Question 1

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

Accepted Answer

A)    
B)    
C)    
D)    
E)  EC is undefined. 
A)    
B)    
C)    
D)    
E)  EC is undefined.

Question 2

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

Accepted Answer

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

Question 3

Evaluate the determinant. ​   ​

Accepted Answer

A)  -56 
B)  -28 
C)  28 
D)  -51 
E)  13 
A)  -56 
B)  -28 
C)  28 
D)  -51 
E)  13

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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. ​

Accepted Answer

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 
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

Question 8

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.

Accepted Answer

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

Question 9

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. ​

Accepted Answer

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 
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

Question 10

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  
And the code matrix   , use matrix A to encode the message "The die is cast."

Accepted Answer

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

Question 11

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? ​

Accepted Answer

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

Question 12

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. ​

Accepted Answer

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

Question 13

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. ​   ​     ;   ​

Accepted Answer

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

Question 14

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 15

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. ​   ​

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

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? ​

Accepted Answer

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% 
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%

Question 18

The King Trucking Company has an order for three products for delivery. The following table gives the particulars for the products. ​
 ​
If the carrier can carry 6,000 cu ft, can carry   , and is insured for   , how many units of each type can be carried?
​
​

Accepted Answer

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

Question 19

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.
​     ,   ,

Accepted Answer

A) True 
 B)False

Question 20

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

Accepted Answer

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

Use the matrices to find EC, if possible.

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

Evaluate the determinant.

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

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

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.

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 And the code matrix , use matrix A to encode the message "The die is cast."

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

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

The King Trucking Company has an order for three products for delivery. The following table gives the particulars for the products. If the carrier can carry 6,000 cu ft, can carry , and is insured for , how many units of each type can be carried?

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. , ,

Use inverse matrices to solve the system of linear equations.

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 the matrices to find EC, if possible. ​ ​ ​ ​

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

Evaluate the determinant. ​ ​

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

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

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.

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 And the code matrix , use matrix A to encode the message "The die is cast."

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

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

The King Trucking Company has an order for three products for delivery. The following table gives the particulars for the products. ​ ​ If the carrier can carry 6,000 cu ft, can carry , and is insured for , how many units of each type can be carried? ​ ​

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. ​ , ,

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

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 the matrices to find EC, if possible.

Evaluate the determinant.

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

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

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

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

The King Trucking Company has an order for three products for delivery. The following table gives the particulars for the products. If the carrier can carry 6,000 cu ft, can carry , and is insured for , how many units of each type can be carried?

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. , ,

Use inverse matrices to solve the system of linear equations.