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Finite Mathematics and Its Applications 12th edition Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair - Solutions

Two n Ã— n matrices A and B are called inverses (of one another) if both products AB and BA equal In.1.2.
The quantities of pants, shirts, and jackets owned by Mike and Don are given by the matrix A, and the costs of these items are given by matrix B.(a) Calculate the matrix AB. (b) Interpret the entries of the matrix AB. (c) Calculate the matrix 1.25B. (d) Interpret the entries of the matrix 1.25B.
Two stores sell the exact same brand and style of a dresser, a nightstand, and a bookcase. Matrix A gives the retail prices (in dollars) for the items. Matrix B gives the number of each item sold at each store in one month.(a) Calculate AB. (b) Interpret the entries of AB. (c) Calculate the matrix
A candy shop sells various items for the price per pound (in dollars) indicated in matrix A. Matrix B gives the number of pounds of coated peanuts, raisins, and espresso beans prepared in a week. Matrix C gives the total number of pounds of white chocolate-covered, milk chocolate-covered, and dark
A company has three appliance stores that sell washers, dryers, and ranges. Matrices W and R give the wholesale and retail prices of these items, respectively. Matrices N and D give the quantities of these items sold by the three stores in November and December, respectively.Determine and interpret
Three professors teaching the same course have entirely different grading policies. The percentage of students given each grade by the professors is summarized in the following matrix:(a) The point values of the grades are A = 4, B = 3, C = 2, D = 1, and F = 0. Use matrix multiplication to
Refer to the 2 Ã— 3 matrix1. Find a12 and a21. 2. Find a23 and a11. 3. For what values of i and j does aij = 6? 4. For what values of i and j does aij = 3?
A professor bases semester grades on four 100-point items: homework, quizzes, a midterm exam, and a final exam. Students may choose one of three schemes summarized in the accompanying matrix for weighting the points from the four items. Use matrix multiplication to determine the most advantageous
In a certain town, the proportions of voters voting Democratic and Republican by various age groups is summarized by this matrix:The population of voters in the town by age group is given by the matrix Interpret the entries of the matrix product BA.
Refer to Exercise 71.The population of voters in the town by age group is given by the matrix (a) According to the data, which party would win and what would be the percentage of the winning vote? (b) Suppose that the population of the town shifted toward older residents, as reflected in the
Suppose that a contractor employs carpenters, bricklayers, and plumbers, working three shifts per day. The number of labor-hours employed in each of the shifts is summarized in the following matrix:Labor in shift 1 costs $20 per hour, in shift 2 $30 per hour, and in shift 3 $40 per hour. Use matrix
A flu epidemic hits a large city. Each resident of the city is either sick, well, or a carrier. The proportion of people in each of the categories is expressed by the following matrix:The population of the city is distributed by age and sex as follows: (a) Compute AB. (b) How many sick males are
Mikey's diet consists of food X and food Y. The matrix N represents the number of units of nutrients 1, 2, and 3 per ounce for each of the foods.The matrices B, L, and D represent the number of ounces of each food that Mikey eats each day for breakfast, lunch, and dinner, respectively. Calculate
A bakery makes three types of cookies, I, II, and III. Each type of cookie is made from the four ingredients A, B, C, and D. The number of units of each ingredient used in each type of cookie is given by the matrix M. The cost per unit of each of the four ingredients (in cents) is given by the
A community fitness center has a pool and a weight room. The admission prices (in dollars) for residents and nonresidents are given by the matrixThe average daily numbers of customers for the fitness center are given by the matrix (a) Compute AP. (b) What is the average amount of money taken in by
A company makes DVD players and TV sets. Each DVD player requires 3 hours of assembly and ½ hour of packaging, while each TV set requires 5 hours of assembly and 1 hour of packaging. (a) Write a matrix T representing the required time for Assembly and packaging of DVD players and TV sets. (b) The
A bakery sells Boston cream pies and carrot cakes. Each Boston cream pie requires 30 minutes preparation time, 30 minutes baking time, and 15 minutes for finishing. Each carrot cake requires 45 minutes preparation time, 50 minutes baking time, and 10 minutes for finishing. (a) Write a matrix T
A beauty salon offers manicures and pedicures. A manicure requires 20 minutes for preparation, 5 minutes for lacquering, and 15 minutes for drying. A pedicure requires 30 minutes for preparation, 5 minutes for lacquering, and 20 minutes for drying. (a) Construct a matrix T representing the time
The J.E. Carrying Company makes two types of backpacks. The larger Huge One backpack requires 2 hours for cutting, 3 hours for sewing, and 2 hours for finishing, and sells for $32. The smaller Regular Joe backpack requires 1.5 hours for cutting, 2 hours for sewing, and 1 hour for finishing, and
A store sells three types of MP3 players. Matrix A contains information about size (in gigabytes), battery life (in hours), and weight (in ounces) of the three MP3 players. Matrix B contains the sales prices (in dollars) of the MP3 players, while matrix C contains the number of each type of player
Find the values of a and b for which A ˆ™ B = I3, where
Determine the matrix B on the basis of the screen shown.1.2. 3. If A is a 3 Ã— 4 matrix and A(BB) is defined, what is the size of matrix B? 4. If B is a 3 Ã— 5 matrix and (AA)B is defined, what is the size of matrix A?
