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#### Mathematical Applications For The Management Life And Social Sciences 11th Edition By Harshbarger – Test Bank

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**Sample Questions**

1. What is the order of matrices A, B, C, D, E?

, , , ,

a. Matrix A is 2×2, matrix B is 3×2, matrix C is 2×3, matrix D is 3×3, matrix E is 5×2.

b. Matrix A is 3×1, matrix B is 3×3, matrix C is 2×5, matrix D is 12×11, matrix E is 3×7.

c. Matrix A is 2×2, matrix B is 2×3, matrix C is 3×2, matrix D is 3×3, matrix E is 2×5.

d. Matrix A is 2×2, matrix B is 3×3, matrix C is 3×3, matrix D is 3×3, matrix E is 5×5.

e. Matrix A is 1×2, matrix B is 3×2, matrix C is 2×3, matrix D is 12×3, matrix E is 3×2.

ANSWER: a

2. Write a zero matrix that is the same order as D.

a.

b.

c.

d.

e.

ANSWER: a

3. Write the matrix that is the negative of matrix B.

a.

b.

c.

d.

e.

ANSWER: e

4. What is element ?

a.

b.

c.

d.

e.

ANSWER: b

5. Write the transpose of matrix A.

a.

b.

c.

d.

e.

ANSWER: d

6. Write the transpose of matrix F.

a.

b.

c.

d.

e.

ANSWER: c

7. What is the sum of matrix M and its negative?

a.

b.

c.

d.

e.

ANSWER: c

8. If matrix M has element , what is ?

a.

b.

c.

d.

e.

ANSWER: a

9. Use the following matrices to perform the indicated matrix operations, if possible.

;

a.

b.

c.

d.

e.

ANSWER: e

10. Use the following matrices to perform the indicated matrix operations, if possible.

;

a.

b.

c.

d.

e.

ANSWER: e

11. Use the following matrices to perform the indicated matrix operations, if possible.

;

a.

b.

c.

d.

e.

ANSWER: b

12. Use the following matrices to perform the indicated matrix operations, if possible.

;

a.

b.

c.

d.

e.

ANSWER: a

13. Use the following matrices to perform the indicated matrix operations, if possible.

a.

b.

c.

d.

e.

ANSWER: c

14. Use the following matrices to perform the indicated matrix operations, if possible.

;

a.

b.

c.

d.

e.

ANSWER: d

15. Use the following matrices to perform the indicated matrix operations, if possible.

;

a.

b.

c.

d.

e. is undefined

ANSWER: e

16. Use the following matrices to perform the indicated matrix operations, if possible.

a.

b.

c.

d.

e.

ANSWER: a

17. Use the following matrices to perform the indicated matrix operations, if possible.

;

a.

b.

c.

d.

e.

ANSWER: b

18. Use the following matrices to perform the indicated matrix operations, if possible.

;

a.

b.

c.

d.

e.

ANSWER: c

19. Use the following matrices to perform the indicated matrix operations, if possible.

;

a.

b.

c.

d.

e.

ANSWER: c

20. Find x, y, z and w.

a. , and

b. , and

c. , and

d. , and

e. , and

ANSWER: d

21. Find x, y, z and w.

a. and

b. and

c. and

d. and

e. and

ANSWER: d

22. Solve for x, y and z.

a. and

b. and

c. and

d. and

e. and

ANSWER: a

23. Find x, y, z, and w .

a. and

b. and

c. and

d. and

e. and

ANSWER: c

24. The tables below give the numbers of some species of threatened and endangered wildlife in the United States and in foreign countries in 2003. Write a matrix A that contains the number of each of these species in the United States in 2003 and a matrix B that contains the number of each of these species outside the United States in 2003.

United States

Mammals Birds Reptiles Amphibians Fish

Endangered

Threatened

Foreign

Mammals Birds Reptiles Amphibians Fish

Endangered

Threatened

a. ,

b. ,

c. ,

d. ,

e. ,

ANSWER: d

25. The tables below give the numbers of some species of threatened and endangered wildlife in the United States and in foreign countries in 2003. Find a matrix with the total number of these species. Assume that U.S. and foreign species are different.

United States

Mammals Birds Reptiles Amphibians Fish

Endangered

Threatened

Foreign

Mammals Birds Reptiles Amphibians Fish

Endangered

Threatened

a.

b.

c.

d.

e.

ANSWER: b

26. The tables below give the numbers of some species of threatened and endangered wildlife in the United States and in foreign countries in 2003. Let A represent the matrix that contains the number of each of these species in the United States in 2003 and B represent the matrix that contains the number of each of these species outside the United States in 2003. Find the matrix B – A. What do the negative entries in matrix B – A mean?

United States

Mammals Birds Reptiles Amphibians Fish

Endangered

Threatened

Foreign

Mammals Birds Reptiles Amphibians Fish

Endangered

Threatened

a.

The negative entries tell us that that particular species is extinct.

b.

The negative entries tell us how many more endangered or threatened species are in foreign countries.

c.

The negative entries tell us that there are more endangered or threatened species in foreign countries.

d.

The negative entries tell us that that particular species is extinct.

e.

The negative entries tell us how many more endangered or threatened species are in the United States.

ANSWER: e

27. The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Form a matrix for the 1995 data and a matrix for the 2002 data.

1995

Breed Rank Number Registered

Labrador retriever 1

Rottweiler 2

German shepherd 3

Golden retriever 4

Beagle 5

Poodle 6

Cocker spaniel 7

Dachshund 8

Pomeranian 9

Yorkshire terrier 10

2002

Breed Rank Number Registered

Labrador retriever 1

Rottweiler 13

German shepherd 3

Golden retriever 2

Beagle 4

Poodle 8

Cocker spaniel 15

Dachshund 5

Pomeranian 12

Yorkshire terrier 6

a. ;

b. ;

c. ;

d. ;

e. ;

ANSWER: c

28. The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Use a matrix operation to find the change from 1995 to 2002 in rank for each breed and in registration numbers for each breed.

1995

Breed Rank Number Registered

Labrador retriever 1

Rottweiler 2

German shepherd 3

Golden retriever 4

Beagle 5

Poodle 6

Cocker spaniel 7

Dachshund 8

Pomeranian 9

Yorkshire terrier 10

2002

Breed Rank Number Registered

Labrador retriever 1

Rottweiler 13

German shepherd 3

Golden retriever 2

Beagle 4

Poodle 8

Cocker spaniel 15

Dachshund 5

Pomeranian 12

Yorkshire terrier 6

a.

b.

c.

d.

e.

ANSWER: a

29. The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Which breed had the greatest improvement in rank?

1995

Breed Rank Number Registered

Labrador retriever 1

Rottweiler 2

German shepherd 3

Golden retriever 4

Beagle 5

Poodle 6

Cocker spaniel 7

Dachshund 8

Pomeranian 9

Yorkshire terrier 10

2002

Breed Rank Number Registered

Labrador retriever

Rottweiler

German shepherd

Golden retriever

Beagle

Poodle

Cocker spaniel

Dachshund

Pomeranian

Yorkshire terrier

a.

b.

c.

d.

e.

ANSWER: b

30. The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Which breed had the largest increase in the number of registrations?

Breed Rank Number Registered

Labrador retriever 1

Rottweiler 2

German shepherd 3

Golden retriever 4

Beagle 5

Poodle 6

Cocker spaniel 7

Dachshund 8

Pomeranian 9

Yorkshire terrier 10

2002

Breed Rank Number Registered

Labrador retriever

Rottweiler

German shepherd

Golden retriever

Beagle

Poodle

Cocker spaniel

Dachshund

Pomeranian

Yorkshire terrier

a.

b.

c.

d.

e.

ANSWER: e

31. The following tables give the capital expenditures and gross operating costs of manufacturing establishments for pollution abatement, in millions of dollars. Use a calculator or spreadsheet to find the sum of the matrices containing these data. This will yield the total costs of manufacturing establishments for air, water, and solid contained waste pollution abatement for the years 2002–2005.

Pollution Abatement Capital Expenditures

Air Water Solid Contained Waste

2002

2003

2004

2005

Pollution Abatement Gross Operating Costs

Air Water Solid Contained Waste

2002

2003

2004

2005

a.

b.

c.

d.

e.

