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All Textbook Solutions for Finite Mathematics for Business, Economics, Life Sciences and Social Sciences

Solve Problems 55-74 using augmented matrix methods: 4x1+6x2=86x19x2=12 Solve Problems 55-74 using augmented matrix methods: 2x1+4x2=43x16x2=6 Solve Problems 75-80 using augmented matrix methods. 3x1x2=72x1+3x2=1 Solve Problems 75-80 using augmented matrix methods. 2x13x2=85x1+3x2=1 Solve Problems 75-80 using augmented matrix methods. 3x1+2x2=42x1x2=5 Solve Problems 75-80 using augmented matrix methods. 4x1+3x2=263x111x2=7 Solve Problems 75-80 using augmented matrix methods. 0.2x10.5x2=0.070.8x10.3x2=0.79 Solve Problems 75-80 using augmented matrix methods. 0.3x10.6x2=0.180.5x10.2x2=0.54 Solve Problems 81-84 using augmented matrix methods. Use a graphing calculator to perform the row operations. 0.8x1+2.88x2=41.25x1+4.34x2=5 Solve Problems 81-84 using augmented matrix methods. Use a graphing calculator to perform the row operations. 2.7x115.12x2=273.25x118.52x2=33 Solve Problems 81-84 using augmented matrix methods. Use a graphing calculator to perform the row operations. 4.8x140.32x2=295.23.75x1+28.7x2=211.2 Solve Problems 81-84 using augmented matrix methods. Use a graphing calculator to perform the row operations. 5.7x18.55x2=35.914.5x1+5.73x2=76.17 Explain why the definition of reduced form ensures that each leftmost variable in a reduced system appears in one and only one equation and no equation contains more than one leftmost variable. Discuss methods for determining whether a consistent system is independent or dependent by examining the reduced form.Refer to Example 6. The rental company charges $19.95 per day for a 10 -foot truck, $29.95 per day for a 14 -foot truck, and $39.95 per day for a 24 -foot truck. Which of the four possible choices in the table would produce the largest daily income from truck rentals?The matrices below are not in reduced form. Indicate which condition in the definition is violated for each matrix. State the row operation(s) required to transform the matrix into reduced form, and find the reduced form. A100326B154012000310 C010100001302D120000001304 Solve by Gauss-Jordan elimination: 3x1+x22x3=2x12x2+x3=32x1x23x3=3 Solve by Gauss-Jordan elimination: 2x14x2x3=84x18x2+3x3=42x1+4x2+x3=11 Solve by Gauss-Jordan elimination: 2x12x24x3=23x13x26x3=32x1+3x2+x3=7 Solve by Gauss-Jordan elimination: x1x2+2x32x5=32x1+2x24x3x4+x5=53x13x2+7x3+x44x5=6 A company that rents small moving trucks wants to purchase 16 trucks with a combined capacity of 19,200 cubic feet. Three different types of trucks are available: a cargo van with a capacity of 300 cubic feet, a 15 foot truck with a capacity of 900 cubic feet, and a 24-foot truck with a capacity of 1,500 cubic feet. How many of each type of truck should the company purchase?In Problems 1-4, write the augmented matrix of the system of linear equations x1+2x2+3x3=12x1+7x25x3=15 In Problems 1-4, write the augmented matrix of the system of linear equations 4x1+x2+=83x15x2=6x1+9x2=4 In Problems 1-4, write the augmented matrix of the system of linear equations. x1++6x3=2x2x3=5x1+3x2=7 In Problems 14, write the augmented matrix of the system of linear equations. 3x1+4x2=10x1+5x3=15x2+x3=20 In Problems 5-8, write the system of linear equations that is represented by the augmented matrix. Assume that the variables are x1,x2.... 133216453 In Problems 5-8, write the system of linear equations that is represented by the augmented matrix. Assume that the variables are x1,x2.... 15240387 In Problems 5-8, write the system of linear equations that is represented by the augmented matrix. Assume that the variables are x1,x2..... 52084 In Problems 5-8, write the system of linear equations that is represented by the augmented matrix. Assume that the variables are x1,x2.... 101110021135 In Problems 9-18, if a matrix is in reduced form, say so. If not, explain why and indicate a row operation that completes the next step of Gauss-Jordan elimination. 100132 In Problems 9-18, if a matrix is in reduced form, say so. If not, explain why and indicate a row operation that completes the next step of Gauss-Jordan elimination. 011032 In Problems 9-18, if a matrix is in reduced form, say so. If not, explain why and indicate a row operation that completes the next step of Gauss-Jordan elimination. 011051 In Problems 9-18, if a matrix is in reduced form, say so. If not, explain why and indicate a row operation that completes the next step of Gauss-Jordan elimination. 100151 In Problems 9-18, if a matrix is in reduced form, say so. If not, explain why and indicate a row operation that completes the next step of Gauss-Jordan elimination. 363022051970 In Problems 9-18, if a matrix is in reduced form, say so. If not, explain why and indicate a row operation that completes the next step of Gauss-Jordan elimination. 51050220511570 In Problems 9-18, if a matrix is in reduced form, say so. If not, explain why and indicate a row operation that completes the next step of Gauss-Jordan elimination. 102012051970 In Problems 9-18, if a matrix is in reduced form, say so. If not, explain why and indicate a row operation that completes the next step of Gauss-Jordan elimination. 11050020001560 In Problems 9-18, if a matrix is in reduced form, say so. If not, explain why and indicate a row operation that completes the next step of Gauss-Jordan elimination. 122002000980 In Problems 9-18, if a matrix is in reduced form, say so. If not, explain why and indicate a row operation that completes the next step of Gauss-Jordan elimination. 1050120511570 In Problems 19-28, write the solution of the linear system corresponding to each reduced augmented matrix. 100010001230 In Problems 19-28, write the solution of the linear system corresponding to each reduced augmented matrix. 10000100001000012013 In Problems 19-28, write the solution of the linear system corresponding to each reduced augmented matrix. 