. Estimate the area of this region.The coordinates given aremeasured in miles.
SAILING In Exercises 55–57,
use the following information.
On a Marconi-rigged sloop, there are two triangular sails, a mainsail and a jib. These sails are shown in acoordinate plane at the right. The coordinates in the plane are measured in feet.
55. Find the area of the mainsail shown.
56. Find the area of the jib shown.
57. Suppose you are making a scale modelof the sailboat with the sails shownusing a scale of l in. = 6 ft. What is thearea of the model’s mainsail?
r rs ts rs u
rrr v w
xmainsail jib
(35, 220)
(0, 0)
(112, 56)
yz {|Each tick mark
represents 50 miles.
} ~ � � � � � �58. The Golden
Triangle refers to a large triangular regionin India. The Taj Mahal is one of the many wonders that lie within the boundaries of this triangle. The triangle is formed by imaginary lines that connect the cities of New Delhi, Jaipur, and Agra. Use the coordinates on the map and a determinant to estimate the area of the Golden Triangle. The coordinates given are measured in miles.
59. BUYING GASOLINE You fill up your car with 10 gallons of premiumgasoline and fill a small gas can with 2 gallons of regular gasoline for your lawnmower. You pay the cashier $13.56. The price of premium gasoline is 12 centsmore per gallon than the price of regular gasoline. Use a linear system andCramer’s rule to find the price per gallon for regular and premium gasoline.
60. The atomic weights of three compounds are shown.
Use a linear system and Cramer’s rule to find the atomic weights of sulphur (S),nitrogen (N), and flourine (F).
61. LOGICAL REASONING Explain what happens to the determinant of a matrixwhen you switch two rows or two columns.
QUANTITATIVE COMPARISON In Exercises 62 and 63, choose the statement
that is true about the given quantities.
¡A The quantity in column A is greater.
¡B The quantity in column B is greater.
¡C The two quantities are equal.
¡D The relationship cannot be determined from the given information.
62.
63.
64. DETERMINANT RELATIONSHIPS Let A = F G and B = F G.
a. How is det AB related to det A and det B?
b. How is det kA related to det A if k is a constant? Check your answer usingmatrix B and several other 2 ª 2 matrices.
5º4
3º2
º12
23
SCIENCE CONNECTION
SOCIAL STUDIES CONNECTION
220 Chapter 4 Matrices and Determinants
TestPreparation
★★Challenge
Column A Column B
The area of a triangle with vertices The area of a triangle with vertices (º3, 4), (4, 2), and (1, º2) (4, 2), (1, º2), and (3, º4)
det F G det F G15
º73
610
º5º2
Compound Formula Atomic weight
Tetrasulphur tetranitride S4N4 184
Sulphur hexaflouride SF6 146
Dinitrogen tetraflouride N2F4 104
EXTRA CHALLENGE
www.mcdougallittell.com
New Delhi(100, 120)
Jaipur(0, 0)
Agra(140, 20)
Each tick markrepresents 20 miles.
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4.3 Determinants and Cramer's Rule 221
EVALUATING FUNCTIONS Find the indicated value of ƒ(x). (Review 2.1)
65. ƒ(x) = x º 10, f (7) 66. ƒ(x) = 3x + 7, ƒ(º2)
67. ƒ(x) = ºx2 + 5, ƒ(º1) 68. ƒ(x) = x2 º 2x º 4, ƒ(7)
69. ƒ(x) = x2 + 4x º 1, ƒS}
12}D 70. ƒ(x) = x5 º 2x º 10, ƒ(3)
GRAPHING SYSTEMS Graph the system of linear inequalities. (Review 3.3)
71. x < 3 72. y ≥ 2x º 3 73. y > ºx º 5x > º2 y > º5x º 8 y > 3x + 1
74. x + y > 3 75. 2x º y ≥ 2 76. 4x º 3y > 14x + y < 4 5x º y ≥ 2 ºx + y ≥ 4
MULTIPLYING MATRICES Find the product. (Review 4.2 for 4.4)
77. F G F G 78. F G F G 79. F G F G80. F G F G 81. F G F G 82. F G F G
Perform the indicated operation(s). (Lessons 4.1, 4.2)
1. F G + F G 2. F G º F G3. º2F G + 2F G 4. F G º 4F G5. F G F G 6. F G F G
Evaluate the determinant of the matrix. (Lesson 4.3)
7. F G 8. F G 9. F G 10. F GUse Cramer’s rule to solve the linear system. (Lesson 4.3)
11. º8x + y = º6 12. 3x º 2y = 10 13. 5x + 4y = 12º5x + 4y = 3 º6x + y = º7 3x º 6y = 3
14. 4x + y + 6z = 2 15. x + y + 4z = 7 16. 3x + 3y º 2z = º182x + 2y + 4z = 1 2x º 3y º z = º24 º5x º 2y º 3z = º1ºx º y + z = º5 º4x + 2y + 2z = 8 7x + y + 6z = 14
17. GARDENING You are planning to turn a triangular region of your yard into a garden. The vertices of the triangle are (0, 0), (5, 2), and (3, 6) where thecoordinates are measured in feet. Find the area of the triangular region. (Lesson 4.3)
º63
º3
520
1221
3º2
1
208
º156
º3º2
96
32
º4º6
0º3º6
194
30
º14
22
70
3º2
º1º2
86
º3º5
1
º160
20
º2
1005
º66
º4
439
º33
6º5
º29
74
7º1
º35
0º2
º85
5º7
º1º8
º3º2
108
5º6
º24
QUIZ 1 Self-Test for Lessons 4.1–4.3
61.9
1.6º4
1.3º3
º21.5
0.60.8
04
31
0.50.2
916
1012
º52
3º7
3º1
º88
º24
110
º38
04
º1º10
74
º1º3
63
º41
º25
MIXED REVIEW
} ~ � � � � � �Developing Concepts
ACTIVITY 4.4Group Activity for use with Lesson 4.4
GROUP ACTIVITY
Work with a partner.
MATERIALS
• pencil
• paper
Investigating Identity and Inverse Matrices
c QUESTION What are some properties of identity and inverse matrices?
c EXPLORING THE CONCEPT
Let A = F G, B = F G, and C = F G. Consider the 2 ª 2
identity matrix I = F G. Find AI, BI, and CI. What do you notice?
Find IA, IB, and IC using the matrices from Step 1. Is multiplication by theidentity matrix commutative?
Let D = F G. The inverse of D is E = F G. Find DE and ED.
What do you notice?
Use matrix multiplication to decide which of the following is the inverse of the
matrix A in Step 1: F G, F G, or F G.
c DRAWING CONCLUSIONS
1. For any 2 ª 2 matrix A, what is true of the products AI and IA where I is the2 ª 2 identity matrix? Justify your answer mathematically.
(Hint: Let A = F G, and compute AI and IA.)
2. How is the relationship between I = F G and other 2 ª 2 matrices similar
to the relationship between 1 and other real numbers?
3. What do you think is the identity matrix for the set of 3 ª 3 matrices? Checkyour answer by multiplying your proposed identity matrix by several 3 ª 3matrices.
4. What is the relationship between a matrix, its inverse, and the identity matrix?How is this relationship like the one that exists between a nonzero real number,its reciprocal, and 1?
5. Does every nonzero matrix have an inverse? Explain. (Hint: Consider a 2 ª 2matrix whose first row contains all nonzero entries and whose second rowcontains all zero entries.)