Table 1 gives the number of public school teachers (elementary and secondary) and the average number of pupils per teacher for three mid-Atlantic states in a recent year. Set up a product of two matrices that gives the total number of pupils in the three states.Table 1 Teachers and Pupils
Table 2 gives the area and 2015 population density for three West Coast states. Set up a product of two matrices that gives total population of the three states.Table 2 State Areas and Densities
Calculate the given expression, where1. A + B 2. B - A 3. BA 4. AB 5. 3A
1. Try adding two matrices of different sizes. How does your technology respond?2. Try multiplying two matrices in which the number of columns of the first matrix differs from the number of rows of the second matrix. How does your technology respond?3. I3 + 2ACalculate the given expression, where
Use the fact that1. Solve 2. Solve
Use a matrix equation to solve the system of linear equations.1.2. 3. 4.
It is found that the number of married and single adults in a certain town are subject to the statistics that follow. Suppose that x and y denote the number of married and single adults, respectively, in a given year (say, as of January 1) and let m and s denote the corresponding numbers for the
A flu epidemic is spreading through a town of 48,000 people. It is found that, if x and y denote the numbers of people sick and well in a given week, respectively, and if s and w denote the corresponding numbers for the following week, then1/3x + 1/4y = s2/3x + 3/4y = w.(a) Write this system of
Statistics show that, at a certain university, 70% of the students who live on campus during a given semester will remain on campus the following semester, and 90% of students living off campus during a given semester will remain off campus the following semester. Let x and y denote the number of
A teacher estimates that, of the students who pass a test, 80% will pass the next test, while of the students who fail a test, 50% will pass the next test. Let x and y denote the number of students who pass and fail a given test, and let u and y be the corresponding numbers for the following
Use the fact that the following two matrices are inverses of each other to solve the system of linear equations.1. 2. 3. 4.
Use the fact that the following two matrices are inverses of each other to solve the system of linear equations.1. 2. 3. 4.
1. Show that if a ‰ 0 and b ‰ 0, then the inverse ofis 2. If B is the inverse of A, then A is the inverse of B.
There are two age groups for a particular species of organism. Group I consists of all organisms aged under 1 year, while group II consists of all organisms aged from 1 to 2 years. No organism survives more than 2 years. The average number of offspring per year born to each member of group I is 1,
Find the inverse of the given matrix.1.2. 3.
1. Show that, if AB is a matrix of all zeros and A has an inverse, then B is a matrix of all zeros.2. Ifwhat is A?
Consider the matricesShow that (AB)-1 = B-1A-1.
Find a 2 × 2 matrix A and a 2 × 1 column matrix B for which AX = B has no solution.
Find a 2 × 2 matrix A and a 2 × 1 column matrix B for which AX = B has infinitely many solutions.
Use the inverse operation to find the inverse of the given matrix.1.2. 3. 4.
Calculate the solution by using a matrix equation.1.2. 3. 4.
Use the Gauss-Jordan method to compute the inverse, if it exists, of the matrix.1.2. 3.
Use an inverse matrix to solve the system of linear equations.1.2.