102011000350 In Problems 19-28, write the solution of the linear system corresponding to each reduced augmented matrix. 120001000350 In Problems 19-28, write the solution of the linear system corresponding to each reduced augmented matrix. 100010001 In Problems 19-28, write the solution of the linear system corresponding to each reduced augmented matrix. 100010530 In Problems 19-28, write the solution of the linear system corresponding to each reduced augmented matrix. 10301257 In Problems 19-28, write the solution of the linear system corresponding to each reduced augmented matrix. 10101146 In Problems 19-28, write the solution of the linear system corresponding to each reduced augmented matrix. 1203001352 In Problems 19-28, write the solution of the linear system corresponding to each reduced augmented matrix. 1023011241 In which of Problems 19,21,23,25and27 is the number of leftmost ones equal to the number of variables?In which of Problems 20,22,24,26and28 is the number of leftmost ones equal to the number of variables?In which of Problems 19,21,23,25and27 is the number of leftmost ones less than the number of variables?In which of Problems 20,22,24,26and28 is the number of leftmost ones less than the number of variables?In Problems 33-38, discuss the validity of each statement about linear systems. If the statement is always true, explain why. If not, give a counter example. If the number of leftmost ones is equal to the number of variables, then the system has exactly one solution.In Problems 33-38, discuss the validity of each statement about linear systems. If the statement is always true, explain why. If not, give a counter example. If the number of leftmost ones is less than the number of variables, then the system has infinitely many solutions.In Problems 33-38, discuss the validity of each statement about linear systems. If the statement is always true, explain why. If not, give a counter example. If the number of leftmost ones is equal to the number of variables and the system is consistent, then the system has exactly one solution.In Problems 33-38, discuss the validity of each statement about linear systems. If the statement is always true, explain why. If not, give a counter example. If the number of leftmost ones is less than the number of variables and the system is consistent, then the system has infinitely many solutions.In Problems 33-38, discuss the validity of each statement about linear systems. If the statement is always true, explain why. If not, give a counter example. If there is an all-zero row, then the system has infinitely many solutions.In Problems 33-38, discuss the validity of each statement about linear systems. If the statement is always true, explain why. If not, give a counter example. If there are no all-zero rows, then the system has exactly one solution.Use row operations to change each matrix in Problems 39-46 to reduced form. 120113 Use row operations to change each matrix in Problems 39-46 to reduced form. 130214 Use row operations to change each matrix in Problems 39-46 to reduced form. 1112341625 Use row operations to change each matrix in Problems 39-46 to reduced form. 111357830 Use row operations to change each matrix in Problems 39-46 to reduced form 103101200036.Use row operations to change each matrix in Problems 39-46 to reduced form. 104013002012 Use row operations to change each matrix in Problems 39-46 to reduced form. 122036012131 Use row operations to change each matrix in Problems 39-46 to reduced form. 126028014421 Solve Problems 47-62 using Gauss-Jordan elimination. 2x1+4x210x3=23x1+9x221x3=0x1+5x212x3=1 Solve Problems 47-62 using Gauss-Jordan elimination. 3x1+5x2x3=7x1+x2+x3=12x1+11x3=7 Solve Problems 47-62 using Gauss-Jordan elimination. 3x1+8x2x3=182x1+x25x3=82x1+4x22x3=4 Solve Problems 47-62 using Gauss-Jordan elimination. 2x1+6x2+15x3=124x1+7x213x3=103x1+6x212x3=9 Solve Problems 47-62 using Gauss-Jordan elimination. 2x1x23x3=8x12x2=7 Solve Problems 47-62 using Gauss-Jordan elimination. 2x1+4x26x3=103x1+3x23x3=6 Solve Problems 47-62 using Gauss-Jordan elimination. 2x1x2=03x1+2x=27x1x2=1 Solve Problems 47-62 using Gauss-Jordan elimination. 2x1x2+=03x1+2x2=7x1x2=2 Solve Problems 47-62 using Gauss-Jordan elimination. 3x14x2x3=12x13x2+x3=1x12x2+3x3=2 Solve Problems 47-62 using Gauss-Jordan elimination. 3x1+7x2x3=11x1+2x2x3=32x1+4x22x3=10 Solve Problems 47-62 using Gauss-Jordan elimination. 3x12x2+x3=72x1+x24x3=0x1+x23x3=1 Solve Problems 4762 using Gauss-Jordan elimination. 2x1+3x2+5x3=21x1x25x3=22x1+x2x3=11 Solve Problems 47-62 using Gauss-Jordan elimination. 2x1+4x22x3=23x1+6x2+3x3=3 Solve Problems 47-62 using Gauss-Jordan elimination. 3x19x2+12x3=62x1+6x28x3=4 Solve Problems 47-62 using Gauss-Jordan elimination. 4x1x2+2x3=34x1+x23x3=108x1+2x2+9x3=1 Solve Problems 47-62 using Gauss-Jordan elimination. 4x12x2+2x3=56x1+3x23x3=210x15x2+9x3=4 Consider a consistent system of three linear equations in three variables. Discuss the nature of the system and its solution set if the reduced form of the augmented coefficient matrix has AOneleftmost1BTwoleftmost1sCThreeleftmost1sDFourleftmost1s Consider a system of three linear equations in three variables. Give examples of two reduced forms that are not row-equivalent if the system is AConsistentanddependentBInconsistent Solve Problems 65-70 using Gauss-Jordan elimination x1+2x24x3x4=72x1+5x29x34x4=16x1+5x27x37x4=13 Solve Problems 65-70 using Gauss-Jordan elimination. 