6. Find the inverse of F = F G by finding values of a, b, c, and d such that
F G F G = F G.01
10
b
d
a
c
74
21
74
21
01
10
b
d
a
c
2º5
º13
3º1
º52
º31
5º2
4
º57
3º4
53
74
3
2
01
10
0.80.3
0.10.6
06
º4º7
35
12
1
222 Chapter 4 Matrices and Determinants
} ~ � � � � � �
4.4 Identity and Inverse Matrices 223
Identity and Inverse Matrices
USING INVERSE MATRICES
The number 1 is the multiplicative identity for real numbers because 1 • a = a anda • 1 = a. For matrices, the n ª n is the matrix that has 1’s on themain diagonal and 0’s elsewhere.
2 ª 2 IDENTITY MATRIX 3 ª 3 IDENTITY MATRIX
I = F G I = 3 4If A is any n ª n matrix and I is the n ª n identity matrix, then IA = A and AI = A.
Two n ª n matrices are of each other if their product (in both orders) is then ª n identity matrix. For example, matrices A and B below are inverses of each other.
AB = F G F G = F G = I BA = F G F G = F G = I
The symbol used for the inverse of A is Aº1.
Finding the Inverse of a 2 32 Matrix
Find the inverse of A = F G.
SOLUTION
Aº1 = }6 º
14
} F G = }
12
}F G = 3 4✓CHECK You can check the inverse by showing that AAº1 = I = Aº1A.
F G 3 4 = F G and 3 4 F G = F G01
10
12
34
º}
21
}
}
32
}
1
º2
01
10
º}
21
}
}
32
}
1
º2
12
34
º}
21
}
}
32
}
1
º2
º13
2º4
º13
2º4
12
34
E X A M P L E 1
01
10
º12
3º5
13
25
01
10
13
25
º12
3º5
inverses
001
010
100
01
10
identity matrix
GOAL 1
Find and use
inverse matrices.
Use inverse
matrices in real-life
situations, such as encoding
a message in Example 5.
. To solve real-life
problems, such as decoding
names of landmarks in
Exs. 44–48.
Why you should learn it
GOAL 2
GOAL 1
What you should learn
4.4R
EAL LIFE
REA
L LIFEThe inverse of the matrix A = 3 4 is
Aº1 = }
|A
1|
} 3 4 = }ad º
1cb
} 3 4 provided ad º cb ≠ 0.ºb
a
d
ºc
ºb
a
d
ºc
b
d
a
c
THE INVERSE OF A 2 X 2 MATRIX
Look Back For help with multiplicativeinverses of real numbers,see p. 5.
STUDENT HELP
The artist Jim Sanborn uses
cryptograms in his work,
such as Kryptos above.
� � � � � � � �
224 Chapter 4 Matrices and Determinants
Solving a Matrix Equation
Solve the matrix equation AX = B for the 2 ª 2 matrix X.
A B
F GX = F GSOLUTION
Begin by finding the inverse of A.
Aº1 = }4 º
13
} F G = F GTo solve the equation for X, multiply both sides of the equation by Aº1 on the left.
F G F GX = F G F G Aº1AX = Aº1B
F GX = F G IX = Aº1B
X = F G X = Aº1B
✓CHECK You can check the solution by multiplying A and X to see if you get B.
. . . . . . . . . .
Some matrices do not have an inverse. You can tell whether a matrix has an inverseby evaluating its determinant. If det A = 0, then A does not have an inverse. Ifdet A ≠ 0, then A has an inverse.
The inverse of a 3 ª 3 matrix is difficult to compute by hand. A calculator that willcompute inverse matrices is useful in this case.
Finding the Inverse of a 3 33 Matrix
Use a graphing calculator to find the inverse of A. Then use the calculator to
verify your result.
A = F GSOLUTION
Enter the matrix A into the graphing calculator and calculate Aº1. Then computeAAº1 and Aº1A to verify that you obtain the 3 ª 3 identity matrix.
0º1º3
º10
º2
116
E X A M P L E 3
º2º3
20
º2º3
20
01
10
º53
8º6
14
13
º11
4º3
14
13
14
13
14
13
º53
8º6
º11
4º3
E X A M P L E 2
[A]-1
[[-2 -3 1]
[-3 -3 1]
[-2 -4 1]]
[A][A]-1
[[1 0 0]
[0 1 0]
[0 0 1]]
[A]-1[A]
[[1 0 0]
[0 1 0]
[0 0 1]]
HOMEWORK HELP
Visit our Web sitewww.mcdougallittell.comfor extra examples.
INT
ERNET
STUDENT HELP
KEYSTROKE HELP
Visit our Web sitewww.mcdougallittell.comto see keystrokes forseveral models ofcalculators.
INT
ERNET
STUDENT HELP
� � � � � � � �
4.4 Identity and Inverse Matrices 225
USING INVERSE MATRICES IN REAL LIFE
A cryptogram is a message written according to a secret code. (The Greek wordkruptos means hidden and the Greek word gramma means letter.) The followingtechnique uses matrices to encode and decode messages.
First assign a number to each letter in the alphabet with 0 assigned to a blank space.
__ = 0 E = 5 J = 10 O = 15 T = 20 Y = 25
A = 1 F = 6 K = 11 P = 16 U = 21 Z = 26
B = 2 G = 7 L = 12 Q = 17 V = 22
C = 3 H = 8 M = 13 R = 18 W = 23
D = 4 I = 9 N = 14 S = 19 X = 24
Then convert the message to numbers partitioned into 1 ª 2 uncoded row matrices.
To encode a message, choose a 2 ª 2 matrix A that has an inverse and multiply theuncoded row matrices by A on the right to obtain coded row matrices.
Converting a Message
Use the list above to convert the message GET HELP to row matrices.
SOLUTION
G E T __ H E L P
f gf gf gf g
Encoding a Message
CRYPTOGRAPHY Use A = F G to encode the message GET HELP.
SOLUTION
The coded row matrices are obtained by multiplying each of the uncoded rowmatrices from Example 4 by the matrix A on the right.
c The coded message is 9, 11, 40, 60, 11, 14, 8, 4.
3º2
2º1
E X A M P L E 5
16125802057
E X A M P L E 4
GOAL 2
NAVAJO CODE
During World War II,a Marine Corps code basedon the complex Navajolanguage was used to sendmessages.
APPLICATION LINK
www.mcdougallittell.com
INT
ERNET
RE
AL LIFE
RE
AL LIFE
FOCUS ON
APPLICATIONS
RE
AL LIFE
RE
AL LIFE
Cryptography
UNCODED ENCODING CODED ROW
ROW MATRIX MATRIX A MATRIX
f7 5g F G = [9 11]
f20 0g F G = [40 60]
f8 5g F G = [11 14]
f12 16g F G = [8 4]3
º22
º1
3º2
2º1
3º2
2º1
3º2
2º1
� � � � � � � �
226 Chapter 4 Matrices and Determinants
DECODING USING MATRICES Decoding the cryptogram created in Example 5would be difficult for people who do not know the matrix A. When larger codingmatrices are used, decoding is even more difficult. But for an authorized receiver whoknows the matrix A, decoding is simple. The receiver only needs to multiply thecoded row matrices by Aº1 on the right to retrieve the uncoded row matrices.
Decoding a Message
CRYPTOGRAPHY Use the inverse of A = F G to decode this message:
º4, 3, º23, 12, º26, 13, 15, º5, 31, º5, º38, 19, º21, 12, 20, 0, 75, º25
SOLUTION
First find Aº1: Aº1 = }3 º
12
} F G = F GTo decode the message, partition it into groups of two numbers to form coded rowmatrices. Then multiply each coded row matrix by Aº1 on the right to obtain theuncoded row matrices.