ANSWER: d

32. The following tables give the capital expenditures and gross operating costs of manufacturing establishments for pollution abatement, in millions of dollars. Which type of pollution abatement was most expensive in 2005?

Pollution Abatement Capital Expenditures

Air Water Solid Contained Waste

2002

2003

2004

2005

Pollution Abatement Gross Operating Costs

Air Water Solid Contained Waste

2002

2003

2004

2005

a. Solid Contained Waste

b. Water

c. Air

ANSWER: a

33. The following tables give the death rates, per 100,000 population, by age for selected years for males and females. If matrix M gives the male data and matrix F gives the female data, use matrix operations to find the death rate per 100,000 for all people in the age categories given and for the given years.

Males

Under

1 1-4 5-14 15-24 25-34 35-44 45-54

1970

1980

1990

1994

2001

Females

Under

1 1-4 5-14 15-24 25-34 35-44 45-54

1970

1980

1990

1994

2001

a.

b.

c.

d.

e.

ANSWER: c

34. The following tables give the death rates, per 100,000 population, by age for selected years for males and females. If matrix M gives the male data and matrix F gives the female data, find matrix M – F and describe what it means.

Males

Under

1 1-4 5-14 15-24 25-34 35-44 45-54

1970

1980

1990

1994

2001

Females

Under

1 1-4 5-14 15-24 25-34 35-44 45-54

1970

1980

1990

1994

2001

a.

M – F represents how much greater the numbers are for males in each category per year.

b.

M – F represents how much greater the numbers are for males in each category per year.

c.

M – F represents how much greater the numbers are for females in each category per year.

d.

M – F represents how much greater the numbers are for females in each category per year.

e.

M – F represents only the male death rates, per 100,000 population by age.

ANSWER: b

35. Let matrix A represent the sales (in thousands of dollars) for the Walbash Company in 2004 in various cities, and let matrix B represent the sales (in thousands of dollars) for the same company in 2005 in the same cities. Write the matrix that represents the total sales by type (wholesale and retail) and city (Chicago, Atlanta, and Memphis) for both years.

;

a.

b.

c.

d.

e.

ANSWER: d

36. Let matrix A represent the sales (in thousands of dollars) for the Walbash Company in 2004 in various cities, and let matrix B represent the sales (in thousands of dollars) for the same company in 2005 in the same cities. Write the matrix that represents the change in sales by type (wholesale and retail) and city (Chicago, Atlanta, and Memphis) from 2004 to 2005.

;

a.

b.

c.

d.

e.

ANSWER: b

37. A poll of 3620 people revealed that of the respondents who were registered Republicans, 893 approved of the president’s job performance, 257 did not, and 135 had no opinion. Of the registered Democrats, 142 approved of the president’s job performance, 426 did not, and 501 had no opinion. Of those registered as Independents, 177 approved, 64 did not approve, and 801 had no opinion. Of the remaining respondents, who were not registered, 142 approved, 38 did not approve, and 44 had no opinion. Represent these data in a 3×4 matrix.

a.

b.

c.

d.

e.

ANSWER: e

38. From the data in the following table, make a matrix A that gives the value (in billions of dollars) of U.S. imports from the various country groupings in the years 2000 and 2002.

Imports to U.S. from Country Groupings

($billions)

2000 2002

North America 386.8 363.0

Western Europe 240.7 246.0

South/Central America 73.3 69.5

Pacific Rim 438.0 393.5

Exports from U.S. to Country Groupings

($billions)

2000 2002

North America 310.3 278.4

Western Europe 201.5 177.0

South/Central America 79.3 51.6

Pacific Rim 222.6 198.6

a.

b.

c.

d.

e.

ANSWER: a

39. Using the data in the following tables, Let matrix A represent imports (in billions of dollars) to U.S. from country groupings, and let matrix B represent exports (in billions of dollars) from U.S. to country groupings. Find the U.S. trade balance with each country grouping in each year by finding matrix B – A.

Imports to U.S. from Country Groupings

($billions)

2000 2002

North America 366.8 343.0

Western Europe 240.7 266.0

South/Central America 73.3 69.5

Pacific Rim 438.0 393.5

Exports from U.S. to Country Groupings

($billions)

2000 2002

North America 290.3 278.4

Western Europe 181.5 177.0

South/Central America 59.3 51.6

Pacific Rim 222.6 198.6

a.

b.

c.

d.

e.

ANSWER: d

40. Using the data in the following tables, Let matrix A represent imports (in billions of dollars) to U.S. from country groupings, and let matrix B represent exports (in billions of dollars) from U.S. to country groupings. Find the total U.S. trade with these groupings by finding A + B.

Imports to U.S. from Country Groupings

($billions)

2000 2002

North America 386.8 343.0

Western Europe 240.7 266.0

South/Central America 93.3 69.5

Pacific Rim 438.0 393.5

Exports from U.S. to Country Groupings

($billions)

2000 2002

North America 310.3 278.4

Western Europe 181.5 177.0

South/Central America 79.3 51.6

Pacific Rim 222.6 198.6

a.

b.

c.

d.

e.

ANSWER: a

41. Ace, Baker, and Champ are being purchased by ALCO, Inc., and their outstanding debts must be paid by the purchaser. The matrix below gives the amounts of debt and the terms for the companies being purchased. If ALCO pays 35% of the amount owed on each debt, write the matrix giving the remaining debts.

a.

b.

c.

d.

e.

ANSWER: e

42. Ace, Baker, and Champ are being purchased by ALCO, Inc., and their outstanding debts must be paid by the purchaser. The matrix below gives the amounts of debt and the terms for the companies being purchased. Suppose ALCO decides to pay 80% of all debts due in 30 days and to increase the debts due in 60 days by 20%. Write a matrix that gives the debts after these transactions are made.

a.

b.

c.

d.

e.

ANSWER: c

43. When a firm buys another company, the company frequently has some outstanding debt that the purchaser must pay. Consider three companies, A, B, and C, that are purchased by Maxx Industries. The following table gives the amount of each company’s debt, classified by the number of days remaining until the debt must be paid. Suppose Maxx Industries pays 20% of the amount owed on each loan. Write the matrix that would give the remaining debt in each category.

Company 30 Days 60 Days More than 60 Days

A

B

C

a.

b.

c.

d.

e.

ANSWER: b

44. When a firm buys another company, the company frequently has some outstanding debt that the purchaser must pay. Consider three companies, A, B, and C, that are purchased by Maxx Industries. The following table gives the amount of each company’s debt, classified by the number of days remaining until the debt must be paid. What payment plan did Maxx Industries use if the outstanding debt is given by the following matrix?

Company 30 Days 60 Days More than 60 Days

A

B

C

a. They paid 30% of the debts due in 30 days, 100% of the debts due in 60 days, and 100% of the debts due in more than 60 days.

b. They paid 30% of the debts due in 30 days, 30% of the debts due in 60 days, and 100% of the debts due in more than 60 days.

c. They paid 100% of the debts due in 30 days, 100% of the debts due in 60 days, and 30% of the debts due in more than 60 days.

d. They paid 100% of the debts due in 30 days, 30% of the debts due in 60 days, and 100% of the debts due in more than 60 days.

e. They paid 100% of the debts due in 30 days, 30% of the debts due in 60 days, and 30% of the debts due in more than 60 days.

ANSWER: d

45. In order to rank the five members of its chess team for play against another school, the coach draws the following diagram. An arrow from 1 to 2 means player 1 has defeated player 2.

1 2

3

4 5

Construct matrix A with elements .

a.

b.

c.

d.

e.

ANSWER: c

46. In order to rank the five members of its chess team for play against another school, the coach draws the following diagram. An arrow from 1 to 2 means player 1 has defeated player 2.

1 2

5

3 4

Construct matrix B with elements .

a.

b.

c.

d.

e.

ANSWER: e

47. In order to rank the five members of its chess team for play against another school, the coach draws the following diagram. An arrow from 1 to 2 means player 1 has defeated player 2.

4 1

5

3 2

Let matrix A represent the matrix with the following elements: .

Let matrix B represent the matrix with the following elements: .

