Question 1

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 2

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. ​

Accepted Answer

A)    
B)    
C)  ​   
D)    
E)    
A)    
B)    
C)  ​   
D)    
E)

Question 3

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. ​

Accepted Answer

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 
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

Question 4

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

Accepted Answer

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 
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

Question 5

Find x, y, z, and w . ​   ​

Accepted Answer

A)    and   
B)    and   
C)    and   
D)    and   
E)    and   
A)    and   
B)    and   
C)    and   
D)    and   
E)    and

Question 6

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 7

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: ​
 
​
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.​

Accepted Answer

A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​   
A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​

Question 8

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. ​

Accepted Answer

A)  ​   
B)  ​   
C)  ​   
D)    
E)    
A)  ​   
B)  ​   
C)  ​   
D)    
E)

Question 9

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 10

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

Accepted Answer

A)    ,   ,   
B)    ,   ,   
C)    ,   ,   
D)    ,   ,   
E)    ,   ,   
A)    ,   ,   
B)    ,   ,   
C)    ,   ,   
D)    ,   ,   
E)    ,   ,

Question 11

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

Accepted Answer

A)  ​   ​ 
B)    
C)    ​ 
D)    ​ 
E)  ​   
A)  ​   ​ 
B)    
C)    ​ 
D)    ​ 
E)  ​

Question 12

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.) ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 13

Use the inverse to solve. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 14

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.
​

Accepted Answer

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 
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

Question 15

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 16

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?
​

Accepted Answer

A)  4.1 
B)  1.3 
C)  41 
D)  13 
E)  46.4 
A)  4.1 
B)  1.3 
C)  41 
D)  13 
E)  46.4

Question 17

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

Accepted Answer

A)    ,   ,   , for any real number   . 
B)    ,   ,   ,   
C)    ,   ,   ,   
D)    ,   ,   , for any real number   . 
E)  no solution 
A)    ,   ,   , for any real number   . 
B)    ,   ,   ,   
C)    ,   ,   ,   
D)    ,   ,   , for any real number   . 
E)  no solution

Question 18

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 19

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.

Accepted Answer

A)    ,   ,   
B)    ,   , for any real number z. 
C)    ,   ,   
D)    ,   , for any real number z. 
E)  no solution 
A)    ,   ,   
B)    ,   , for any real number z. 
C)    ,   ,   
D)    ,   , for any real number z. 
E)  no solution

Question 20

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?
​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Use inverse matrices to solve the system of linear equations.

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

Find x, y, z, and w .

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

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

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

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

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.)

Use the inverse to solve.

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.

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

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

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

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.

Linear Equations and Functions

Quadratic and Other Special Functions

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability and Data Description

Derivatives

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Functions of Two or More Variables

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

Exam 3: Matrices

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

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

Find x, y, z, and w . ​ ​

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

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

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

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

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.) ​

Use the inverse to solve. ​ ​

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. ​

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

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

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

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.

Linear Equations and Functions

Quadratic and Other Special Functions

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability and Data Description

Derivatives

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Functions of Two or More Variables

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

Use inverse matrices to solve the system of linear equations.

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

Find x, y, z, and w .

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

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

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

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

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.)

Use the inverse to solve.

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.

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