From the uncoded row matrices you can read the message as follows.
f gf gf gf gf gf gf gf gf gB E A M __ M E __ U P __ S C O T T Y __
025202015319016210513013152
13
12
13
12
º11
3º2
E X A M P L E 6
ALAN TURING,
an English mathema-tician, helped break codesused by the German militaryduring World War II.
RE
AL LIFE
RE
AL LIFE
FOCUS ON
PEOPLE
CODED ROW DECODING UNCODED
MATRIX MATRIX Aº1 ROW MATRIX
f g F G = f g
f g F G = f g
f g F G = f g
f g F G = f g
f g F G = f g
f g F G = f g
f g F G = f g
f g F G = f g
f g F G = f g02513
12
º2575
202013
12
020
15313
12
12º21
19013
12
19º38
162113
12
º531
0513
12
º515
13013
12
13º26
13113
12
12º23
5213
12
3º4
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4.4 Identity and Inverse Matrices 227
1. What is the identity matrix for 2 ª 2 matrices? for 3 ª 3 matrices?
2. For two 2 ª 2 matrices A and B to be inverses of each other, what must be trueof AB and BA?
3. Explain how to find the inverse of a 2 ª 2 matrix.
4. How do you know that the matrix X in Example 2 must be 2 ª 2?
5. If B = F G, does B have an inverse? Explain.
Find the inverse of the matrix.
6. F G 7. F G 8. F G9. F G 10. F G 11. F G
12. DECODING A MESSAGE Use the coding information on pages 225 and226 and the inverse of the matrix D to decode the following message.
D = F G º71, 39, º35, 20, º118, 69, º84, 49, º95, 57
FINDING INVERSES Find the inverse of the matrix.
13. F G 14. F G 15. F G16. F G 17. F G 18. F G19. F G 20. F G 21. F G22. F G 23. 3 4 24. 3 4SOLVING EQUATIONS Solve the matrix equation.
25. F GX = F G 26. F GX = F G27. F GX = F G 28. F GX = F G29. F GX + F G = F G 30. F GX + F G = F G31. F GX º F G = F G 32. F GX º F G = F G6
248
17
º15
º3º2
46
º2º1
31
10
23
26
º1º4
9º6
16
4º3
34
º95
º74
º3º9
7º2
515
81
74
31
18º13
º5º3
º124
º31
º54
65
0º1
43
41
20
2020
1726
º12
58
10
3º4
º135
º50
}
43
}
}
52
}
}
54
}
º1
2.510
2.28
}
21
}
1
}
23
}
º2
º33
11º9
º44
5º4
º72
º62
º21
º7º4
21
73
17º3
º61
87
11
23
68
º54
4º3
PRACTICE AND APPLICATIONS
31
º5º2
20.2
1.63.2
34
0.52.5
4
}
14
}
}
21
}
º2
04
º16
2º1
º30
32
º4º3
º41
8º2
GUIDED PRACTICE
Vocabulary Check ✓Concept Check ✓
Skill Check ✓
Extra Practiceto help you masterskills is on p. 945.
STUDENT HELP
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 13–24, 33Example 2: Exs. 25–32Example 3: Exs. 34–39Examples 4, 5: Exs. 40–43Example 6: Exs. 44–48
� � � � � � � �
228 Chapter 4 Matrices and Determinants
IDENTIFYING INVERSES Tell whether the matrices are inverses of each other.
33. F G and F G 34. F G and F G35. F G and F G 36. F G and F G
FINDING INVERSES Use a graphing calculator to find the inverse of the
matrix A. Check the result by showing that AAº1 = I and Aº1
A = I.
37. A = F G 38. A = F G 39. A = F GENCODING Use the code on page 225 and the matrix to encode the message.
40. JOB WELL DONE 41. STAY THERE
A = F G A = F G42. COME TO DINNER 43. HAPPY BIRTHDAY
A = F G A = F GTRAVEL In Exercises 44–48, use the following
information.
Your friend is traveling abroad and is sending youpostcards with encoded messages. You must decipherwhat landmarks your friend has visited. Use the inverseof matrix D to decode each message. Each messagerepresents a landmark in the country where your friendis traveling. Use the coding information on pages 225and 226 to help you.
D = F G44. º1, 4, 30, º41, 39, º58, 22, º33, 31, º46, 23, º34, 1, 1
45. 21, º31, 22, º26, º9, 19, º20, 40, º3, 11, 20, º24, 10, º15
46. 39, º58, º2, 12, 0, 9, º19, 38, 13, º9, º16, 33, 10, º15
47. 32, º44, 10, º15, º4, 15, 9, º13, 40, º60, 22, º25, 7, º6, 4, 6
48. Using the decoded messages, tell what country your friend is visiting.
49. Use the matrices shown. The columns of matrix T givethe coordinates of the vertices of a triangle. Matrix A is a transformation matrix.
A = F G T = F Ga. Find AT and AAT. Then draw the original triangle and the two transformed
triangles. What transformation does A represent?
b. Suppose you start with the triangle determined by AAT and want to reverse thetransformation process to produce the triangle determined by AT and then thetriangle determined by T. Describe how you can do this.
32
24
11
º10
01
GEOMETRY CONNECTION
º32
2º1
º22
5º4
º11
4º3
23
11
º25
1º2
º208
133
254
º63
º9
01
º3
º7º411
502
452
º315
502
452
3º2
1
º25º10
23
21
º2
104
º9
213
º42
º5
304
º8º3
6
21
º1
114
º8
8º5
º10
º107
12
º2111
º134
223
057
3º10
13
º3º1
103
Skills Review For help withtransformations, see p. 921.
STUDENT HELP
� � � � � � � �
4.4 Identity and Inverse Matrices 229
50. Writing Describe the process used to solve a matrix equation.
51. MULTIPLE CHOICE What is the inverse of F G?
¡A 3 4 ¡B F G ¡C 3 4 ¡D F G52. MULTIPLE CHOICE What is the solution of F GX = F G?
¡A F G ¡B F G ¡C F G ¡D F G ¡E F G53. CODE BREAKER You are a code breaker and intercept the encoded message
45, º35, 38, º30, 18, º18, 35, º30, 81, º60, 42, º28, 75, º55, 2, º2, 22,º21, 15, º10 that you know is being sent to someone named John. You canconclude that f gAº1 = f g and f gAº1 = f gwhere Aº1 is the inverse of the encoding matrix A, 10 represents J, 15 representsO, 8 represents H, and 14 represents N.
Let Aº1 = F G.
a. Write and solve two systems of equations to find w, x, y, and z.
b. Find Aº1, and decode the rest of the message.
SOLVING SYSTEMS Solve the system of linear equations using any algebraic
method. (Review 3.2, 3.6 for 4.5)
54. 3x + 5y = 12 55. 4x º 12y = 2 56. º5x + 7y = 33x + 4y = 11 º2x + 6y = º1 4x º 9y = º40
57. 7x + y + 3z = 22 58. x + 3z = 6 59. 2x + y º 4z = 42x º 2y + 9z = º10 º2x + 3y + z = º11 4x º 3y + 8z = º8º3x º 5y º 10z = 8 3x º y + 2z = 13 º2x + 7y º 12z = 24
MATRIX OPERATIONS Perform the indicated operation, if possible. If not
possible, state the reason. (Review 4.1)
60. F G + F G 61. F G º F G62. º8F G 63. F G º F G64. F G + F G 65. }
12
}F G66. CATERING You are in charge of catering for a school function. To limit the cost,
you will serve only two entrees. One is a vegetarian dish that costs $6 and the other isa chicken dish that costs $8. If there will be 150 people at the function and yourbudget for the food is $1000, how many of each type of entree will be served?(Review 3.1, 3.2)
216
108
46
º1º9
48
3º2
º14
º25
6º3
9º15
º2º12
º10º9
816
º513
7º9
40
38
º1º6
º53
º3º1
º22
º65
8º2
º30
4º2
2º1
º4º5
MIXED REVIEW
x
z
w
y
148º30381510º3545
72
04
47
02
24
07
74
0º2
74
02
4325
42
21
53
1
º1
3
º}
72
}
º}
113}
}
113}
}
133}
º}
276}
1
1
º3
º}
72
}
º}
113}
º}
113}
}
133}
}
276}
º26
º27
TestPreparation
★★Challenge
EXTRA CHALLENGE
www.mcdougallittell.com
� � � � � � � �
230 Chapter 4 Matrices and Determinants
Solving Systems Using Inverse Matrices
SOLVING SYSTEMS USING MATRICES
In Lesson 4.3 you learned how to solve a system of linear equations using Cramer’srule. Here you will learn to solve a system using inverse matrices.