The player i is the top-ranked player if the sum of row i in the matrix A + B is the largest. What is the number of this player?

a. player 3

b. player 2

c. player 5

d. player 1

e. player 4

ANSWER: a

48. The figure that follows depicts the mean (average) flow time for a job when the critical-ratio rule is used to dispatch workers at several machines required to complete the job. In the figure, WHN 1, WHN 2, and WHN 3 represent three different rules for determining when a worker should be transferred to another machine, and Where represents six rules for determining the machine to which the worker is transferred. Construct a 3×6 matrix A to represent these data, with entries rounded to the nearest integer.

a.

b.

c.

d.

e.

ANSWER: b

49. The matrix that follows represents work efficiency for workers at nine different work centers. Suppose that the work centers changes in rules cause efficiency to decrease by 0.04 at work centers 2 and 5 and to decrease by 0.01 at work center 7. Write the matrix that describes the new efficiencies.

Work Centers

1 2 3 4 5 6 7 8 9

a.

b.

c.

d.

e.

ANSWER: d

50. Use a computer spreadsheet to find the average efficiency for each work center over the first five workers. Which work center is the most efficient?

Work Centers

1 2 3 4 5 6 7 8 9

a. Number 4

b. Number 3

c. Number 8

d. Number 5

e. Number 1

ANSWER: c

51. Multiply the matrices.

a.

b.

c.

d.

e.

ANSWER: b

52. Multiply the matrices.

a.

b.

c.

d.

e.

ANSWER: e

53. Multiply the matrices.

a.

b.

c.

d.

e.

ANSWER: a

54. Use the matrices to find CD, if possible.

a.

b.

c.

d.

e.

ANSWER: d

55. Use the matrices to find DE, if possible.

a.

b.

c.

d.

e.

ANSWER: c

56. Use the matrices to find CF, if possible.

a.

b.

c.

d.

e. CF is undefined.

ANSWER: b

57. Use the matrices to find EC, if possible.

a.

b.

c.

d.

e. EC is undefined.

ANSWER: e

58. Use the matrices to find BA, if possible.

a.

b.

c.

d.

e. BA is undefined.

ANSWER: e

59. Use the matrices to find , if possible.

a.

b.

c.

d.

e. is undefined.

ANSWER: c

60. Use the matrices to find BE, if possible.

a.

b.

c.

d.

e. BE is undefined.

ANSWER: e

61. Use the matrices to find , if possible.

a.

b.

c.

d.

e. is undefined.

ANSWER: a

62. Use matrix A to find A2, if possible.

a.

b.

c.

d.

e. A2 is undefined.

ANSWER: d

63. Use matrix A to find A3, if possible.

a.

b.

c.

d.

e. A3 is undefined.

ANSWER: c

64. Use matrix F to find F2, if possible.

a.

b.

c.

d.

e. F2 is undefined.

ANSWER: e

65. (CD)•E and C•(DE) are equal.

a. True

b. False

ANSWER: True

66. CD = DC.

a. True

b. False

ANSWER: False

67. and are equal.

a. True

b. False

ANSWER: True

68. Use the matrices below. Perform the indicated operations.

a.

b.

c.

d.

e.

ANSWER: c

69. Use the matrices below. Perform the indicated operations.

BA

a.

b.

c.

d.

e.

ANSWER: d

70. Use the matrices below. Perform the indicated operations.

DC

a.

b.

c.

d.

e.

ANSWER: c

71. Use the matrices below. Perform the indicated operations.

AZ

a.

b.

c.

d.

e.

ANSWER: b

72. Use the matrices below. Perform the indicated operations.

AI

a.

b.

c.

d.

e.

ANSWER: e

73. Use the matrices below. Perform the indicated operations.

IA

a.

b.

c.

d.

e.

ANSWER: d

74. Use the matrices below. Perform the indicated operations.

I•F•Z

a.

b.

c.

d.

e.

ANSWER: a

75. Is it true for matrices (as it is for real numbers) that multiplication by a zero matrix only sometimes gives is a result of a zero matrix provided the multiplication is defined?

a. True

b. False

ANSWER: False

76. Find AB and BA if and , with .

a. ,

b. ,

c. ,

d. ,

e. ,

ANSWER: e

77. Using the matrices A and B below, what condition must be satisfied in order that AB and BA are defined?

and

a.

b.

c.

d.

e.

ANSWER: e

78. For , are FFT and FTF defined?

a. Yes

b. No

ANSWER: Yes

79. For , what size is each product FFT and FTF?

a. FFT is 2×2; FTF is 2×2

b. FFT is 2×2; FTF is 4×4

c. FFT is 4×2; FTF is 4×4

d. FFT is 4×4; FTF is 2×2

e. FFT is 2×4; FTF is 4×2

ANSWER: b

80. For , can FFT = FTF? Explain.

a. No, because FFTand FTF have the same dimensions but different values.

b. Yes, because FFT and FTF have the same dimensions.

c. Yes, because FFT and FTF have the same values.

d. No, because FFT and FTF have different dimensions.

e. Yes, because FFT and FTF have the same dimensions and the same values.

ANSWER: d

81. Substitute the given values of x, y, and z into the matrix equation and use matrix multiplication to determine if the values satisfy the equation.

, ,

a. True

b. False

ANSWER: True

82. Substitute the given values of x, y, and z into the matrix equation and use matrix multiplication to determine if the values satisfy the equation.

, ,

a. True

b. False

ANSWER: True

83. Substitute the given values of x, y, and z into the matrix equation and use matrix multiplication to determine if the values satisfy the equation.

, ,

a. True

b. False

ANSWER: True

84. Substitute the given values of x, y, and z into the matrix equation and use matrix multiplication to determine if the values satisfy the equation.

, ,

a. True

b. False

ANSWER: False

85. Use technology to find the product AB of the following matrices.

a.

b.

c.

d.

e.

ANSWER: c

86. A car dealer can buy midsize cars for 11% under the list price, and he can buy luxury cars for 14% under the list price. The following table gives the list prices for two midsize and two luxury cars.

Midsize 36,000 42,000

Luxury 50,000 56,000

Write these data in a matrix and multiply it on the left by the matrix

a.

b.

c.

d.

e.

ANSWER: e

87. A clothing manufacturer has factories in Atlanta, Chicago, and New York. Sales (in thousands) during the first quarter are summarized in the matrix below.

During this period, the selling price of a coat was $125, of a shirt $50, of a pair of pants $50, and of a tie $25. Use matrix multiplication to find the total revenue received by each factory.

a. Atlanta $13,100,000; Chicago $11,300,000; New York $6,750,000;

b. Atlanta $9,725,000; Chicago $11,825,000; New York $4,950,000;

c. Atlanta $13,100; Chicago $11,300; New York $6,750;

d. Atlanta $9,725; Chicago $11,825; New York $4,950;

e. Atlanta $975,000; Chicago $1,157,500; New York $495,000.

ANSWER: b

88. Matrix A below gives the fraction of the Earth’s area and the projected fraction of its population for five continents in 2050. Matrix B gives the Earth’s area (in square miles) and its projected 2050 population. Find the area and population of each given continent by finding AB.

,

a.

b.

c.

d.

e.

ANSWER: a

89. Suppose the weights (in grams) and lengths (in centimeters) of three groups of laboratory animals are given by matrix A, where column 1 gives the lengths and each row corresponds to one group.

If the increase in both weight and length over the next two weeks is 19% for group I, 9% for group II, and 3% for group III, then the increases in the measures during the 2 weeks can be found by computing GA, where

What are the increases in respective weights and measures at the end of these 2 weeks?

a.

b.

c.

d.

e.

ANSWER: c

90. Suppose the weights (in grams) and lengths (in centimeters) of three groups of laboratory animals are given by matrix A, where column 1 gives the lengths and each row corresponds to one group.

If the increase in both weight and length over the next two weeks is 12% for group I, 5% for group II, and 3% for group III, then the increases in the measures during the 2 weeks can be found by computing GA, where

Find the matrix that gives the new weights and measures at the end of this period by computing , where is the 3×3 identity matrix.

a.

b.

c.

d.

e.

ANSWER: e

91. Multiplication by a matrix can be used to encode messages (and multiplication by its inverse can be used to decode messages). Given the code

a b c d e f g h i j k l m n o p q r s t u v w x y z blank

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

and the code matrix , use matrix A to encode the message “The die is cast.”

a.

b.

c.

d.

e.