2x1+4x+25x3+4x4=8x1+2x+22x3+x4=3 Solve Problems 65-70 using Gauss-Jordan elimination. x1x+23x32x4=12x1+4x23x3+x4=0.53x1x+210x34x4=2.94x13x2+8x32x4=0.6 Solve Problems 65-70 using Gauss-Jordan elimination. x1+x+24x3+x4=1.3x1+x2x3=1.12x1+x3+3x4=4.42x+15x2+11x3+3x4=5.6 Solve Problems 65-70 using Gauss-Jordan elimination. x12x2+x3+x4+2x5=22x1+4x2+2x3+2x42x5=03x16x2+x3+x4+5x5=4x1+2x2+3x3+x4+x5=3 Solve Problems 65-70 using Gauss-Jordan elimination. x13x+2x3+x4+2x5=2x1+5x+22x3+2x42x5=02x16x+22x3+2x4+4x5=4x+13x2x3+x5=3 Find a,bandc so that the graph of the quadratic equation y=ax2+bx+c passes through the points 2,9,1,9and4,9.Find a,b,andc so that the graph of the quadratic equation y=ax2+bx+c passes through the points 1,5,2,7and5,1.Boat production. A small manufacturing plant makes three types of inflatable boats: one-person, two-person, and four-person models. Each boat requires the services of three departments, as listed in the table. The cutting, assembly, and packaging departments have available a maximum of 380,330and120 labor-hours per week, respectively (A) How many boats of each type must be produced each week for the plant to operate at full capacity? (B) How is the production schedule in part (A) affected if the packaging department is no longer used? (C) How is the production schedule in part (A) affected if the four-person boat is no longer produced?Production scheduling. Repeat Problem 73 assuming that the cutting, assembly, and packaging departments have available a maximum of 350,330and115 labor-hours per week, respectively.Tank car leases. A chemical manufacturer wants to lease a fleet of 24 railroad tank cars with a combined carrying capacity of 520,000 gallons. Tank cars with three different carrying capacities are available: 8,000 gallons, 16,000 gallons, and 24,000 gallons. How many of each type of tank car should be leased?Airplane leases. A corporation wants to lease a fleet of 12 airplanes with a combined carrying capacity of 220 passengers. The three available types of planes carry 10,15and20 passengers, respectively. How many of each type of plane should be leased?Tank car leases. Refer to Problem 75. The cost of leasing an 8,000 -gallon tank car is $450 per month, a 16,000 -gallon tank car is $650 per month, and a 24,000 -gallon tank car is $1,150 per month. Which of the solutions to Problem 75 would minimize the monthly leasing cost?Airplane leases. Refer to Problem 76. The cost of leasing a 10 -passenger airplane is $8,000 per month, a 15-passenger airplane is $14,000 per month, and a 20-passenger airplane is $16,000 per month. Which of the solutions to Problem 76 would minimize the monthly leasing cost?Income tax. A corporation has a taxable income of $7,650,000. At this income level, the federal income tax rate is 50 the state lax rate is 20, and the local tax rate is 10. If each tax rate is applied to the total taxable income, the resulting tax liability for the corporation would be 80 of taxable income. However, it is customary to deduct taxes paid to one agency before computing taxes for the other agencies. Assume that the federal taxes are based on the income that remains after the state and local taxes are deducted, and that state and local taxes are computed in a similar manner. What is the tax liability of the corporation (as a percentage of taxable income) if these deductions are taken into consideration?Income tax. Repeat Problem 79 if local taxes are not allowed as a deduction for federal and state taxes.Taxable income. As a result of several mergers and acquisitions, stock in four companies has been distributed among the companies. Each row of the following table gives the percentage of stock in the four companies that a particular company owns and the annual net income of each company (in millions of dollars): So company A holds 71 of its own stock, 8 of the stock in company B, 3 of the stock in company C.etc. For the purpose of assessing a state tax on corporate income, the taxable income of each company is defined to be its share of its own annual net income plus its share of the taxable income of each of the other companies, as determined by the percentages in the table. What is the taxable income of each company (to the nearest thousand dollars)?Taxable income. Repeat Problem 81 if tax law is changed so that the taxable income of a company is defined to be all of its own annual net income plus its share of the taxable income of each of the other companies.Nutrition. A dietitian in a hospital is to arrange a special diet composed of three basic foods. The diet is to include exactly 340 units of calcium, 180 units of iron, and 220 units of vitamin A. The number of units per ounce of each special ingredient for each of the foods is indicated in the table. (A) How many ounces of each food must be used to meet the diet requirements? (B) How is the diet in part (A) affected if food C is not used? (C) How is the diet in part (A) affected if the vitamin A requirement is dropped?Nutrition. Repeat Problem 83 if the diet is to include exactly 400 units of calcium, 160 units of iron, and 240 units of vitamin A.Plant food. A farmer can buy four types of plant food. Each barrel of mix A contains 30 pounds of phosphoric acid. 50 pounds of nitrogen, and 30 pounds of potash; each barrel of mix B contains 30 pounds of phosphoric acid. 75 Pounds of nitrogen, and 20 pounds of potash; each barrel of mix C contains 30 pounds of phosphoric acid. 25 pounds of nitrogen, and 20 pounds of potash; and each barrel of mix D contains 60 pounds of phosphoric acid. 25 pounds of nitrogen, and 50 pounds of potash. Soil tests indicate that a particular field needs 900 pounds of phosphoric acid, 750 pounds of nitrogen, and 700 pounds of potash. How many barrels of each type of food should the farmer mix together to supply the necessary nutrients for the field?Animal feed. In a laboratory experiment, rats are to be fed 5 packets of food containing a total of 80 units of vitamin E. There are four different brands of food packets that can be used. A packet of brand A contains 5 units of vitamin E. a packet of brand B contains 10 units of vitamin E, a packet of brand C contains 15 units of vitamin E, and a packet of brand D contains 20 units of vitamin E. How many packets of each brand should be mixed and fed to the rats?Plant food. Refer to Problem 85. The costs of the four mixes are Mix A,$46 ; Mix B,$72 ; Mix C,$57 ; and Mix D,$63. Which of the solutions