In the activity you learned that a linear system can be written as a matrix equation AX = B. The matrix A is the coefficient matrix of the system, X is the and B is the
Writing a Matrix Equation
Write the system of linear equations as a matrix equation.
º3x + 4y = 5 Equation 1
2x º y = º10 Equation 2
SOLUTION
A X B
F G F G = F G. . . . . . . . . .
Once you have written a linear system as AX = B, you can solve for X by multiplyingeach side of the matrix by Aº1 on the left.
AX = B Write original matrix equation.
Aº1AX = Aº1B Multiply each side by Aº1.
IX = Aº1B Aº1A = I
X = Aº1B IX = X
5º10
xy
4º1
º32
E X A M P L E 1
matrix of constants.matrix of variables,
GOAL 1
Solve systems of
linear equations using
inverse matrices.
Use systems of
linear equations to solve
real-life problems, such as
determining how much
money to invest in Example 4.
. To solve real-lifeproblems, such as planning
a stained glass project in
Ex. 42.
Why you should learn it
GOAL 2
GOAL 1
What you should learn
4.5R
EAL LIFE
REA
L LIFE
Investigating Matrix Equations
Write the left side of the matrix equation as a single matrix. Then F GF G = F G Matrix equation
equate corresponding entries of the matrices. What do you obtain?
Use what you learned in Step 1 to 2x º y = º4 Equation 1
write the following linear system º4x + 9y = 1 Equation 2
as a matrix equation.
2
86
xy
º42
51
1
DevelopingConcepts
ACTIVITY
� � � � � � �
4.5 Solving Systems Using Inverse Matrices 231
SOLUTION OF A LINEAR SYSTEM Let AX = B represent a system of linearequations. If the determinant of A is nonzero, then the linear system has exactly one solution, which is X = Aº1B.
Solving a Linear System
Use matrices to solve the linear system in Example 1.
º3x + 4y = 5 Equation 1
2x º y = º10 Equation 2
SOLUTION
Begin by writing the linear system in matrix form, as in Example 1. Then find theinverse of matrix A.
Aº1 = }3 º1
8} F G = 3 4Finally, multiply the matrix of constants by Aº1.
X = Aº1B = 3 4F G = F G = F Gc The solution of the system is (º7, º4). Check this solution in the original
equations.
Using a Graphing Calculator
Use a matrix equation and a graphing calculator to solve the linear system.
2x + 3y + z = º1 Equation 1
3x + 3y + z = 1 Equation 2
2x + 4y + z = º2 Equation 3
SOLUTION
The matrix equation that represents the system is F G F G = F G.
Using a graphing calculator, you can solve the system as shown.
Enter matrix A. Enter matrix B. Multiply B by Aº1.
c The solution is (2, º1, º2). Check this solution in the original equations.
[A]-1[B]
[[2 ]
[-1]
[-2]]
MATRIX [B] 3X1
[-1 ]
[1 ]
[-2 ]
MATRIX [A] 3X3
[2 3 1 ]
[3 3 1 ]
[2 4 1 ]
º11
º2
xyz
111
334
232
E X A M P L E 3
xy
º7º4
5º10
}54
}
}
35}
}51
}
}
25}
}54
}
}
35}
}51
}
}
25}
º4º3
º1º2
E X A M P L E 2
STUDENT HELP
Study TipRemember that you canuse the method shown inExamples 2 and 3 providedA has an inverse. If Adoes not have an inverse,then the system has eitherno solution or infinitelymany solutions, and youshould use a differenttechnique.
Look Back For help with systems of linear equations, see p. 150.
STUDENT HELP
� � � � � � � USING LINEAR SYSTEMS IN REAL LIFE
Writing and Using a Linear System
INVESTING You have $10,000 to invest. You want to invest the money in a stockmutual fund, a bond mutual fund, and a money market fund. The expected
annual returns for these funds are given in the table.
You want your investment to obtain an overall annual return of 8%. A financial planner recommends that you invest the same amount in stocks as in bonds and the money market combined. How much should you invest in each fund?
SOLUTION
+ + =
0.10 + 0.07 + 0.05 = 0.08
= +
Stock amount = Money market amount =
Bond amount = Total invested = 10,000
+ + = 10,000 Equation 1
0.10 + 0.07 + 0.05 = 0.08 (10,000) Equation 2
= + Equation 3
First rewrite the equations above in standard form and then in matrix form.
s + b + m = 10,0000.10s + 0.07b + 0.05m = 800 F G F G = F Gs º b º m = 0
Enter the coefficient matrix A and the matrix of constants B into a graphingcalculator. Then find the solution X = Aº1B.
c You should invest $5000 in the stock mutual fund, $2500 in the bond mutual fund,and $2500 in the money market fund.
[A]-1[B]
[[5000]
[2500]
[2500]]
MATRIX [B] 3X1
[10000 ]
[800 ]
[0 ]
MATRIX [A] 3X3
[1 1 1 ]
[.1 .07 .05 ]
[1 -1 -1 ]
10,000800
0
sbm
10.05º1
10.07º1
10.1
1
mbs
mbs
mbs
b
ms
MMamount
Bondamount
Stockamount
Totalinvested
MMamount
Bondamount
Stockamount
Totalinvested
MMamount
Bondamount
Stockamount
E X A M P L E 4
GOAL 2
232 Chapter 4 Matrices and Determinants
INVESTING
Each year studentsacross the country in grades4 through 12 invest ahypothetical $100,000 instocks to compete in theStock Market Game. Studentscan enter their transactionsusing the internet.
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Investment Expected return
Stock mutual fund 10%
Bond mutual fund 7%
Money market (MM) fund 5%
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4.5 Solving Systems Using Inverse Matrices 233
1. What are a matrix of variables and a matrix of constants, and how are they usedto solve a system of linear equations?
2. If |A|≠ 0, what is the solution of AX = B in terms of A and B?
3. Explain why the solution of AX = B is not X = BAº1.
Write the linear system as a matrix equation.
4. x + y = 8 5. x + 3y = 9 6. x + y + z = 102x º y = 6 4x º 2y = 7 5x º y = 1
3x + 4y + z = 8
Use an inverse matrix to solve the linear system.
7. x + y = 2 8. ºx º 2y = 3 9. 4x + 3y = 67x + 8y = 21 2x + 8y = 1 6x º 2y = 10
10. INVESTING Look back at Example 4 on page 232. Suppose you have $60,000 to invest and you want an overall annual return of 9%. Use the expected annual returns shown to determine how much you should invest in each fund. Assume you are investing as much in stocks as in bonds and the money market combined.
WRITING MATRIX EQUATIONS Write the linear system as a matrix equation.