ANSWER: b

92. Multiplication by a matrix can be used to encode messages (and multiplication by its inverse can be used to decode messages). Given the code

a b c d e f g h i j k l m n o p q r s t u v w x y z blank

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

and the code matrix , use matrix A to encode the message “To be or not to be.”

a.

b.

c.

d.

e.

ANSWER: a

93. Suppose products A and B are made from plastic, steel, and glass, with the number of units of each raw material required for each product given by the table below.

Plastic Steel Glass

Product A 3 1 0.50

Product B 5 0.50 2

Because of transportation costs to the firm’s two plants, X and Y, the unit costs for some of the raw materials are different. The table below gives the unit costs for each of the raw materials at the two plants.

Plant X Plant Y

Plastic 11 9

Steel 22 27

Glass 15 14

Using the information just given, find the total cost of producing each of the products at each of the factories.

a.

b.

c.

d.

e.

ANSWER: d

94. Consider the original budget matrix. Assume there is a 20% increase in manufacturing, a 4% increase in office, a 6% increase in sales, a 21% increase in shipping, a 5% increase in accounting, and a 4% decrease in management. Find the new budget matrix by developing a matrix A to represent these departmental increases and then computing the matrix BA.

Mfg. Office Sales Shp Act. Mgt.

Supplies 0.7 8.5 10.2 1.1 5.6 3.6

Phone 0.5 0.2 6.1 1.3 0.2 1.0

Transp. 2.2 0.4 8.8 1.2 1.2 4.8

Salaries 251.8 63.4 81.6 35.2 54.3 144.2

Utilities 30.0 1.0 1.0 1.0 1.0 1.0

Materials 788.9 0 0 0 0 0

Mfg. = Manufacturing; Shp = Shipping;

Act. = Accounting; Mgt. = Management

a.

b.

c.

d.

e.

ANSWER: e

95. Use the indicated row operation to change matrix A, where .

Add times row 1 to row 3 of matrix A and place the result in row 3 to get 0 in row 3, column 1.

a.

b.

c.

d.

e.

ANSWER: e

96. Write the augmented matrix associated with the system.

a.

b.

c.

d.

e.

ANSWER: c

97. Write the augmented matrix associated with the system.

a.

b.

c.

d.

e.

ANSWER: a

98. The given matrix is an augmented matrix used in the solution of a system of linear equations. What is the solution of the system?

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: d

99. The given matrix is an augmented matrix used in the solution of a system of linear equations. What is the solution of the system?

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: b

100. The given matrix is an augmented matrix used in the solution of a system of linear equations. What is the solution of the system?

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: d

101. The given matrix is an augmented matrix representing a system of linear equations. Find the solution of the system.

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: c

102. The given matrix is an augmented matrix representing a system of linear equations. Find the solution of the system.

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: e

103. The given matrix is an augmented matrix representing a system of linear equations. Find the solution of the system.

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: a

104. Use row operations on the augmented matrix to solve the given system of linear equations.

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: b

105. Use row operations on the augmented matrix to solve the given system of linear equations.

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: d

106. Use row operations on the augmented matrix to solve the given system of linear equations.

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: a

107. Use row operations on the augmented matrix to solve the given system of linear equations.

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: c

108. Use row operations on the augmented matrix to solve the given system of linear equations.

a. , , ,

b. , , ,

c. , , ,

d. , , ,

e. , , ,

ANSWER: b

109. Use row operations on the augmented matrix to solve the given system of linear equations.

a. , , ,

b. , , ,

c. , , ,

d. , , ,

e. , , ,

ANSWER: c

110. A system of linear equations and a reduced matrix for the system are given. Use the reduced matrix to find the general solution of the system, if one exists

a. , ,

b. , ,

c. , ,

d. , ,

e. no solution

ANSWER: e

111. A system of linear equations and a reduced matrix for the system are given. Use the reduced matrix to find the general solution of the

system, if one exists.

a. , ,

b. , ,

c. , ,

d. , ,

e. no solution

ANSWER: e

112. A system of linear equations and a reduced matrix for the system are given. Use the reduced matrix to find the general solution of the

system, if one exists.

a. , , for any real number z.

b. , , for any real number z.

c. , , for any real number z.

d. , , for any real number z.

e. no solution

ANSWER: b

113. A system of linear equations and a reduced matrix for the system are given. Use the reduced matrix to find the general solution of the

system, if one exists.

a. , , for any real number z.

b. , , for any real number z.

c. , , for any real number z.

d. , , for any real number z.

e. no solution

ANSWER: a

114. Describe the procedure for finding the general solution of a system of three linear equations involving p, q, and r if its reduced matrix indicate has one row of zeros at the bottom. You may assume that the coefficients in the augmented matrix correspond to variables in alphabetical order from left to right.

a. Use the reduced matrix to find equations for each variable p, q, and r. These equations can be used as the general solution to the system of equations.

b. Use the reduced matrix to find a general equation for p in terms of q and r. This equation can be used as the general solution to the system of equations.

c. Use the reduced matrix to solve for p and q in terms of r. These solutions will be valid for any real value of r.

d. Use the reduced matrix to solve for r in terms of p and q. This solution will be valid for any real value of p and q.

e. It is not possible to find a general solution for a system of three linear equations if its reduced matrix indicates an infinite number of solutions.

ANSWER: c

115. A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system.

a. , ,

b. , , for any real number z.

c. , ,

d. , , for any real number z.

e. no solution

ANSWER: d

116. A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system.

a. , , for any real number z.

b. , ,

c. , , for any real number z.

d. , , for any real number z.

e. no solution

ANSWER: c

117. A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

a. , ,

b. , ,

c. , , for any real number z.

d. , , for any real number z.

e. no solution

ANSWER: c

118. A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

a. , ,

b. , ,

c. , ,

d. , ,

e. no solution

ANSWER: a

119. A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

a. , ,

b. , ,

c. , ,

d. , , for any real number of z.

e. no solution

ANSWER: d

120. A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

a. , , , for any real number z.

b. , ,

c. , ,

d. , ,

e. no solution

ANSWER: a

121. A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

a. , ,

b. , , for any real number z.

c. , , for any real number z.

d. , , for any real number z.

e. no solution

ANSWER: b

122. A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

a. , , for any real number .

b. , , for any real number .

c. , , for any real number .

d. , , for any real number .

e. no solution

ANSWER: c

123. A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.

a. , , for any real number .

b. , ,

c. , , for any real number .

d. , , for any real number .

e. No solution

ANSWER: c

124. Use technology to solve the system of equations, if a solution exists.

a. , , ,

b. , , , for any real number .

c. , , ,

d. , , , for any real number .

e. no solution

ANSWER: d

125. Use technology to solve the system of equations, if a solution exists.

a. , , , ,

b. , , , ,

c. , , , ,

d. , , , ,

e. no solution

ANSWER: b

126. Use technology to solve the system of equations, if a solution exists.

a. , , ,

b. , , ,

c. , , ,

d. , , ,

e. no solution

ANSWER: e

127. Use technology to solve the system of equations, if a solution exists.

a. , , , for any real number .

b. , , ,

c. , , ,

d. , , , for any real number .

e. no solution

ANSWER: d

128. Use technology to solve the system of equations, if a solution exists.

a. , , , for any real number and .

b. , , , ,

c. , , , ,

d. , , , for any real number and .

e. no solution

ANSWER: a

129. Solve the following system of equations for y. This gives a formula for solving two equations in two variables for y.

a.

b.

c.

d.

e.

ANSWER: c

130. A man has $282,000 invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is $27,000 and the amount invested at 8% is twice that invested at 12%. How much is invested in each property?

a. $33,840 at 12%, $28,200 at 10%, $22,560 at 8%

b. $88,800 at 12%, $44,400 at 10%, $148,800 at 8%

c. $76,800 at 12%, $68,400 at 10%, $136,800 at 8%

d. $55,500 at 12%, $111,000 at 10%, $115,500 at 8%

e. $60,000 at 12%, $102,000 at 10%, $120,000 at 8%

ANSWER: e

131. A man has invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is and the amount invested at 8% is twice that invested at 12%. What is the annual income from each property?

a.

b.

c.

d.

e.