Find a 2 Ã— 2 matrix A for which
Find a 2 Ã— 2 matrix A for which
Figure 1 gives the responses of a group of 100 randomly selected college freshmen when asked for the highest academic degree that they intended to obtain. Twice as many students intended to obtain master's degrees as their highest degree than intended to obtain bachelor's degrees as their highest
Figure 2 gives the responses of a group of 100 randomly selected college freshmen when asked whether the college they were attending was their first choice or second choice. The number of students who attended their first-choice college was 16 more than the students who did not. The number of
Figure 3 gives the responses of a group of 100 randomly selected college freshmen when asked for the type of high school attended. The number of students who attended public schools was 5 times the number who attended private schools minus 3 times the number who were homeschooled. The number of
Figure 4 gives the responses of a group of randomly selected college freshmen when asked for the types of placement tests taken. A total of 82 placement tests were taken. The number of mathematics placement tests taken was 2 more than twice the number of writing placement tests taken. Eight more
1. How many cents of energy are required to produce $1 worth of manufactured goods?2. How many cents of energy are required to produce $1 worth of services?Suppose that a simplified economy consisting of the three sectors Manufacturing, Energy, and Services has the input-output matrix
1. Repeat Exercise 11 for the consumer demand matrix2. Given that find the production matrix when the consumer demand matrix is Suppose that a simplified economy consisting of the three sectors Manufacturing, Energy, and Services has the input-output matrix
Suppose that, in the economy of Example 1, the demand for electricity triples and the demand for coal doubles, whereas the demand for steel increases by only 50%. At what levels should the various industries produce in order to satisfy the new demand?
Suppose that the conglomerate of Example 2 is faced with an increase of 50% in demand for computers, a doubling in demand for semiconductors, and a decrease of 50% in demand for business forms. At what levels should the various divisions produce in order to satisfy the new demand?
Suppose that the conglomerate of Example 2 experiences a doubling in the demand for business forms. At what levels should the computer and semiconductor divisions produce?
Suppose that the economy of Example 1 experiences a 20% increase in the demand for coal. At what levels should the three industries produce?
In the economy of Example 1, suppose that $4 billion worth of coal, $2 billion worth of steel, and $5 billion worth of electricity are produced. How much of each industry's output will be available for consumption?
In the conglomerate of Example 2, suppose that $400,000,000 worth of computers, $200,000,000 worth of semiconductors, and $300,000,000 worth of business forms are produced. How much of each division's output will be available for consumption?
A simplified economy consists of the two sectors Transportation and Energy. For each $1 worth of output, the transportation sector requires $.25 worth of input from the transportation sector and $.20 of input from the energy sector. For each $1 worth of output, the energy sector requires $.30 from
Rework Exercise 19 under the condition that the consumer demand for transportation is $4 billion and the consumer demand for energy is $7 billion. A simplified economy consists of the two sectors Transportation and Energy. For each $1 worth of output, the transportation sector requires $.25 worth
A corporation has a plastics division and an industrial equipment division. For each $1 worth of output, the plastics division needs $.02 worth of plastics and $.10 worth of equipment. For each $1 worth of output, the industrial equipment division needs $.01 worth of plastics and $.05 worth of
Rework Exercise 21 under the condition that the consumer demand for plastics is $1,860,000 and the demand for industrial equipment is $2,790,000. A corporation has a plastics division and an industrial equipment division. For each $1 worth of output, the plastics division needs $.02 worth of
In an economic system, each of three industries depends on the others for raw materials. To make $1 worth of processed wood requires 30 ¢ worth of wood, 20 ¢ steel, and 10 ¢ coal. To make $1 worth of steel requires no wood, 30 ¢ steel, and 20 ¢ coal. To make $1 worth of coal requires 10 ¢
Rework Exercise 23 under the condition that the consumer demand is $200 for wood, $300 for steel, and $800 for coal. In an economic system, each of three industries depends on the others for raw materials. To make $1 worth of processed wood requires 30 ¢ worth of wood, 20 ¢ steel, and 10 ¢ coal.
An industrial system involves manufacturing, transportation, and agriculture. The interdependence of the three industries is given by the input-output matrixAt what levels must the industries produce to satisfy a demand for $100 million worth of manufactured goods, $80 million of transportation,
An economy consists of the three sectors agriculture, energy, and manufacturing. For each $1 worth of output, the agriculture sector requires $.08 worth of input from the agriculture sector, $.10 worth of input from the energy sector, and $.20 worth of input from the manufacturing sector. For each
A town has a merchant, a baker, and a farmer. To produce $1 worth of output, the merchant requires $.30 worth of baked goods and $.40 worth of the farmer's products. To produce $1 worth of output, the baker requires $.50 worth of the merchant's goods, $.10 worth of his own goods, and $.30 worth of
A multinational corporation does business in the United States, Canada, and England. Its branches in one country purchase goods from the branches in other countries according to the matrixwhere the entries in the matrix represent proportions of total sales by the respective branch. The external
Consider the matrix (I - A)-1 from Example 1. Show that if the consumer demand for coal is increased by $1 billion, then the additional amounts (in billions of dollars) that must be produced by each of the three industries is given by the first column of (I - A)-1.