11. x + y = 5 12. x + 2y = 6 13. 5x º 3y = 93x º 4y = 8 4x º y = 5 º4x + 2y = 10
14. 2x º 5y = º11 15. x + 8y = 4 16. 2x º 5y = 4º3x + 7y = 15 4x º 5y = º11 x º 3y = 1
17. x º 4y + 5z = º4 18. 3x º y + 4z = 16 19. 0.5x + 3.1y º 0.2z = 5.92x + y º 7z = º23 2x + 4y º z = 10 1.2x º 2.5y + 0.7z = 2.2º4x + 5y + 2z = 38 x º y + 3z = 31 0.3x + 4.8y º 4.3z = 4.8
20. x + z = 9 21. 8y º 10z = º23 22. x + y º z = 0ºx º y + 2z = 6 6y º 12z = 14 2x º z = 12x + 7y º z = º4 º9x + 5z = 0 y + z = 2
SOLVING SYSTEMS Use an inverse matrix to solve the linear system.
23. 3x + y = 8 24. x + y = º1 25. 2x + 7y = º535x + 2y = 11 11x + 12y = 8 x + 3y = º22
26. 7x + 5y = 8 27. 5x º 7y = 54 28. º5x º 7y = º94x + 3y = 4 2x º 4y = 30 2x + 3y = 3
29. x + 2y = º9 30. 2x + 4y = º26 31. 9x º 5y = 43º2x º 3y = 14 2x + 5y = º31 º2x + 2y = º22
PRACTICE AND APPLICATIONS
GUIDED PRACTICE
Vocabulary Check ✓
Concept Check ✓
Skill Check ✓
Investment Expected return
Stock mutual fund 12%
Bond mutual fund 8%
Money market fund 5%
Extra Practiceto help you masterskills is on p. 945.
STUDENT HELP
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 11–22Example 2: Exs. 23–31Example 3: Exs. 32–39Example 4: Exs. 40–44
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234 Chapter 4 Matrices and Determinants
SOLVING SYSTEMS Use the given inverse of the coefficient matrix to solve
the linear system.
32. 2y º z = º2 33. x º y º 3z = 95x + 2y + 3z = 4 5x + 2y + z = º307x + 3y + 4z = º5 º3x º y = 4
Aº1 = F G Aº1 = F GSOLVING SYSTEMS Use an inverse matrix and a graphing calculator to
solve the linear system.
34. 3x + 2y = 13 35. ºx + y º 3z = º4 36. 3x + 5y º 5z = 213x + 2y + z = 13 3x º 2y + 8z = 14 º4x + 8y º 5z = 12x + y + 3z = 9 2x º 2y + 5z = 7 2x º 5y + 6z = º16
37. 2x + z = 2 38. 4x + 3y + z = 14 39. x + y º 3z = º175x º y + z = 5 6x + y = 9 2x + z = 12ºx + 2y + 2z = 0 3x + 5y + 3z = 21 º7x º 2y + z = º11
40. SKATING PARTY You are planning a birthday party for your youngerbrother at a skating rink. The cost of admission is $3.50 per adult and $2.25 perchild, and there is a limit of 20 people. You have $50 to spend. Use an inversematrix to determine how many adults and how many children you can invite.
41. DENTAL FILLINGS Dentists use various amalgams for silver fillings. Thematrix shows the percents (expressed as decimals) of powdered alloys used inpreparing three different amalgams. Suppose a dentist has 5483 grams of silver,2009 grams of tin, and 129 grams of copper. How much of each amalgam can be made?
PERCENT ALLOY BY WEIGHT
Amalgam
A B C
F G42. STAINED GLASS You are making mosaic tiles from three types of stained
glass. You need 6 square feet of glass for the project and you want there to be asmuch iridescent glass as red and blue glass combined. The cost of a sheet ofglass having an area of 0.75 square foot is $6.50 for iridescent, $4.50 for red, and$5.50 for blue. How many sheets of each type should you purchase if you plan tospend $45 on the project?
43. WALKWAY LIGHTING A walkway lighting package includes a transformer,a certain length of wire, and a certain number of lights on the wire. The price ofeach lighting package depends on the length of wire and the number of lights onthe wire.
• 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.
Write and solve a system of equations to find the cost of a transformer, the costper foot of wire, and the cost of a light. Assume the cost of each item is the samein each lighting package.
0.730.270.00
0.720.250.03
0.700.260.04
Silver
Tin
Copper
5º16
7
3º9
4
1º3
1
8º5
º10
º117
14
º111
DENTIST
Dentists diagnose,prevent, and treat problemsof the teeth and mouth.Dental amalgams have beenused for more than 150 yearsto restore the teeth of over100 million Americans.
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AL LIFE
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Visit our Web sitewww.mcdougallittell.comfor help with Exs. 32 and 33.
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4.5 Solving Systems Using Inverse Matrices 235
44. CONSTRUCTION BUSINESS You are an accountant for a constructionbusiness and are planning next year’s budget. You have $200,000 to spend onsalaries, equipment maintenance, and other general expenses. Based on previousfinancial records of the business, you expect to spend five times as much onsalaries as on equipment maintenance, and you expect general expenses to be10% of the amount spent on the other two categories combined. Write and solvea system of equations to find the amount you should budget for each category.
45. MULTI-STEP PROBLEM A company sells different sizes of gift baskets witha varying assortment of meat and cheese. A basic basket with 2 cheeses and 3 meats costs $15, a big basket with 3 cheeses and 5 meats costs $24, and asuper basket with 7 cheeses and 10 meats costs $50.
a. Write and solve a system of equations using the information about the basicand big baskets.
b. Write and solve a system of equations using the information about the big andsuper baskets.
c. Writing Compare the results from parts (a) and (b) and make a conjectureabout why there is a discrepancy.
46. SOLVING SYSTEMS OF FOUR EQUATIONS Solve the linear system using thegiven inverse of the coefficient matrix.
w + 6x + 3y º 3z = 22w + 7x + y + 2z = 5w + 5x + 3y º 3z = 3º6x º 2y + 3z = 6
EVALUATING FUNCTIONS Evaluate ƒ(x) or g(x) for the given value of x.
(Review 2.7)
ƒ(x) = g(x) =
47. ƒ(8) 48. ƒ(11) 49. ƒ(º2) 50. ƒ(0)
51. g(3) 52. g(0) 53. g(º1) 54. g(º3)
GRAPHING FUNCTIONS Graph the function and label the vertex.
(Review 2.8 for 5.1)
55. y = |x º 5| 56. y = |x| + 8 57. y = º|x º 8| º 9
58. y = |x º 5| + 4 59. y = º|x + 3| + 4 60. y = º|x + 6| º 2
FINDING INVERSES Find the inverse of the matrix. (Review 4.4)
61. F G 62. F G 63. F G64. F G 65. F G 66. F Gº2
º267
42
73
º5º1
113
17º2
8º1
21
52
º43
7º5
}
18}x + 8, if x < º1
2x º 1, if x ≥ º1
}
34}x º 8, if x ≤ 8
ºx + 6, if x > 8
MIXED REVIEW
TestPreparation
★★Challenge
Aº1 = 3 490
º8º5
º33º13320
º3032
401
º39º24
EXTRA CHALLENGE
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236 Chapter 4 Matrices and Determinants
Find the inverse of the matrix. (Lesson 4.4)
1. F G 2. F G 3. F G 4. F GUse an inverse matrix to solve the linear system. (Lesson 4.5)
5. 4x + 7y = 24 6. º9x + 13y = 3 7. 8x + 7y = 3x + 2y = 7 2x º 3y = º1 º2x º 2y = 0
8. BUYING FLATWARE The price of flatware varies depending on the numberof place settings you buy as well as other items included in the set. Suppose a setwith 4 place settings costs $142 and a set with 8 place settings and a serving setcosts $351. Find the cost of a place setting and a serving set. Assume that thecost of each item is the same for each flatware set. (Lesson 4.5)
57
68
1º3
º69
5º3
º74
12
47
QUIZ 2 Self-Test for Lessons 4.4 and 4.5
Systems of Equations
THENTHENOVER 2000 YEARS AGO, a method for solving systems of equations using rectangulararrays of counting rods was presented in Nine Chapters on the Mathematical Art, anearly Chinese mathematics text. A problem from this book is given below.