ANSWER: b

132. A bank lent million for the development of three new products, with one loan at 6%, one loan at 7%, and the third loan at 8%. The amount lent at 8% was equal to the sum of the amounts lent at the other two rates, and the bank’s annual income from the loans was . How much was lent at each rate?

a.

b.

c.

d.

e.

ANSWER: a

133. A psychologist studying the effects of nutrition on the behavior of laboratory rats is feeding one group a combination of three foods: I, II, and III. Each of these foods contains three additives, A, B, and C, that are being used in the study. Each additive is a certain percentage of each of the foods as follows:

Foods

I II III

Additive A 10% 30 % 60%

Additive B 0 % 4 % 5%

Additive C 2 % 2 % 12 %

If the diet requires 53 g per day of A, g per day of B, and g per day of C, find the number of grams per day of each food that must be used.

a.

b.

c.

d.

e.

ANSWER: d

134. A brokerage house offers three stock portfolios. Portfolio I consists of 2 blocks of common stock and 1 municipal bond. Portfolio II consists of 4 blocks of common stock, 2 municipal bonds, and 3 blocks of preferred stock. Portfolio III consists of 7 blocks of common stock, 3 municipal bonds, and 3 blocks of preferred stock. A customer wants blocks of common stock, municipal bonds, and blocks of preferred stock. How many units of each portfolio should be offered?

a.

b.

c.

d.

e.

ANSWER: c

135. Each ounce of substance A supplies of the required nutrition a patient needs. Substance B supplies of the required nutrition per ounce and substance C supplies of required nutrition per ounce. If digestive restrictions require that substances A and C be given in equal amounts and that the amount of substance B be one-fifth of these other amounts, find the number of ounces of each substance that should be in the meal to provide 100% nutrition.

a.

b.

c.

d.

e.

ANSWER: e

136. The King Trucking Company has an order for three products for delivery. The following table gives the particulars for the products.

Type I Type II Type III

Unit volume (cubic feet) 10 8 20

Unit weight (pounds) 10 20 40

Unit value (dollars) 100 20 200

If the carrier can carry 6,000 cu ft, can carry , and is insured for , how many units of each type can be carried?

a.

b.

c.

d.

e.

ANSWER: a

137. A botanist can purchase plant food of four different types, I, II, III, and IV. Each food comes in the same size bag, and the following table summarizes the number of grams of each of three nutrients that each bag contains.

Food(grams)

I II III IV

Nutrient A 5 5 10 5

Nutrient B 10 5 30 10

Nutrient C 5 15 10 25

The botanist wants to use a food that has these nutrients in a different proportion and determines that he will need a total of of A, of B, and of C. Let x = Type I bags, y = Type II bags, z = Type III bags, and w = Type IV bags, where x, y, z, and w are non-negative amounts.

Find a general solution that will determine the number of bags of each type of food that should be ordered.

a.

b.

c.

d.

e.

ANSWER: d

138. In the analysis of traffic flow, a certain city estimates the following situation for the “square” of its downtown district. In the following figure, the arrows indicate the flow of traffic. If x1 represents the number of cars traveling between intersections A and B, x2 represents the number of cars traveling between B and C, x3 the number between C and D, and x4 the number between D and A, we can formulate equations based on the principle that the number of vehicles entering an intersection equals the number leaving it. That is, for intersection A we obtain

Formulate equations for the traffic at B, C, and D.

a.

b.

c.

d.

e.

ANSWER: c

139. In the analysis of traffic flow, a certain city estimates the following situation for the “square” of its downtown district. In the following figure, the arrows indicate the flow of traffic. If x1 represents the number of cars traveling between intersections A and B, x2 represents the number of cars traveling between B and C, x3 the number between C and D, and x4 the number between D and A, we can formulate equations based on the principle that the number of vehicles entering an intersection equals the number leaving it. That is, for intersection A we obtain

Formulate equations for the traffic at B, C, and D. Solve the system of these four equations.

a.

b. ,

c.

d.

e.

ANSWER: b

140. Three different bacteria are cultured in one dish and feed on three nutrients. Each individual of species I consumes 1 unit of each of the first and second nutrients and 2 units of the third nutrient. Each individual of species II consumes 2 units of the first nutrient and 2 units of the third nutrient. Each individual of species III consumes 2 units of the first nutrient, 3 units of the second nutrient, and 5 units of the third nutrient. Let x1, x2 and x3 represent species I, II and III respectively. If the culture is given units of the first nutrient, units of the second nutrient, and units of the third nutrient, how many of each species can be supported such that all of the nutrients are consumed?

a.

b.

c.

d.

e.

ANSWER: a

141. An irrigation system allows water to flow in the pattern shown in the figure below. Water flows into the system at A and exits at B, C, D, and E with the amounts shown. Using the fact that at each point the water entering equals the water leaving, formulate an equation for water flow at each of the five points and solve the system.

a. , , , , (in thousands of gallons)

b. , , , , (in thousands of gallons)

c. , , , , (in thousands of gallons)

d. , , , , (in thousands of gallons)

e. , , , , , (in thousands of gallons)

ANSWER: e

142. A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,300, and risk factors make it prudent to invest some money in all three investments. Let x denote the amount invested at 8 %, y denote the amount invested at 7%, and z denote the amount invested at 10%. Find a general description for the amounts invested at the three rates.

a. , with

b. , with

c. , with

d. , with

e. , with

ANSWER: b

143. A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,600, and risk factors make it prudent to invest some money in all three investments. If $30,000 is invested at 10%, how much will be invested at each of the other rates?

a. $50,000 at 8%; $140,000 at 7%;

b. $20,000 at 8%; $170,000 at 7%;

c. $30,000 at 8%; $160,000 at 7%;

d. $10,000 at 8%; $180,000 at 7%;

e. $40,000 at 8%; $150,000 at 7%

ANSWER: c

144. A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,400, and risk factors make it prudent to invest some money in all three investments. What is the minimum amount that will be invested at 7%, and in this case how much will be invested at the other rates?

a. $100,000 at 8%, $0 at 7%, $120,000 at 10%

b. $90,000 at 8%, $110,000 at 7%, $20,000 at 10%

c. $95,000 at 8%, $115000 at 7%, $10,000 at 10%

d. $100,000 at 8%, $120,000 at 7%, $0 at 10%

e. $90,000 at 8%, $130,000 at 7%, $0 at 10%

ANSWER: d

145. A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,300, and risk factors make it prudent to invest some money in all three investments. What is the maximum amount that will be invested at 7%, and in this case how much will be invested at the other rates?

a. $220,000 at 7%, $0 at 8%, $0 at 10%

b. $180,000 at 7%, $10,000 at 8%, $30,000 at 10%

c. $190,000 at 7%, $0 at 8%, $30,000 at 10%

d. $30,000 at 7%, $0 at 8%, $190,000 at 10%

e. $190,000 at 7%, $20,000 at 8%, $10,000 at 10%

ANSWER: c

146. A brokerage house offers three stock portfolios. Portfolio I consists of 2 blocks of common stock and 1 municipal bond. Portfolio II consists of 4 blocks of common stock, 2 municipal bonds, and 3 blocks of preferred stock. Portfolio III consists of 2 blocks of common stock, 1 municipal bond, and 3 blocks of preferred stock. A customer wants 16 blocks of common stock, 8 municipal bonds, and 6 blocks of preferred stock. Let x equal the number of units of Portfolio I, y equal the number of units of Portfolio II, and z equal the number of units of Portolio III. If the numbers of the three portfolios offered must be integers, find the general solution that can be used to determine all possible offerings.

a. , ,

b. , ,

c. , ,

d. , ,

e. , ,

ANSWER: e

147. If A is a matrix and B is its inverse, what does the product AB equal?

a. zero matrix

b. identity matrix

c. identity matrix

d. zero matrix

e. matrix A

ANSWER: b

148. If and , are C and D inverse matrices?

a. Yes

b. No

ANSWER: Yes

149. If and , does ?

a. Yes

b. No

ANSWER: Yes

150. If and , does ?

a. Yes

b. No

ANSWER: No

151. Find the inverse matrix, if one exists.

a.

b.

c.

d.

e. no inverse exists

ANSWER: d

152. Find the inverse matrix, if one exists.

a.

b.

c.

d.

e. no inverse exists

ANSWER: e

153. Find the inverse matrix, if one exists.

a.

b.

c.

d.

e. no inverse exists

ANSWER: a

154. Find the inverse of matrix A and check it by calculating .

a.

b.

c.

d.

e.