Which sector of the economy requires the greatest amount of services in order to produce $1 worth of output?Suppose that a simplified economy consisting of the three sectors Manufacturing, Energy, and Services has the input-output matrix
Refer to Exercise 29. Interpret the significance of the second and third columns of (I - A)-1.Consider the matrix (I - A)-1 from Example 1. Show that if the consumer demand for coal is increased by $1 billion, then the additional amounts (in billions of dollars) that must be produced by each of the
Use the input-output matrix A and the consumer- demand matrix D to find the production matrix X for the Leontieff model. Round your answers to two decimal places.1.2.
Which sector of the economy requires the least amount of manufacturing in order to produce $1 worth of output?Suppose that a simplified economy consisting of the three sectors Manufacturing, Energy, and Services has the input-output matrix
What is the dollar amount of the energy costs needed to produce $10 million worth of goods from each sector?Suppose that a simplified economy consisting of the three sectors Manufacturing, Energy, and Services has the input-output matrix
What is the dollar amount of the costs for services needed to produce $10 million worth of goods from each sector?Suppose that a simplified economy consisting of the three sectors Manufacturing, Energy, and Services has the input-output matrix
On what sector are services most dependent?Suppose that a simplified economy consisting of the three sectors Manufacturing, Energy, and Services has the input-output matrix
On what sector is manufacturing least dependent?Suppose that a simplified economy consisting of the three sectors Manufacturing, Energy, and Services has the input-output matrix
1. Determine the internal consumption when the production matrix is2. Show that Suppose that a simplified economy consisting of the three sectors Manufacturing, Energy, and Services has the input-output matrix
1. What is meant by a solution to a system of linear equations? 2. What is a matrix? 3. State the three elementary row operations on equations or matrices.
Define the product of two matrices.
1. Define scalar product. 2. Define the inverse of a matrix, A-1. 3. Give the formula for the inverse of a 2 × 2 matrix.
Explain how to use the inverse of a matrix to solve a system of linear equations.
Describe the steps of the Gauss-Jordan method for calculating the inverse of a matrix.
What are an input-output matrix and a consumer-demand matrix?
Explain how to solve an input-output analysis problem.
1. What does it mean for a system of equations or a matrix to be in diagonal form?2. What is meant by pivoting a matrix about a nonzero entry?
State the Gauss-Jordan elimination method for solving a system of linear equations.
What is a row matrix? Column matrix? Square matrix? Identity matrix, In?
1. What is meant by aij, the ijth entry of a matrix? 2. Define the sum and difference of two matrices.
1. How many subadult females are there after one year?2. Did the total population of females increase or decrease during the year?Let A denote the matrixAccording to the mathematical model, subsequent population distributions are generated by multiplication on the left by A. That is, A ˆ™
Pivot each matrix about the circled element.1.2.
Find the inverse of the appropriate matrix, and use it to solve the system of equations
The matrices are inverses of each other.Use these matrices to solve the following systems of linear equations: (a) (b)
Use the Gauss-Jordan method to calculate the inverse of the matrix.1.2.
Farmer Brown has 1000 acres of land on which he plans to grow corn, wheat, and soybeans. The cost of cultivating these crops is $357 per acre for corn, $127 per acre for wheat, and $181 per acre for soybeans. If Farmer Brown wishes to use all of his available land and his entire budget of $269,000,
A company makes backyard playground equipment such as swing sets, slides, and play sets. The cost (in dollars) to make a specific style of each piece is given in matrix C. The sales price (in dollars) for each piece is given in matrix S. Two stores sell these specific pieces, and matrix A gives the
A person wants to invest money in three different college savings plans. Matrix A contains the percentages (in decimal form) invested in bonds, stocks, and a conservative fixed income fund for each of the three different plans. Matrix B gives the total amount invested in each of the three savings
Sara, Quinn, Tamia, and Zack are working at the pool this summer. One week, they spend the following amounts of time at three different tasks:The hourly pay (in dollars) for the three different tasks is given by (a) Calculate and interpret the matrix AB. (b) Who earned the most that week? Who
The produce department at a grocery store makes fruit baskets of apples, bananas, and oranges. The cost for each apple is $.85, for each banana is $.20, and for each orange is $.76. Suppose that each basket needs to have 18 pieces of fruit and costs $9 to make. How many apples, bananas, and oranges

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