to Problem 85 would minimize the cost of the plant food?Animal feed. Refer to Problem 86.The costs of the four brands are Brand A,$1.50 ; Brand B,$3.00 ; Brand C,$3.75 ; and Brand D,$2.25. Which of the solutions to Problem 86 would minimize the cost of the rat food?Population growth. The U.S. population was approximately 75 million in 1900, 150 million in 1950 and 275 million in 2000. Construct a model for this data by finding a quadratic equation whose graph passes through the points 0,75,50,150and100,275 Use this model to estimate the population in 2050.Population growth. The population of California was approximately 30 million in 1990, 34 million in 2000 and 37 million in 2010. Construct a model for this data by finding a quadratic equation whose graph passes through the points 0,30,10,34and20,37. Use this model to estimate the population in 2030. Do you think the estimate is plausible? Explain. (Source: US Census Bureau)Female life expectancy. The life expectancy for females born during 1980-1985 was approximately 77.6 years. This grew to 78 years during 1985-1990 and to 78.6 years during 1990-1995. Construct a model for this data by finding a quadratic equation whose graph passes through the points 0.77,6,5,78and10.78,6. Use this model to estimate the life expectancy for females born between 1995 and 2000 and for those born between 2000and2005.Male life expectancy. The life expectancy for males born during 1980-1985 was approximately 70.7 years. This grew to 71.1 years during 1985-1990 and to 71.8 years during 1990-1995. Construct a model for this data by finding a quadratic equation whose graph passes through the points 0,70.7,5,71.1and10,71.8. Use this model to estimate the life expectancy far males born between 1995and000 and for those born between 2000and2005.Female life expectancy. Refer to Problem 91. Subsequent data indicated that life expectancy grew to 79.1 years for females born during 1995-2000 and to 79.7 years for females born during 2000-2005. Add the points 15,79.1,and20,79.7 to the data set in Problem 91. Using a graphing calculates to find a quadratic regression model for all foe data points. Graph the data and the model in the same viewing window.Male life expectancy. Refer to Problem 92. Subsequent data indicated that life expectancy grew to 73.2 years for males born during 1995-2000 and to 74.3 years for males born during 2000-2005. Add the points 15,73.2and20,74.3 to the data set in Problem 92. Use a graphing calculator to find a quadratic regression model for all five data points. Graph the data and the model in the same viewing window.Sociology. Two sociologists have grant money to study school busing in a particular city. They wish to conduct an opinion survey using 600 telephone contacts and 400 house contacts. Survey company A has personnel to do 30 telephone and 10 house contacts per hour: survey company B can handle 20 telephone and 20 house contacts per hour. How many hours should be scheduled for each firm to produce exactly the number of contacts needed?Sociology. Repeat Problem 95 if 650 telephone contacts and 350 house contacts are needed.Traffic Wow. The rush-hour traffic flow for a network of four one-way streets m a city is shown in the figure. The numbers next to each street indicate the number of vehicles per hour that enter and leave the network on that street. The variables x1,x2andx3 represent the flow of traffic between the four intersections in the network. (A) For a smooth traffic the number of vehicles entering each intersection should always equal the number leaving. For example, since 1,500 vehicles enter the intersection of 5th Street and Washington Avenue each hour and x1+x4 vehicles leave this intersection, we see that x1+x4=1,500. Find the equations determined by the traffic flow at each of the other three intersections. (B) Find (the solution to the system in part (A). (C) What is the maximum number of vehicles that can travel from Washington Avenue to Lincoln Avenue on 5th Street? What is the minimum number? (D) If traffic lights are adjusted so that 1,000 vehicles per hour travel from Washington Avenue to Lincoln Avenue on 5th Street, determine the flow around the rest of the network.Traffic flow. Refer to Problem 97. Closing Washington Avenue east of 6th Street for construction changes the traffic flow for the network as indicated m the figure. Repeat parts (A)-(D) of Problem 97 for this traffic flow.In addition to the commutative and zero properties, there are other significant differences between real number multiplication and matrix multiplication. (A) In real number multiplication, the only real number whose square is 0 is the real number 002=0. Find at least one 22 matrix A with all elements nonzero such that A2=0, where 0 is the 22 zero matrix. (B) In real number multiplication, the only nonzero real number that is equal to its square is the real number 112=1. Find at least one 22 matrix B with all elements nonzero such that B2=B.Add: 321013+231212 Subtract: 235321 Find a,b,c,andd so that abcd4213=2582 Find: 101.30.23.5 Repeat Example 5 with A=$45,000$77,000$106,000$22,000andB=$190,000$345,000$266,000$276,000 10322341=?If the factory in Example 7 also produces a trick water ski that requires 5 labor-hours in the assembly department and 1.5 labor-hours in the finishing department, write a product between appropriate row and column matrices that will give the total labor cost for this ski. Compute the cost.Find each product, if it is defined: A11023220121130B12113011023220 C11222142D21421122 E321423F423321 Find a,b,c, and d so that 6053acbd=1624646 Refer to Example 10. The company wants to know how many hours to schedule in each department in order to produce 2,000 trick skis and 1,000 slalom skis. These production requirements can be represented by either of the following matrices: TrickskisSlalomskisP=2,0001,000Q=2,0001,000TrickskisSlalomskis Using the labor-hour matrix L from Example 10, find PL or LQ, whichever has a meaningful interpretation for this problem, and label the rows and columns accordingly,In Problems 1-14, perform the indicated operation, if possible. 