ANSWER: b

155. Find the inverse of matrix A and check it by calculating .

a.

b.

c.

d.

e.

ANSWER: c

156. Find the inverse of matrix A and check it by calculating .

a.

b.

c.

d.

e.

ANSWER: e

157. Find the inverse matrix, if one exists.

a.

b.

c.

d.

e. no inverse exists

ANSWER: d

158. Find the inverse matrix, if one exists.

a.

b.

c.

d.

e. no inverse exists

ANSWER: a

159. Find the inverse matrix, if one exists.

a.

b.

c.

d.

e. no inverse exists

ANSWER: e

160. Find the inverse matrix, if one exists.

a.

b.

c.

d.

e. no inverse exists

ANSWER: e

161. Use a graphing calculator to find the inverse of .

a.

b.

c.

d.

e.

ANSWER: b

162. The inverse of matrix A is given. Use the inverse to solve for X.

Solve .

a.

b.

c.

d.

e.

ANSWER: c

163. The inverse of matrix A is given. Use the inverse to solve for X.

Solve .

a.

b.

c.

d.

e.

ANSWER: a

164. The inverse of matrix A is given. Use the inverse to solve for X.

Solve .

a.

b.

c.

d.

e.

ANSWER: d

165. Use the inverse to solve.

a.

b.

c.

d.

e.

ANSWER: e

166. Use inverse matrices to solve the system of linear equations.

a.

b.

c.

d.

e.

ANSWER: c

167. Use inverse matrices to solve the system of linear equations.

a.

b.

c.

d.

e.

ANSWER: b

168. Use inverse matrices to solve the system of linear equations.

a.

b.

c.

d.

e.

ANSWER: a

169. Use inverse matrices to find the solution of the system of equations.

a.

b.

c.

d.

e.

ANSWER: d

170. Use inverse matrices to find the solution of the system of equations.

a.

b.

c.

d.

e.

ANSWER: c

171. Use inverse matrices to find the solution of the system of equations.

a.

b.

c.

d.

e.

ANSWER: e

172. Use inverse matrices and the graphing calculator to find the solution of the system of equations.

a.

b.

c.

d.

e.

ANSWER: e

173. Evaluate the determinant.

a. –56

b. –28

c. 28

d. –51

e. 13

ANSWER: b

174. Use determinants to decide whether the matrix has an inverse.

a. True

b. False

ANSWER: True

175. Evaluate the determinant.

a. 31

b. 1

c. 8

d. 23

e. 10

ANSWER: a

176. Evaluate the determinant.

a. 33

b. –16

c. –2

d. –17

e. –6

ANSWER: d

177. Use technology to find the determinant.

a. 9

b. 45

c. 25

d. 27

e. 15

ANSWER: c

178. Use technology to find the determinant.

a. 11

b. 6

c. 4

d. 2

e. 8

ANSWER: c

179. Use determinants to decide whether the matrix has an inverse.

a. True

b. False

ANSWER: True

180. Use technology to find the determinant.

a. 387

b. 368

c. 374

d. 366

e. 348

ANSWER: b

181. We have encoded messages by assigning the letters of the alphabet the numbers 1–26, with a blank assigned the number 27, and by using an encoding matrix A that converts the numbers to the coded message. To decode any message encoded by A, we must find the inverse of A, denoted by A-1, and multiply A-1 times the coded message. Use this code and the given encoding matrix A to find the matrix that can be used to decode the given message.

Code matrix is

The message is

138, 59, 72, 29, 143, 58, 57, 24, 101, 44

a.

b.

c.

d.

e.

ANSWER: e

182. If the code matrix is , find the matrix that can be used to decode the message 47, 36, 35, 28, 92, 69, 63, 93, 74, 56, 71, 57, 17, 17, 16. (Because A and A-1 are 3 × 3 matrices, use triples of numbers.)

a.

b.

c.

d.

e.

ANSWER: d

183. We have encoded messages by assigning the letters of the alphabet the numbers 1–26, with a blank assigned the number 27, and by using an encoding matrix A that converts the numbers to the coded message. To decode any message encoded by A, we must find the inverse of A, denoted by A-1, and multiply A-1 times the coded message. Use this code and the given encoding matrix A to find the matrix that can be used to decode the given message.

The code matrix is . The message is 33, 132, 214, 22, 110, 203, 34, 132, 201.

a.

b.

c.

d.

e.

ANSWER: c

184. Suppose that a government study showed that 80% of urban families remained in an urban area in the next generation (and 20% moved to a rural area), whereas 70% of rural families remained in a rural area in the next generation (and 30% moved to an urban area). This means that if represents the current percents of urban families u0 and of rural families r0, then represents the percents of urban families u and rural families r one generation from now. Currently the population is 70% urban and 30% rural. Find the percents in each group one generation before this one.

a.

b.

c.

d.

e.

ANSWER: c

185. Set up the system of equations and then solve it by using inverse matrices.

Transportation The Ace Freight Company has an order for two products to be delivered to two stores of a company. The table below gives information regarding the two products.

Product I Product II

Unit volume (cu ft) 20 30

Unit Weight (lb) 100 400

If truck A can carry 2300 cu ft and 23,000 lb, how many of each product can it carry?

a.

b.

c.

d.

e.

ANSWER: d

186. Set up the system of equations and then solve it by using inverse matrices.

Transportation The Ace Freight Company has an order for two products to be delivered to two stores of a company. The table below gives information regarding the two products.

Product I Product II

Unit volume (cu ft) 20 30

Unit Weight (lb) 100 400

If truck B can carry 2250 cu ft and 20,000 lb, how many of each product can it carry?

a.

b.

c.

d.

e.

ANSWER: a

187. Set up the system of equations and then solve it by using inverse matrices.

Nutrition A biologist has a 40% solution and a 10% solution of the same plant nutrient. How many cubic centimeters of each solution should be mixed to obtain 35 cc of a 28% solution?

a.

b.

c.

d.

e.

ANSWER: e

188. A manufacturer of table saws has three models: Deluxe, Premium, and Ultimate, which must be painted, assembled, and packaged for shipping. The table gives the number of hours required for each of these operations for each type of table saw. If the manufacturer has 98 hours available per day for painting, 150 hours for assembly, and 36 hours for packaging, how many of each type of saw can be produced each day?

a.

b.

c.

d.

e.

ANSWER: b

189. Set up the system of equations and then solve it by using inverse matrices.

Transportation Ace Trucking Company has an order for three products, A, B, and C to be delivered. The table gives the volume in cubic feet, the weight in pounds, and the value for insurance in dollars for a unit of each of the products. If the carrier can carry 8,000 cu ft and 12,400 lb and is insured for $55,800, how many units of each product can be carried?

Product Product Product

A B C

Unit volume (cu ft) 24 20 40

Weight (lb) 40 30 60

Value ($) 150 180 200

a.

b.

c.

d.

e.

ANSWER: c

190. Set up the system of equations and then solve it by using inverse matrices.

Investment A company offers three mutual fund plans for its employees. Plan I consists of 4 blocks of common stock and 2 municipal bonds. Plan II consists of 8 blocks of common stock, 4 municipal bonds, and 6 blocks of preferred stock. Plan III consists of 14 blocks of common stock, 6 municipal bonds, and 6 blocks of preferred stock. An employee wants to combine these plans so that she has 82 blocks of common stock, 38 municipal bonds, and 36 blocks of preferred stock. How many units of each plan does she need?

a.

b.

c.

d.

e.

ANSWER: d

191. Set up the system of equations and then solve it by using inverse matrices.

Bee ancestry Because a female bee comes from a fertilized egg and a male bee comes from an unfertilized egg, the number of ancestors of a male t + 2 bee generations before the present generation is the sum of the number of ancestors t and t + 1 generations before the present. If the numbers of ancestors of a male bee in a given generation t and in the previous generation are given by , then there is a matrix M such that the numbers of ancestors in the two generations preceding generation t are given by .