Three bundles of top-grade ears of rice, two bundles of medium-grade ears ofrice, and one bundle of low-grade ears of rice make 39 dou (of rice by volume);two bundles of top-grade ears of rice, three bundles of medium-grade ears ofrice, and one bundle of low-grade ears of rice make 34 dou; one bundle of top-grade ears of rice, two bundles of medium-grade ears of rice, and three bundlesof low-grade ears of rice make 26 dou. How many dou are there in a bundle ofeach grade of rice?
Top-grade ears of rice
Medium-grade ears of rice
Low-grade ears of rice
Total produce
1. Use matrices to organize the given information and solve the problem.
2. How is the arrangement of counting rods similar to your matrices? How is it different?
TODAY, computers use matrices representing systems with many variables to solve complicated problems like predicting weather and designing aircraft.NOWNOW
Great Wall of
China planned
Chinese method of
solving systems described
Mechanical
calculator,
France1642
c. 220 B .C .
Electronic
computers, USA
c. 1940A .D . 50
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Chapter 4 Extension 237
Solving Systems UsingAugmented Matrices
USING ELEMENTARY ROW OPERATIONS
A matrix containing the coefficient matrix and the matrix of constants for a system oflinear equations is called the of the system.
LINEAR SYSTEM AUGMENTED MATRIX
F GRecall from Chapter 3 that equations in a system can be multiplied by a constant, or amultiple of one equation can be added to another equation. Similar operations can beperformed on the rows of an augmented matrix to solve the corresponding system.
To solve a system, use elementary row operations to transform the originalaugmented matrix into a matrix having 1’s along the main diagonal and 0’s below the main diagonal. A matrix of this form is said to be in triangular form.
Using Row Operations to Solve a Two-Variable System
LINEAR SYSTEM
x º 2y = 7º3x + 5y = º4
Add 3 times the first equation to the second equation. You get this system:
x º 2y = 7ºy = 17
Multiply the second equation by º1.
x º 2y = 7y = º17
The second row of the matrix tells you that y = º17. Substitute º17 for y in theequation for the first row: x º 2(º17) = 7, or x = º27. The solution is (º27, º17).
E X A M P L E 1
7º4
::
º25
1º3
x º 2y = 7º3x + 5y = º4
augmented matrix
GOAL
Solve systems of
linear equations using
elementary row operations
on augmented matrices.
The use of augmented
matrices allows you to solve
a linear system by
suppressing the variables
and working only with the
coefficients and constants.
Why you should learn it
GOAL
What you should learn
Extension
Two augmented matrices are row-equivalent if their corresponding systems
have the same solution(s). Any of these row operations performed on an
augmented matrix will produce a matrix that is row-equivalent to the original:
• Interchange two rows.
• Multiply a row by a nonzero constant.
• Add a multiple of one row to another row.
ELEMENTARY ROW OPERATIONS
.
AUGMENTED MATRIX
F GAdd 3 times the first row, R1, to thesecond row, R2.
3R1 + R2 ˘ F GMultiply the second row by º1.
(º1)R2 ˘ F G7º17
::
º21
10
717
::
º2º1
10
7º4
::
º25
1º3
C H A P T E R 4
.
.
.
± ² ³ ´ µ ¶ · ¸
238 Chapter 4 Matrices and Determinants
Using Row Operations to Solve a Three-Variable System
LINEAR SYSTEM
2x + 4y + 5z = 5x + 3y + 3z = 2
2x + 4y + 6z = 2
Add º1 times the first equation to the third equation. You get this system:
2x + 4y + 5z = 5x + 3y + 3z = 2
z = º3
Add º0.5 times the first equation to the second equation.
2x + 4y + 5z = 5y + 0.5z = º0.5
z = º3
Multiply the first equation by 0.5.
x + 2y + 2.5z = 2.5y + 0.5z = º0.5
z = º3
The third row of the matrix tells you that z = º3. Substitute º3 for z in the equationfor the second row, y + 0.5z = º0.5, to obtain y + 0.5(º3) = º0.5, or y = 1. Thensubstitute º3 for z and 1 for y in the equation for the first row, x + 2y + 2.5z = 2.5,to obtain x + 2(1) + 2.5(º3) = 2.5, or x = 8. The solution is (8, 1, º3).
EXERCISES
SOLVING SYSTEMS Use an augmented matrix to solve the linear system.
1. 6x + 4y = 8 2. x + y = 2 3. x + 2y = º93x + 3y = 9 7x + 8y = 21 º2x º 3y = 14
4. x º 3y = 5 5. 3x + 2y = 2 6. x + y = º1º2x º 4y = 20 5x º 6y = 50 7x + 9y = º19
7. º2x º y = º5 8. 9x º 4y = 2 9. º12x + 15y = 36x + 5y = 17 º6x º 16y = º6 º7x º 20y = º4
10. 2x + 6y + 3z = 2 11. 2x + 6y + 3z = 8 12. 2x + 10y = 28x + 3y + z = 1 x + 5y + 5z = 1 x + 3y + 4z = 22x + 5y + 2z = º1 x + 3y + z = 3 x + 5y º z = 10
13. x + 4y º 2z = 3 14. x º y + 3z = 6 15. x + 2z = 4x + 3y + 7z = 1 x º 2y = 5 x + y + z = 62x + 9y + z = 8 2x º 2y + 5z = 9 3x + 3y + 4z = 28
16. CRITICAL THINKING Try using an augmented matrix to solve the given system.What happens? What can you say about the system’s solution(s)?
x º 2y + 7z = 65x º 10y + 35z = 303x º 6y + 21z = 18
E X A M P L E 2
AUGMENTED MATRIX
F GAdd º1 times the first row to the third row.
(º1)R1 + R3 ˘F G
Add º0.5 times the first row to the second row.
º0.5R1 + R2 ˘ F GMultiply the first row by 0.5.
0.5R1 ˘ F G2.5º0.5
º3
:::
2.50.5
1
210
100
5º0.5
º3
:::
50.5
1
410
200
52
º3
:::
531
430
210
522
:::
536
434
212
.
.
.
.
.
.
.
.
± ² ³ ´ µ ¶ · ¸Chapter SummaryCHAPTER
4
WHAT did you learn?
Use matrices and determinants in real-life situations. (4.1–4.5)
Perform matrix operations.• add and subtract matrices (4.1)
• multiply a matrix by a scalar (4.1)
• multiply two matrices (4.2)
Evaluate the determinant of a matrix. (4.3)
Find the inverse of a matrix. (4.4)
Solve matrix equations. (4.1, 4.4, 4.5)
Solve systems of linear equations.• using Cramer’s rule (4.3)
• using inverse matrices (4.5)
Use systems of linear equations to solve real-lifeproblems. (4.3, 4.5)
How does Chapter 4 fit into the BIGGER PICTURE of algebra?
