Question 1

Evaluate the determinant. ​   ​

Accepted Answer

A) 33 
B) -16 
C) -2 
D) -17 
E) -6 
A) 33 
B) -16 
C) -2 
D) -17 
E) -6

Question 2

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 3

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 4

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

Accepted Answer

A) shipping = 21,500, printing = 14,000 
B) shipping = 19,500, printing = 16,000 
C) shipping = 21,500, printing = 15,000 
D) shipping = 14,000, printing = 21,500 
E) shipping = 19,500, printing = 15,000 
A) shipping = 21,500, printing = 14,000 
B) shipping = 19,500, printing = 16,000 
C) shipping = 21,500, printing = 15,000 
D) shipping = 14,000, printing = 21,500 
E) shipping = 19,500, printing = 15,000

Question 5

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

Accepted Answer

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

Question 6

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 7

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

Accepted Answer

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

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

Question 9

Find the inverse matrix, if one exists.

Accepted Answer

A)    
B)    
C)    
D)    
E) no inverse exists 
A)    
B)    
C)    
D)    
E) no inverse exists

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

An underdeveloped country has an agricultural industry and a manufacturing industry, with technology matrix A. ​   ​
Surpluses of 10 units of agricultural products and 609 units of manufactured products are desired. Find the gross production of each industry.
​

Accepted Answer

A) agriculture = 160, manufacturing = 1210 
B) agriculture = 1340, manufacturing = 1210 
C) agriculture = 160, manufacturing = 1180 
D) agriculture = 1340, manufacturing = 1180 
E) agriculture = 170, manufacturing = 1210 
A) agriculture = 160, manufacturing = 1210 
B) agriculture = 1340, manufacturing = 1210 
C) agriculture = 160, manufacturing = 1180 
D) agriculture = 1340, manufacturing = 1180 
E) agriculture = 170, manufacturing = 1210

Question 14

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

Accepted Answer

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

Question 15

Set up the system of equations and then solve it by using inverse matrices. ​
Nutrition A biologist has a 40% solution and a 10% solution of the same plant nutrient. How many cubic centimeters of each solution should be mixed to obtain 35 cc of a 28% solution?
​

Accepted Answer

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

Question 16

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 
A) True 
B) False

Question 17

(CD)·E and C·(DE) are equal. ​

Accepted Answer

A) True 
B) False 
A) True 
B) False

Question 18

Use the matrices below. Perform the indicated operations. ​
AZ

Accepted Answer

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

Question 19

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

Accepted Answer

A)   
,  
,  
,   
B)   
,  
,  
,   
C)   
,  
,  
,   
D)   
,  
,  
,   
E) no solution 
A)   
,  
,  
,   
B)   
,  
,  
,   
C)   
,  
,  
,   
D)   
,  
,  
,   
E) no solution

Question 20

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

Accepted Answer

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

Evaluate the determinant.

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

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

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

A 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.

Find the inverse matrix, if one exists.

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

An underdeveloped country has an agricultural industry and a manufacturing industry, with technology matrix A. Surpluses of 10 units of agricultural products and 609 units of manufactured products are desired. Find the gross production of each industry.

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

Set up the system of equations and then solve it by using inverse matrices. Nutrition A biologist has a 40% solution and a 10% solution of the same plant nutrient. How many cubic centimeters of each solution should be mixed to obtain 35 cc of a 28% solution?

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. , ,

(CD)·E and C·(DE) are equal.

Use the matrices below. Perform the indicated operations. AZ

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

Algebraic Concepts

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; Data Description

Derivatives

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Functions of Two or More Variables

Filters

Exam 4: Matrices

Evaluate the determinant. ​ ​

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

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

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. ​

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

A 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.

Find the inverse matrix, if one exists.

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

An underdeveloped country has an agricultural industry and a manufacturing industry, with technology matrix A. ​ ​ Surpluses of 10 units of agricultural products and 609 units of manufactured products are desired. Find the gross production of each industry. ​

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

Set up the system of equations and then solve it by using inverse matrices. ​ Nutrition A biologist has a 40% solution and a 10% solution of the same plant nutrient. How many cubic centimeters of each solution should be mixed to obtain 35 cc of a 28% solution? ​

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. ​ , ,

(CD)·E and C·(DE) are equal. ​

Use the matrices below. Perform the indicated operations. ​ AZ

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

Algebraic Concepts

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; Data Description

Derivatives

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Functions of Two or More Variables

Filters

Evaluate the determinant.

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

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

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.

An underdeveloped country has an agricultural industry and a manufacturing industry, with technology matrix A. Surpluses of 10 units of agricultural products and 609 units of manufactured products are desired. Find the gross production of each industry.

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

Set up the system of equations and then solve it by using inverse matrices. Nutrition A biologist has a 40% solution and a 10% solution of the same plant nutrient. How many cubic centimeters of each solution should be mixed to obtain 35 cc of a 28% solution?

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. , ,

(CD)·E and C·(DE) are equal.

Use the matrices below. Perform the indicated operations. AZ