For a given male bee, the numbers of ancestors 5 and 6 generations back are given by .

Find the numbers of ancestors 4 and 5 generations back by multiplying both sides of by the inverse of M.

a.

b.

c.

d.

e.

ANSWER: b

192. The following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of 1 unit of product.

For each 100 units of manufactured products produced, how many units of fuels are required?

a. 33

b. 2.9

c. 1.1

d. 29

e. 11

ANSWER: e

193. The following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of 1 unit of product.

How many units of utilities are required to produce 30 units of agricultural products?

a. 3

b. 6

c. 9.3

d. 12

e. 3.6

ANSWER: a

194. For the economy, what industry is most dependent on utilities?

a. Agriculture

b. Utilities

c. Manufacturing

d. Fules

e. All industries equally depend on utilities.

ANSWER: c

195. The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries)

For each 1,000 units of raw materials produced, how many units of agricultural and food products were required?

a. 4.1

b. 1.3

c. 41

d. 13

e. 46.4

ANSWER: d

196. The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries)

For each 1,000 units of raw materials produced, how many units of service were required?

a. 14.4

b. 144

c. 9

d. 0.9

e. 159.3

ANSWER: c

197. The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries)

How many units of fuels were required to produce 1,000 units of manufactured goods?

a. 12.1

b. 8

c. 0.8

d. 3.3

e. 33

ANSWER: e

198. The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries)

How many units of fuels were required to produce 1,000 units of power (utilities’ goods)?

a. 0.5

b. 5

c. 54

d. 5.4

e. 59.9

ANSWER: b

199. The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries)

Which industry is most dependent on its own goods for its operations?

a. raw materials

b. agriculture and food

c. service

d. utilities

e. fuels

ANSWER: a

200. The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries)

Which industry is least dependent on its own goods?

a. raw materials

b. manufacturing

c. fules

d. utilites

e. service

ANSWER: e

201. The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries)

Which is most dependent on the service industry?

a. manufacturing

b. utilities

c. service

d. fuels

e. agriculture and food

ANSWER: c

202. The following technology matrix describes the relationship of certain industries within the economy to each other. (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries)

Which industry would be most affected by a rise in the cost of utilites?

a. manufacturing

b. utilities

c. fuels

d. service

e. raw materials

ANSWER: d

203. Suppose a primitive economy consists of two industries, farm products and farm machinery. Suppose also that its technology matrix is

If surpluses of 97 units of farm products and 18 units of farm machinery are desired, find the gross production of each industry.

a. 205 units of farm product, 60 units of machinery

b. 260 units of farm product, 60 units of machinery

c. 210 units of farm product, 55 units of machinery

d. 205 units of farm product, 55 units of machinery

e. 260 units of farm product, 270 units of machinery

ANSWER: d

204. Suppose an economy has two industries, agriculture and minerals. Suppose further that the technology matrix for this economy is A.

If surpluses of 40 agricultural units and 70 mineral units are desired, find the gross production to the nearest unit for each industry.

a. agriculture = 167, minerals = 103

b. agriculture = 65, minerals = 167

c. agriculture = 71, minerals = 96

d. agriculture = 71, minerals = 103

e. agriculture = 65, minerals = 96

ANSWER: c

205. A primitive economy has a mining industry and a fishing industry, with technology matrix A.

If surpluses of 132 units of mining output and 27 units of fishing output are desired, find the gross production of each industry.

a. mining = 180, fishing = 60

b. mining = 180, fishing = 240

c. mining = 200, fishing = 60

d. mining = 200, fishing = 70

e. mining = 240, fishing = 70

ANSWER: a

206. Suppose the economy of an underdeveloped country has an agricultural industry and an oil industry, with technology matrix A.

If surpluses of 0 units of agricultural products and 610 units of oil products are desired, find the gross production of each industry to the nearest unit.

a. agricultural = 800, oil = 825

b. agricultural = 110, oil = 700

c. agricultural = 110, oil = 715

d. agricultural = 100, oil = 715

e. agricultural = 100, oil = 700

ANSWER: e

207. Suppose a simple economy with only an agricultural industry and a steel industry has the following technology matrix.

If surpluses of 15 units of agricultural products and 34 units of steel are desired, find the gross production to the nearest unit of each industry.

a. agriculture = 74, steel = 103

b. agriculture = 175, steel = 174

c. agriculture = 174, steel = 175

d. agriculture = 71, steel = 103

e. agriculture = 74, steel = 101

ANSWER: d

208. One sector of an economy consists of a mining industry and a manufacturing industry, with technology matrix A.

Surpluses of 32 units of mining output and 263 units of manufacturing output are desired. Find the gross production of each industry.

a. mining = 790, manufacturing = 500

b. mining = 290, manufacturing = 500

c. mining = 285, manufacturing = 790

d. mining = 790, manufacturing = 495

e. mining = 290, manufacturing = 495

ANSWER: b

209. An underdeveloped country has an agricultural industry and a manufacturing industry, with technology matrix A.

Surpluses of 10 units of agricultural products and 609 units of manufactured products are desired. Find the gross production of each industry.

a. agriculture = 160, manufacturing = 1210

b. agriculture = 1340, manufacturing = 1210

c. agriculture = 160, manufacturing = 1180

d. agriculture = 1340, manufacturing = 1180

e. agriculture = 170, manufacturing = 1210

ANSWER: c

210. A simple economy has an electronic components industry and a computers industry. Each unit of electronic components output requires inputs of 0.3 unit of electronic components and 0.2 unit of computers. Each unit of computer industry output requires inputs of 0.8 unit of electronic components and 0.1 unit of computers. Write the technology matrix for this simple economy. (E, electronics components industry; C, computer industry)

a.

b.

c.

d.

e.

ANSWER: c

211. A simple economy has an electronic components industry and a computers industry. Each unit of electronic components output requires inputs of 0.3 unit of electronic components and 0.6 unit of computers. Each unit of computer industry output requires inputs of 0.2 unit of electronic components and 0.2 unit of computers. If surpluses of 641 units of electronic components and 10 units of computers are desired, find the gross production to the nearest unit of each industry.

a. electronics = 1170, computers = 865

b. electronics = 1170, computers = 890

c. electronics = 1207, computers = 2072

d. electronics = 1207, computers = 865

e. electronics = 2072, computers = 2060

ANSWER: b

212. An economy has an agricultural industry and a textile industry. Each unit of agricultural output requires 0.4 unit of agricultural input and 0.2 unit of textiles input. Each unit of textiles output requires 0.1 unit of agricultural input and 0.2 unit of textiles input. Write the technology matrix for this economy. (A, agricultural industry; T, textile industry)

a.

b.

c.

d.

e.

ANSWER: e

213. An economy has an agricultural industry and a textile industry. Each unit of agricultural output requires 0.4 unit of agricultural input and 0.1 unit of textiles input. Each unit of textiles output requires 0.1 unit of agricultural input and 0.2 unit of textiles input. If surpluses of 5 units of agricultural products and 195 units of textiles are desired, find the gross production of each industry.

a. agriculture = 60, textiles = 240

b. agriculture = 50, textiles = 250

c. agriculture = 60, textiles = 250

d. agriculture = 300, textiles = 250

e. agriculture = 60, textiles = 300

ANSWER: b

214. An economy has a manufacturing industry and a banking industry. Each unit of manufacturing output requires inputs of 0.4 unit of manufacturing and 0.1 unit of banking. Each unit of banking output requires inputs of 0.3 unit each of manufacturing and banking. Write the technology matrix for this economy. (M, manufacturing industry; B, banking industry)

a.

b.

c.

d.

e.