15+310 In Problems 1-14, perform the indicated operation, if possible. 3274 In Problems 1-14, perform the indicated operation, if possible. 20360410 In Problems 1-14, perform the indicated operation, if possible. 9280+9008 In Problems 1-14, perform the indicated operation, if possible. 36+19 In Problems 1-14, perform the indicated operation, if possible. 4286513 7E In Problems 1-14, perform the indicated operation, if possible. 1012+43 In Problems 1-14, perform the indicated operation, if possible. 341212 In Problems 1-14, perform the indicated operation, if possible. 112342 In Problems 1-14, perform the indicated operation, if possible. 23121102 In Problems 1-14, perform the indicated operation, if possible. 32412513 In Problems 1-14, perform the indicated operation, if possible. 11022312 14E In Problems 15-22, find the matrix product. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 50051234 In Problems 15-22, find the matrix product. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 30031357 In Problems 15-22, find the matrix product. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 12345005 In Problems 15-22, find the matrix product. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 13573003 In Problems 15-22, find the matrix product. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 01003579 In Problems 15-22, find the matrix product. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 00102468 In Problems 15-22, find the matrix product. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 35790100 In Problems 15-22, find the matrix product. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 24680010 Find the products in Problems 23-30. 3254 Find the products in Problems 23-30. 3524 Find the products in Problems 23-30. 5432 Find the products in Problems 23-30. 2435 Find the products in Problems 23-30. 101213 Find the products in Problems 23-30. 113201 Find the products in Problems 23-30. 213101 Find the products in Problems 23-30. 201113 Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 AC Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 CA Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 AB Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 BA Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 B2 Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 C2 Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 B+AD Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 C+DA Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 0.1DB Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 0.2CD Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 3BA+4AC Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 2DB+5CD Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 2BA+6CD Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 1AC+3DB Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 ACD Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 CDA Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 DBA Problems 31-48 refer to the following matrices: A=201432B=3125C=102431235D=301212 BAD If aandb are nonzero real numbers A=aabb,andB=aaaa find AB and BA.If aandb are nonzero real numbers, A=abab,andB=aaaa find AB and BA.If aandb are nonzero real numbers and A=abb2a2ab Find A2.If aandb are nonzero real numbers and A=abab1ab2ab Find A2.In Problems 53-54, use a graphing calculator to calculate B,B2,B3...andAB,AB2,AB3 describe any patterns you observe in each sequence of matrices. A=0.30.7andB=0.40.60.20.8 In Problems 53-54, use a graphing calculator to calculate B,B2,B3...andAB,AB2,AB3 describe any patterns you observe in each sequence of matrices. A=0.40.6andB=0.90.10.30.7 Find a,b,c, and d so that abcd+2301=1234 Find w,x,y, and z so that 4230+wxyz=2305 Find a,b,c, and d so that 1223abcd=1032 Find a,b,c, and d so that 1314abcd=6577 In Problems 59-62, determine whether the statement is true or false. There exist two 11 matrices AandB such that ABBA.In Problems 59-62, determine whether the statement is true or false. There exist two 22 matrices AandB such that ABBA.In Problems 59-62, determine whether the statement is true or false. There exist two 22 matrices AandB such that AB is the 22 zero matrix.In Problems 59-62, determine whether the statement is true or false. There exist two 11 matrices AandB such that AB is the 11 zero matrix.A square matrix is a diagonal matrix if all elements not on the principal diagonal are zero. So a 22 diagonal matrix has the form A=a00d Where a and d are real numbers. Discuss the validity of each of the following statements. If the statement is always true, explain why. If not, give examples. (A) IfAandBare22 diagonal matrices, then A+B is a 22 diagonal matrix. (B) IfAandBare22 diagonal matrices, then AB is a 22 diagonal matrix. (C) IfAandBare22 diagonal matrices, then AB=BA.A square matrix is an upper triangular matrix if all elements below the principal diagonal are zero. So a 22 upper triangular matrix has the form A=ab0d Where a,bandd, are real numbers. Discuss the validity of each of the following statements. If the statement is always true, explain why If not, give examples. (A) IfAandBare22 upper triangular matrices, then A+B is a 22 upper triangular matrix. (B) IfAandBare22 upper triangular matrices, then AB is a 22 upper triangular matrix. (C) IfAandBare22 upper triangular matrices, then AB=BA.Cost analysis. A company with two different plants manufactures guitars and banjos. Its production costs for each instrument are given in the following matrices: PlantXPlantYGuitarBanjoGuitarBanjoMaterialsLabor$47$39$90$125=A$56$42$84$115=B Find 12A+B the average cost of production for the two plants Cost analysis. If both labor and materials at plant X in Problem 65 are increased by 20, find 121.2A+B the new average cost of production for