Your study of linear algebra has continued with Chapter 4. Matrices are usedthroughout linear algebra, especially for solving linear systems. This introductionto matrices, their uses, and properties of matrix operations also connects to yourpast. For example, instead of multiplying real numbers, or variables that representreal numbers, you multiplied matrices. You also saw properties, such as thecommutative property of multiplication, that apply to real numbers but notto matrices.
Did you write out the steps?
Here is an example of severalsteps you can write whenmultiplying two matrices,following the Study Strategy
on page 198.
STUDY STRATEGY
239
Writing Out the Steps
F GF G= F G= F G= F Gº42
14
323
0 º 420 + 14
º9 + 1227 º 4
1(0) + (º6)7º3(0) + 2(7)
1(º9) + (º6)(º2)º3(º9) + 2(º2)
07
º9º2
º62
1º3
WHY did you learn it?
Organize data, such as the number and dollar value ofHispanic music products shipped. (p. 204)
Find the total cost of college tuition plus room andboard. (p. 205)
Write U.S. population data as percents of totalpopulation. (p. 205)
Calculate the cost of softball equipment. (p. 210)
Find the area of the Golden Triangle. (p. 220)
Encode or decode a cryptogram. (pp. 225, 226)
Extend the process of equation solving to equationswhose solutions are matrices. (pp. 201, 224)
Find the cost of gasoline. (p. 220)
Calculate a budget. (p. 235)
Decide how much money to invest in each of threetypes of mutual funds. (p. 232)
± ² ³ ´ µ ¶ · ¸
240 Chapter 4 Matrices and Determinants
Chapter ReviewCHAPTER
4
• matrix, p. 199
• dimensions of a matrix, p. 199
• entries of a matrix, p. 199
• row matrix, p. 199
• column matrix, p. 199
• square matrix, p. 199
• zero matrix, p. 199
• equal matrices, p. 199
• scalar, p. 200
• determinant, p. 214
• Cramer’s rule, p. 216
• coefficient matrix, p. 216
• identity matrix, p. 223
• inverse matrix, p. 223
• matrix of variables, p. 230
• matrix of constants, p. 230
VOCABULARY
4.1 MATRIX OPERATIONS
You can add or subtract matrices that have the same dimensions byadding or subtracting corresponding entries.
F G + F G = F G = F GYou cannot subtract these matrices because they have different dimensions.
F G º F GTo do scalar multiplication, multiply each entry in the matrix by the scalar.
º3F G = F G = F GTo solve this matrix equation, equate corresponding entries and solve for x and y.
F G = F G23y
º6º1
29
x + 2º1
18º3
º24
36º9º6
(º3)(º6)(º3)(1)(º3)(8)
(º3)(º12)(º3)(3)(º3)(2)
º618
º1232
º41
º57
12
1º3
º20
º110
14º4
º2 + 16 + 4
5 + 90 + (º4)
14
9º4
º26
50
Examples onpp. 199–202
Perform the indicated operation if possible. If not possible, state the reason.
1. F G º F G 2. F G º F G 3. F G + F G4. F G +F G 5. 2F G 6. }
12}F G
Solve the matrix equation for x and y.
7. F G = F G 8. F G + F G = F G9. F G + F G = F G 10. F G º F G = F Gº7
86
º25
º31x
º25
7y
º311
º323
º4º2
0x
3y
º124
º713
º3x
50
º68
4y
133
º11410
y º 95
1410
1º5x
08
º2
º24
º6
º121
6º511
4100
º21
º3
142
51
º4
132
0º2
1
177
204
106
º1
694
5º3
º21
3º4
97
02
412
153
EXAMPLES
x + 2 = º6 3y = 9x = º8 y = 3
± ² ³ ´ µ ¶ · ¸
Chapter Review 241
4.2 MULTIPLYING MATRICES
You can multiply a matrix with n columns by a matrix with n rows.
F GF G = F G = F Gº1713
º3630
(º6)(3) + (1)(1)(5)(3) + (º2)(1)
(º6)(6) + (1)(0)(5)(6) + (º2)(0)
31
60
1º2
º65
Examples onpp. 208–210
Write the product. If it is not defined, state the reason.
11. F Gf g 12. F GF G 13. F GF G88
º1º4
32
79
10
120
º51
1510
2º3
º7º1012
º4
4.3 DETERMINANTS AND CRAMER’S RULE
You can evaluate the determinant of a 2 ª 2 or a 3 ª 3 matrix. Find products of the entries on the diagonals and subtract.
det F G = * * = º2(4) º 1(º6) = º8 + 6 = º2
det F G = * * = (24 + 6 + 20) º [60 + (º24) + (º2)] = 16
You can find the area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) using
Area = 6 }21
} * *where 6 indicates you should choose the sign that yields a positive value.
You can use Cramer’s rule to solve a system of linear equations. First find thedeterminant of the coefficient matrix and then use Cramer’s rule to solve for x and y.
3x º 4y = 12x + 2y = 14
det F G = * * = 3(2) º 1(º4) = 6 + 4 = 10
x = }10} = = }81
00} = 8 y = }10} = }
3(14)1º0
1(12)} = }
31
00} = 3
12(2) º 14(º4)}}10
º42
31
º42
31
111
y1y2y3
x1x2x3
16
º4
2º1
2
532
16
º4
2º1
2
532
16
º4
2º1
2
º64
º21
º64
º21
Examples onpp. 214–217
Evaluate the determinant of the matrix.
14. F G 15. F G 16. F G 17. F G18. Find the area of a triangle with vertices A(0, 1), B(2, 4), and C(1, 8).
Use Cramer’s rule to solve the linear system.
19. 7x º 4y = º3 20. 2x + y = º2 21. 5x º 4y + 4z = 182x + 5y = º7 x º 2y = 19 ºx + 3y º 2z = 0
4x º 2y + 7z = 3
4º2º3
º312
201
014
113
320
º31
62
12
º93
EXAMPLE
EXAMPLES
* *1214
31* *º4
21214
¹ º » ¼ ½ ¾ ¿ À
242 Chapter 4 Matrices and Determinants
IDENTITY AND INVERSE MATRICESExamples onpp. 222–226
Find the inverse of the matrix.
22. F G 23. F G 24. F G 25. F GSolve the matrix equation.
26. F GX = F G 27. F GX + F G = F Gº32
96
º21
86
º53
º74
94
0º1
32
53
º11
6º5
6º4
º32
23
21
311
27
4.5
4.4
SOLVING SYSTEMS USING INVERSE MATRICES
You can use inverse matrices to solve a system of linear equations.
x + 3y = 10 F G F G = F G2x + 5y = º2
A X = B
Then X = Aº1B = }1(5) º
12(3)
} F G F G = º1F G = F G.
The solution is (º56, 22).
º5622
56º22
10º2
º31
5º2
10º2
x
y
35
12
Examples onpp. 230–232
Use an inverse matrix to solve the linear system.
28. 9x + 8y = º6 29. x º 3y = º2 30. 4x º 14y = º15ºx º y = 1 5x + 3y = 17 18x º 12y = 9
Use an inverse matrix and a graphing calculator to solve the linear system.
31. x º y º 4z = 3 32. 4x + 10y º z = º3 33. 5x º 3y + 5z = º1ºx + 3y º z = º1 11x + 28y º 4z = 1 3x + 2y + 4z = 11x º y + 5z = 3 º6x º 15y + 2z = º1 2x º y + 3z = 4
You can find the inverse of an n ª n matrix provided its determinantdoes not equal zero.
The inverse of A = F G is Aº1 = F G.
If A = F G, then Aº1 = }7(2) º
15(3)
} F G = º1F G = F G.