ANSWER: d

215. An economy has a manufacturing industry and a banking industry. Each unit of manufacturing output requires inputs of 0.5 unit of manufacturing and 0.2 unit of banking. Each unit of banking output requires inputs of 0.3 unit each of manufacturing and banking. If surpluses of 133 units of manufacturing and 100 units of banking are desired, find the gross production of each industry.

a. manufacturing =424, banking =264

b. manufacturing = 390, banking = 320

c. manufacturing = 380, banking = 310

d. manufacturing = 380, banking = 320

e. manufacturing = 390, banking = 310

ANSWER: a

216. Suppose the development department of a firm charges 10% of its total monthly costs to the promotional department, and the promotional department charges 5% of its total monthly costs to the development department. The direct costs of the development department are $25,400, and the direct costs of the promotional department are $9,400. The solution to the matrix equation or the column matrix gives the total costs for each department. Note that this equation can be written in the form or , which is the form for an open economy. Find the total costs for each department in this (micro)economy.

a. development = 27,000, promotional = 11,000

b. development = 27,000, promotional = 12,000

c. development = 26,000, promotional = 12,000

d. development = 38,000, promotional = 12,000

e. development = 26,000, promotional = 11,000

ANSWER: c

217. Suppose the shipping department of a firm charges 20% of its total monthly costs to the printing department and that the printing department charges 10% of its total monthly costs to the shipping department. If the direct costs of the shipping department are $18,000 and the direct costs of the printing department are $11,000, find the total costs for each department. Round your answer to the nearest hundredths.

a. shipping = 21,500, printing = 14,000

b. shipping = 19,500, printing = 16,000

c. shipping = 21,500, printing = 15,000

d. shipping = 14,000, printing = 21,500

e. shipping = 19,500, printing = 15,000

ANSWER: e

218. The sales department of an auto dealership charges 10% of its total monthly costs to the service department, and the service department charges 20% of its total monthly costs to the sales department. During a given month, the direct costs are $87,100 for sales and $49,600 for service. Find the total costs of each department.

a. sales = 69,500, service = 59,000

b. sales = 99,000, service = 59,500

c. sales = 87,100, service = 49,600

d. sales = 98,000, service = 59,000

e. sales = 99,000, service = 59,000

ANSWER: b

219. Suppose that an economy can be represented by the technology matrix below. If surpluses of 180 units of fishing output, 93 units of agricultural goods, and 40 units of mining goods are desired, find the gross production to the nearest unit of each industry.

a. fishing = 632, agriculture = 796, mining = 570

b. fishing = 633, agriculture = 796, mining = 568

c. fishing = 632, agriculture = 796, mining = 568

d. fishing = 632, agriculture = 793, mining = 568

e. fishing = 633, agriculture = 793, mining = 570

ANSWER: d

220. Suppose that the economy of a small nation has an electronics industry, a steel industry, and an auto industry, with the following technology matrix.

If the nation wishes to have surpluses of 104 units of electronics production, 271 units of steel production, and 200 automobiles, find the gross production of each industry.

a. electronics = 1254, steel = 1262, autos = 722

b. electronics = 1252, steel = 1262, autos = 724

c. electronics = 1254, steel = 1269, autos = 722

d. electronics = 1254, steel = 1269, autos = 724

e. electronics = 1252, steel = 1262, autos = 722

ANSWER: e

221. Suppose an economy has the technology matrix below. If surpluses of 525 units of electronics, 31 units of steel, and 140 autos are desired, find the gross production for each industry.

a. electronics = 2150, steel = 985, autos = 690

b. electronics = 2150, steel = 2152, autos = 690

c. electronics = 2150, steel = 690, autos = 985

d. electronics = 2152, steel = 985, autos = 690

e. electronics = 2152, steel = 690, autos = 992

ANSWER: a

222. Suppose that an economy has the technology matrix below. If surpluses of 56 units of manufactured goods and 104 units of agricultural goods are desired, find the gross production of each industry.

a. service = 75, manufacturing = 220, agriculture = 200

b. service = 75, manufacturing = 220, agriculture = 210

c. service = 70, manufacturing = 220, agriculture = 200

d. service = 70, manufacturing = 250, agriculture = 200

e. service = 75, manufacturing = 250, agriculture = 210

ANSWER: c

223. Suppose the technology matrix for a closed model of a simple economy is given by

Find the gross productions for the industries.

a. products = households, machinery = households

b. products = households, machinery = households

c. products = households, machinery = households

d. products = households, machinery = households

e. products = households, machinery = households

ANSWER: b

224. Suppose the technology matrix for a closed model of a simple economy is given by matrix A. Find the gross productions for the industries.

a. government = households, industry = households

b. government = households, industry = households

c. government = households, industry = households

d. government = households, industry = households

e. government = households, industry = households

ANSWER: e

225. Suppose the technology matrix for a closed model of an economy is given by matrix A. Find the gross productions for the industries.

a. shipping = households, manufacturing = households

b. shipping = households, manufacturing = households

c. shipping = households, manufacturing = households

d. shipping = households, manufacturing = households

e. shipping = households, manufacturing = households

ANSWER: d

226. A closed model for an economy has a manufacturing industry, utilities industry, and households industry. Each unit of manufacturing output uses 0.6 unit of manufacturing input, 0.4 unit of utilities input, and 0.1 unit of households input. Each unit of utilities output requires 0.3 unit of manufacturing input, 0.5 unit of utilities input, and 0.1 unit of households input. Each unit of household output requires 0.2 unit each of manufacturing and utilities input and 0.4 unit of households input. Write the augmented matrix for this closed model of the economy.

a.

b.

c.

d.

e.

ANSWER: b

227. A closed model for an economy has a manufacturing industry, utilities industry, and households industry. Each unit of manufacturing output uses 0.5 unit of manufacturing input, 0.4 unit of utilities input, and 0.1 unit of households input. Each unit of utilities output requires 0.4 unit of manufacturing input, 0.5 unit of utilities input, and 0.1 unit of households input. Each unit of household output requires 0.4 unit each of manufacturing and utilities input and 0.2 unit of households input. Find the gross production for each industry.

a. manufacturing = 6 households, utilities = 7 households

b. manufacturing = households, utilities = households

c. manufacturing = 2 households, utilities = 2 households

d. manufacturing = 3 households, utilities = 3 households

e. manufacturing = 4 households, utilities = 4 households

ANSWER: e

228. A closed model for an economy identifies government, the profit sector, the nonprofit sector, and households as its industries. Each unit of government output requires 0.3 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.4 unit of households input. Each unit of profit sector output requires 0.2 unit of government input, 0.3 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.4 unit of households input. Each unit of nonprofit sector output requires 0.2 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.6 unit of households input. Each unit of households output requires 0.05 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.25 unit of households input. Write the augmented matrix for this closed model of the economy.

a.

b.

c.

d.

e.

ANSWER: c

229. A closed model for an economy identifies government, the profit sector, the nonprofit sector, and households as its industries. Each unit of government output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input. Each unit of profit sector output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input. Each unit of nonprofit sector output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input. Each unit of households output requires unit of government input, unit of profit sector input, unit of nonprofit sector input, and unit of households input. Find the gross production for each industry.

a. government = households, profit = households, nonprofit = households

b. government = households, profit = households, nonprofit = households

c. government = households, profit = households, nonprofit = households

d. government = households, profit = households, nonprofit = households

e. government = households, profit = households, nonprofit = households

ANSWER: a

230. Card tables are made by joining 4 legs and a top using 4 bolts. The legs are each made from a steel rod. The top has a frame made from 4 steel rods. A cover and four special clamps that brace the top and hold the legs are joined to the frame using a total of 8 bolts. The parts-listing matrix for the card table assembly is given by

If an order is received for 10 card tables, 4 legs, 1 top, 2 covers, 6 clamps, and 13 bolts, how many of each primary assembly item are required to fill the order?

a. 88 rods, 13 covers, 50 clamps, 141 bolts

b. 88 rods, 88 covers, 141 clamps, 141 bolts

c. 50 rods, 141 covers, 50 clamps, 141 bolts

d. 88 rods, 13 covers, 50 clamps, 141 bolts

e. 141 rods, 13 covers, 88 clamps, 88 bolts

ANSWER: d

231. A log carrier has a body made from a 4-ft length of reinforced cloth having a patch on each side and a dowel slid through each end to act as handles. The parts-listing matrix for the log carrier is given by

How many of each primary assembly items are required to fill an order for 500 log carriers and 30 handles?

a. 2060 handles, 970 cloths, 500 patches

b. 1030 handles, 2060 cloths, 1000 patches

c. 1030 handles, 500 cloths, 500 patches

d. 1030 handles, 970 cloths, 2060 patches

e. 1030 handles, 500 cloths, 2060 patches

ANSWER: c

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