the two plants.Markup. An import car dealer sells three models of a car. The retail prices and the current dealer invoice prices (costs) for the basic models and options indicated are given in the following two matrices (where “Air" means air-conditioning): RetailPriceBasicAM/FMCruiseCarAirradiocontrolModelAModelBModelC35,0752,5601,07064039,0451,84077046045,5353,4001,415850=M DealerinvoicepriceBasicAM/FMCruiseCarAirradiocontrolModelAModelBModelC30,9962,05085051034,8571,58566039541,6672,8901,200725=N We define the markup matrix to be MN (markup is the difference between the retail price and the dealer invoice price). Suppose that the value of the dollar has had a sharp decline and the dealer invoice price is to have an across the board 15 increase next year. To stay competitive with domestic cars, the dealer increases the retail prices 10. Calculate a markup matrix for next year’s models and the options indicated. (Compute results to the nearest dollar.)Markup. Referring to Problem 67, what is the markup matrix resulting from a 20 increase in dealer invoice prices and an increase in retail prices of 15 ? (Compute results to the nearest dollar.)Labor costs. A company with manufacturing plants located in Massachusetts (MA) and Virginia (VA) has labor-hour and wage requirements for the manufacture of three types of inflatable boats as given in the following two matrices: Labor hours per boat CuttingdepartmentAssemblydepartmentPackagingdepartmentM=0.6hr0.6hr0.2hr1.0hr0.9hr0.3hr1.5hr1.2hr0.4hrOne-personthatTwo-personboatFour-personboat Hourly wages MAVAN=17.3014.6512.2210.6310.299.66cuttingdepartmentAssemblydepartmentPackagingdepartment (A) Find the labor costs for a one-person boat manufactured at the Massachusetts plant. (B) Find the labor costs for a four-person boat manufactured at the Virginia plant. (C) Discuss possible interpretations of the elements in the matrix products MNandNM. (D) If either of the products MNorNM has a meaningful interpretation. Find the product and label its rows and columns.Inventory value. A personal computer retail company sells five different computer models through three stores. The inventory of each model on hand in each store is summarized in matrix M. Wholesale W and retail R values of each model computer are summarized in matrix N ModelABCDEM=4237123506104343Store1Store2Store3. WRN=7001,4001,8002,7003,5008401,8002,4003,3004,900ABCDEModel (A) What is the retail value of the inventory at store 2 ? (B) What is the wholesale value of the inventory at store 3 ? (C) If either product MN or NM has a meaningful interpretation. Find the product and label its rows and columns. What do the entries represent? (D) Discuss methods of matrix multiplication that can be used to find the total inventory of each model on hand at all three stores. State the matrices that can be used and perform the necessary operations.Cereal. A nutritionist for a cereal company blends two cereals in three different mixes. The amounts of protein, carbohydrate and fat (in grams per ounce) in each cereal are given by matrix M. The amounts of each cereal used in the three mixes are given by matrix N. CerealACerealBM=4g/oz2g/oz20g/oz3g/oz16g/oz1g/ozProteinCarbohydrateFat MixXMixYMixZN=15oz5oz10oz10oz5oz15ozCerealACerealB (A) Find the amount of protein in mix X. (B) Find the amount of fat in mix Z . (C) Discuss possible interpretations of the elements in the matrix products MN and NM. (D) If either of the products MN or NM has a meaningful interpretation. Find the product and label its rows and columns.Heredity. Gregor Mendel (1822-1884) made discoveries that revolutionized the science of genetics. In one experiment he crossed dihybrid yellow round peas (yellow and round are dominant characteristics; the peas also contained genes for the recessive characteristics green and wrinkled) and obtained peas of the types indicated in the matrix: RoundWrinkledYellowGreen31510110832=M Suppose he carried out a second experiment of the same type and obtained peas of the types indicated in this matrix: RoundWrinkledYellowGreen37012811036=N If the results of the two experiments are combined, discuss matrix multiplication methods that can be used to find the following quantities. State the matrices that can be used and perform the necessary operations. (A) The total number of peas in each category. (B) The total number of peas in all four categories. (C) The percentage of peas in each category Politics. In a local California election, a public relations firm promoted its candidate in three ways: telephone calls, house calls, and letters. The cost per contact is given in matrix M, and the number of contacts of each type made in two adjacent cities is given in matrix N. CostpercontactM=1.203.00$1.45TelephonecallHouseLetter TelephonecallHousecallLetterN=1,0002,0005008005,0008,000BerkeleyOakland (A) Find the total amount spent in Berkeley. (B) Find the total amount spent in Oakland. (C) If either product MNorNM has a meaningful interpretation. Find the product and label its rows and columns. What do the entries represent? (D) Discuss methods of matrix multiplication that can be used to find the total number of telephone calls, house calls, and letters. State the matrices that can be used and perform the necessary operations.Test averages. A teacher has given four tests to a class of five students and stored the results in the following matrix: Tests 1234AnnBobCarolDanEric7884818691658492959092917582879183888176=M Discuss methods of matrix multiplication that the teacher can use to obtain the information indicated below. In each case, state the matrices to be used and then perform the necessary operations. (A) The average on all four tests for each student, assuming that all four tests are given equal weight. (B) The average on all four tests for each student, assuming that the first three tests are given equal weight and the fourth is given twice this weight. (C) The class average on each of the four tests.The only real number solutions to the equation x2=1 are x=1andx=1. (A) Show that A=0110 satisfies A2=I where I is the 22 identity. (B) Show that B=0110 satisfies B2=I. (C) Find a 22 matrix with all elements nonzero whose square is the 22 identity matrix.