You can use the inverse of a matrix A to solve a matrix equation AX = B: X = Aº1B.
F GX = F GAº1 = }7 º
16} F G = F G
X = F GF G = F Gº62
6º1
02
35
º31
7º2
º31
7º2
º31
7º2
02
35
37
12
3º7
º25
º37
2º5
º37
2º5
32
75
ºb
a
d
ºc
1}
|A|b
d
a
c
EXAMPLES
EXAMPLE
Write in matrix form.
¹ º » ¼ ½ ¾ ¿ À
Chapter Test 243
Chapter TestCHAPTER
4
ÁÁ(1, 7)
(2, 2)
(4, 5) ÂÃ
Perform the indicated operation(s).
1. F G + F G 2. 0.25F G 3. º4SF G º F GD4. F GF G 5. F G F G 6. F GF G
Solve the matrix equation for x and y.
7. F G = F G 8. F G = F G 9. 3F G = F GEvaluate the determinant of the matrix.
10. F G 11. F G 12. F G 13. F GFind the area of the triangle with the given vertices.
14. A(2, 1), B(5, 3), C(7, 1) 15. A(º1, 0), B(º3, 3), C(0, 4) 16. A(º3, 2), B(º1, 4), C(º4, 3)
Use Cramer’s rule to solve the linear system.
17. 2x + y = 12 18. º4x + 5y = º10 19. x + y = 2 20. 5x º 2y + 7z = 125x + 3y = 27 5x º 6y = 13 2y º z = 0 2x + 5y + 3z = 10
ºx º y + z = º1 3x º y + 4z = 8Find the inverse of the matrix.
21. F G 22. F G 23. F G 24. F GSolve the matrix equation.
25. F GX = F G 26. F GX = F G 27. F GX = F GUse an inverse matrix to solve the linear system.
28. x º y = 5 29. 3x + 2y = º8 30. 2x º 7y = 6º2x + 3y = º9 º2x + 5y = 18 º3x + 11y = º10
31. STAINED GLASS You are making a stained glass panel using different colors asshown. The coordinates given are measured in inches. Find the area of the red triangle.
32. DECODING Use the inverse of A = F G and the coding information on
pages 225 and 226 to decode the message below.
44, º15, 3, º1, 80, º32, 39, º17, 3, º1, 12, º4, 77, º26
33. BUDGETING MEALS You have $18 to spend for lunch during a 5 day workweek. It costs you about $1.50 to make a lunch at home and about $5 to buy alunch. How many times each work week should you make a lunch at home?
º1º1
23
82
612
104
02
1º6
01
10
56
22
º69
3º2
71
81
0º5
10
4º5
º66
º21
º11
59
43
4º2
1
30
º5
º160
130
052
412
º1º1
º21
º94
7º3
3º12
º15y
1º4
x
894
2x
19
ºy
º221
83
º1º9
y + 63
º1x º 4
001
260
º141
01
º1
1º1
2
020
04
3º5
12
º69
º226
4º3
0
183
4º1
4
8º8
4º3
10º6
1º4
º1236
20º4
8º8
77
2º5
3º2
º4º2
50
23
Ä Å Æ Ç È É Ê Ë
244 Chapter 4 Matrices and Determinants
Chapter Standardized TestCHAPTER
4
1. MULTIPLE CHOICE Which matrix equals
2SF G + F GD?
¡A F G ¡B F G¡C F G ¡D F G¡E F G
2. MULTIPLE CHOICE What are the values of x and yin the matrix equation?
2xF G = F G¡A x = 8, y = º80 ¡B x = º4, y = º80
¡C x = 4, y = 80 ¡D x = 4, y = º80
¡E x = 2, y = º40
3. MULTIPLE CHOICE What is the product of
F G and F G?
¡A F G ¡B F G¡C F G ¡D F G¡E F G
4. MULTIPLE CHOICE What is the determinant of
F G?
¡A º78 ¡B º34 ¡C º16
¡D 34 ¡E 78
5. MULTIPLE CHOICE What is the inverse of
F G?
¡A F G ¡B F G¡C F G ¡D F G¡E F G
6. MULTIPLE CHOICE Which matrix has no inverse?
¡A F G ¡B F G¡C F G ¡D F G¡E F G
7. MULTIPLE CHOICE What is the area of the trianglein square units?
¡A 40 ¡B 30 ¡C 26
¡D 20 ¡E 13
8. MULTIPLE CHOICE What is the solution of thelinear system?
7x + 5y = 64x + 3y = 3
¡A x = 3, y = 3 ¡B x = º3, y = º3
¡C x = 3, y = º3 ¡D x = º3, y = 3
¡E x = 8, y = º10
(23, 4)
(2, 21)
(6, 3)ÌÃÍÎ
41
º42
11
10
46
º23
6º9
4º6
05
60
59
º4º7
º5º9
47
º5º9
º4º7
59
º47
º5º9
4º7
º5º4
97
52
º2
1º1
4
2º3
0
º28
º3
2113
º2
º22
º3
19132
º22
º7
1913
º2
º2º6º3
1913
º2
º22
º3
1913
º2
º21
º1
105
43
º1
012
º1º2
3
º840
º16y
º15
º2º10
º10476
42º18
6030
º24º6
º2220
6º12
º2418
4º14
º418
4º2
º56
0º3
º73
2º4
TEST-TAKING STRATEGY During a test it is important to stay mentally focused, but also
physically relaxed. If you start to get tense, put your pencil down and take some deep breaths.
This may help you regain control.
Ä Å Æ Ç È É Ê Ë
Chapter Standardized Test 245
9. QUANTITATIVE COMPARISON Choose the statement that is true about thegiven quantities.
¡A The quantity in column A is greater.
¡B The quantity in column B is greater.
¡C The two quantities are equal.
¡D The relationship cannot be determined from the given information.
10. MULTI-STEP PROBLEM The School Spirit club ordered shirts to sell atbasketball games. The number of each size and type of shirt they ordered is shown in the matrix at the right.
a. At the first basketball game the club sold 17 T-shirts (4 small, 5 medium, 6 large, and 2 extra large) and 12 sweatshirts (3 medium, 5 large, and 4 extralarge). Write a matrix that gives the number of each size and type of shirt sold at the game. Then write a matrix that shows the number of shirts left.
b. The wholesale price of a T-shirt is $8, and the club sells them for $10 each.The wholesale price of a sweatshirt is $20, and the club sells them for $25each. Write a row matrix for the number of each type of shirt (T-shirt orsweatshirt) sold at the game from part (a). Write a column matrix for the profit on each type of shirt. Multiply the row matrix by the column matrix and interpret the result.
11. MULTI-STEP PROBLEM You and a friend are planning a secret meeting.
You agree that A = F G will be your coding matrix. Use the coding
information on pages 225 and 226.
a. Use matrix A to encode the message SATURDAY.
b. Use the inverse of A to decode your answer to part (a).
c. Your friend replied with the message below. Use the inverse of A to decode it.
34, 49, 6, 9, 23, 41, 6, 12, 14, 19, 16, 20
d. Use matrix A to encode your answer to part (c).
12. MULTI-STEP PROBLEM Use the following linear system.
x º 2y = 13x º 5y = 4
a. Use Cramer’s rule to solve the system.
b. Use inverse matrices to solve the system.
c. Use the substitution or linear combination method to solve the system.
d. Solve the system by graphing. Label the solution on your graph.
e. Writing Which method do you prefer for solving this linear system?
12
11
Column A Column B
det F G det F Gº15
º23
12
0º8
FUNDRAISING
T-shirt Sweatshirt
3 420204020
2525
10050
S
M
L
XL