(A) Suppose that the square matrix M has a row of all zeros. Explain why M has no inverse. (B) Suppose that the square matrix M has a column of all zeros. Explain why M has no inverse.The inverse of A=abcd is A1=dadbcbadbccadbcaadbc=1DdbcaD=adbc Provided that D0. (A) Use matrix multiplication to verify this formula. What can you conclude about A1ifD=0? (B) Use this formula to find the inverse of matrix M in Example 3.Multiply (a) 10012357and23571001 (b) 100010001423658and4236581001 Let M=311110101 (A) Form the augmented matrix MI. (B) Use row operations to transform MI into IB. (C) Verify by multiplication that B=M1 (that is, show that BM=I ).Find M1, given M=2612 Find N1,givenN=3162.The message below was also encoded with the matrix A in Example 5. Decode this message: 468485284746324178254253253763437183193725 In Problems 1-4, find the additive inverse and lies multiplicative inverse, if defined, of each real number. (If necessary, review Section A.1) (A) 4 (B) 3 (C) 0 In Problems 1-4, find the additive inverse and lies multiplicative inverse, if defined, of each real number. (If necessary, review Section A.1) (A) 7 (B) 2 (C) 1 3E 4E In Problems 5-8, does the given matrix have a multiplicative inverse? Explain your answer. 25 In Problems 5-8, does the given matrix have a multiplicative inverse? Explain your answer. 48 In Problems 5-8, does the given matrix have a multiplicative inverse? Explain your answer. 0000 In Problems 5-8, does the given matrix have a multiplicative inverse? Explain your answer. 1001 In Problems 9-18, find the matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. (A) 10002345 (B) 23451000 In Problems 9-18, find the matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. (A) 10001652 (B) 16521000 In Problems 9-18, find the matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. (A) 00012345 (B) 23450001 In Problems 9-18, find the matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. (A) 00011652 (B) 16520001 In Problems 9-18, find the matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. (A) 10012345 (B) 23451001 In Problems 9-18, find the matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. (A) 10011652 (B) 16521001 In Problems 9-18, find the matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 100010001213242510 In Problems 9-18, find the matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 100010001340125631 In Problems 9-18, find the matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 213242510100010001 In Problems 9-18, find the matrix products. Note that each product can be found mentally, without the use of a calculator or pencil-and-paper calculations. 340125631100010001 In Problems 19-28, examine the product of the two matrices to determine if each is the inverse of the other. 3423;3423 In Problems 19-28, examine the product of the two matrices to determine if each is the inverse of the other. 2142;1122 In Problems 19-28, examine the product of the two matrices to determine if each is the inverse of the other. 2211;1111 In Problems 19-28, examine the product of the two matrices to determine if each is the inverse of the other. 5723;3725 In Problems 19-28, examine the product of the two matrices to determine if each is the inverse of the other. 5283;3285 In Problems 19-28, examine the product of the two matrices to determine if each is the inverse of the other. 7453;3457 In Problems 19-28, examine the product of the two matrices to determine if each is the inverse of the other. 120010111;120010110 In Problems 19-28, examine the product of the two matrices to determine if each is the inverse of the other. 101312001;101311001 In Problems 19-28, examine the product of the two matrices to determine if each is the inverse of the other. 111021230;331221452 In Problems 19-28, examine the product of the two matrices to determine if each is the inverse of the other. 101311000;101312001 Without performing any row operations, explain why each of the matrices in Problems 29-38 does not have an inverse. 132201 Without performing any row operations, explain why each of the matrices in Problems 29-38 does not have an inverse. 243011 Without performing any row operations, explain why each of the matrices in Problems 29-38 does not have an inverse. 123021 Without performing any row operations, explain why each of the matrices in Problems 29-38 does not have an inverse. 012213 Without performing any row operations, explain why each of the matrices in Problems 29-38 does not have an inverse. 0102 Without performing any row operations, explain why each of the matrices in Problems 29-38 does not have an inverse. 0102 Without performing any row operations, explain why each of the matrices in Problems 29-38 does not have an inverse. 1200 Without performing any row operations, explain why each of the matrices in Problems 29-38 does not have an inverse. 1200 Without performing any row operations, explain why each of the matrices in Problems 29-38 does not have an inverse. 1224 Without performing any row operations, explain why each of the matrices in Problems 29-38 does not have an inverse. 1212 Given M in Problems 39-48, Find M1 and show that M1M=I. 1031 Given M in Problems 39-48, Find M1 and show that M1M=I. 1501 Given M in Problems 39-48, Find M1 and show that M1M=I. 1213 Given M in Problems 39-48, Find M1 and show that M1M=I. 2153 Given M in Problems 39-48, Find M1 and show that M1M=I. 1327 Given M in Problems 39-48, Find M1 and show that M1M=I. 2111 Given M in Problems 39-48, Find M1 and show that M1M=I. 130011214 Given M in Problems 39-48, Find M1 and show that M1M=I. 230123015 Given M in Problems 39-48, Find M1 and show that M1M=I. 110231102 Given M in Problems 39-48, Find M1 and show that M1M=I. 101210112 Find the inverse of each matrix in Problems 49-56, if it exists 4332 Find the inverse of each matrix in Problems 49-56, if it exists 4354 Find the inverse of each matrix in Problems 49-56, if it exists 2639 Find the inverse of each matrix in Problems 49-56, if it exists 2436 Find the inverse of each matrix in Problems 49-56, if it exists 2143 Find the inverse of each matrix in Problems 49-56, if it exists 5322 Find the inverse of each matrix in Problems 49-56, if it exists 31535 Find the inverse of each matrix in Problems 49-56, if it exists 510224 In Problems 57-60, find the inverse. Note that each inverse be found mentally, without the use of a calculator or pencil paper calculations. 3003 In Problems 57-60, find the inverse. Note that each inverse be found mentally, without the use of a calculator or pencil paper calculations. 5005 In Problems 57-60, find the inverse. Note that each inverse be found mentally, without the use of a calculator or pencil paper calculations. 20012 In Problems 57-60, find the inverse. Note that each inverse be found mentally, without the use of a calculator or pencil paper calculations. 10013 Find the inverse of each matrix in Problems 61-68, If it exists 522210101 Find the inverse of each matrix in Problems 61-68, If it exists 224111101 Find the inverse of each matrix in Problems 61-68, If it exists 211110110 Find the inverse of each matrix in Problems 61-68, If it exists 110211011 Find the inverse of each matrix in Problems 61-68, If it exists 122430404 Find the inverse of each matrix in Problems 61-68, If it exists 422420505 Find the inverse of each matrix in Problems 61-68, If it exists 212428621 Find the inverse of each matrix in Problems 6168, If it exists 11433222119 Show that A11=Afor;A=4332 Show that AB1=B1A1for A=4332andB=2537 Discuss the existence of M1 for 22 diagonal matrices of the form M=a00d Generalize your conclusions to nn diagonal matrices.Discuss the existence of M1for22 for upper triangular matrices of the form M=ab0d Generalize your conclusions to nn upper triangular matrices.73E In Problems 73-75, find A1 and A2. 5465 In Problems 73-75, find A1 and A2. 5385 Based on your observations in Problems 73-75, if A=A1 for a square matrix A, what is A2 ? Give a mathematical argument to support your conclusion.Problems 77-80 refer to the encoding matrix A=1213 Cryptography. Encode the message "WINGARDIUMLEVIOSA" using matrix A .Problems 77-80 refer to the encoding matrix A=1213 Cryptography. Encode the message "FINITEINCANTATEM” using matrix A.Problems 77-80 refer to the encoding matrix A=1213 Cryptography. The following message was encoded with matrix A. Decode this message: 5270172155294344527025352933151855 Problems 77-80 refer to the encoding matrix A=1213 Cryptography. The following message was encoded with matrix A. Decode this message: 364455385655751823567522333755274053795981 Problems 81-84 require the use of a graphing calculator or computer. Use the 44 encoding matrix B given below. Form a matrix with 4 row’s and as many columns as necessary to accommodate the message. B=2213122111012323 Cryptography. Encode the message "DEPART ISTANBUL ORIENT EXPRESS" using matrix B.Problems 81-84 require the use of a graphing calculator or computer. Use the 44 encoding matrix B given below. Form a matrix with 4 row’s and as many columns as necessary to accommodate the message. B=2213122111012323 Cryptography. Encode the message "SAIL FROM LISBON IN MORNING” using matrix B.Problems 81-84 require the use of a graphing calculator or computer. Use the 44 encoding matrix B given below. Form a matrix with 4 row’s and as many columns as necessary to accommodate the message. B=2213122111012323 Cryptography. The following message was encoded with matrix B. Decode this message: 85742710931271340139735815461701893695923871813922 Problems 81-84 require the use of a graphing calculator or computer. Use the 44 encoding matrix B given below. Form a matrix with 4 row’s and as many columns as necessary to accommodate the message. B=2213122111012323 Cryptography. The following message was encoded with matrix B. Decode this message: 756128943522134049211652424519643855106569752410267491982105510 Problems 85-88 require the use of a graphing calculator or a computer. Use the 55 encoding matrix C given below. Form a matrix with 5 rows and as many columns as necessary to accommodate the message. C=1010101103211110010211121 Cryptography. Encode the message 'THE EAGLE HAS LANDED" using matrix C.Problems 85-88 require the use of a graphing calculator or a computer. Use the 55 encoding matrix C given below. Form a matrix with 5 rows and as many columns as necessary to accommodate the message. C=1010101103211110010211121 Cryptography. Encode the message "ONE IF BY LAND AND TWO IF BY SEA" using matrix C.Problems 85-88 require the use of a graphing calculator or a computer. Use the 55 encoding matrix C given below. Form a matrix with 5 rows and as many columns as necessary to accommodate the message. C=1010101103211110010211121 Cryptography. The following message was encoded with matrix C. Decode this messag: 377258455630675046602777414539282452143732587036762238701267 Problems 85-88 require the use of a graphing calculator or a computer. Use the 55 encoding matrix C given below. Form a matrix with 5 rows and as many columns as necessary to accommodate the message. C=1010101103211110010211121 Cryptography. The following message was encoded with matrix C. Decode this message: 257555355043835460532513599531535401545336060365115737022 Refer to the mathematical model in Example 4: A110.050.1Xx1x2=Bk1k2 (A) Does matrix equation (3) always have a solution for any constant matrix B ? (B) Do all these solutions make sense for the original problem? If not, give examples. (C) If the total investment is k1=$10,000, describe all possible annual returns k2.

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