Question 1

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 2

A man has   invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is   and the amount invested at 8% is twice that invested at 12%. What is the annual income from each property? ​

Accepted Answer

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

Question 3

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 4

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 $16,600, and risk factors make it prudent to invest some money in all three investments. If $30,000 is invested at 10%, how much will be invested at each of the other rates? ​

Accepted Answer

A) $50,000 at 8%; $140,000 at 7%; 
B) $20,000 at 8%; $170,000 at 7%; 
C) $30,000 at 8%; $160,000 at 7%; 
D) $10,000 at 8%; $180,000 at 7%; 
E) $40,000 at 8%; $150,000 at 7% 
A) $50,000 at 8%; $140,000 at 7%; 
B) $20,000 at 8%; $170,000 at 7%; 
C) $30,000 at 8%; $160,000 at 7%; 
D) $10,000 at 8%; $180,000 at 7%; 
E) $40,000 at 8%; $150,000 at 7%

Question 5

In order to rank the five members of its chess team for play against another school, the coach draws the following diagram. An arrow from 1 to 2 means player 1 has defeated player 2. ​   Construct matrix A with elements   .
​

Accepted Answer

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

Question 6

A botanist can purchase plant food of four different types, I, II, III, and IV. Each food comes in the same size bag, and the following table summarizes the number of grams of each of three nutrients that each bag contains. ​   ​
The botanist wants to use a food that has these nutrients in a different proportion and determines that he will need a total of   of A,   of B, and   of C. Let x = Type I bags, y = Type II bags, z = Type III bags, and w = Type IV bags, where x, y, z, and w are non-negative amounts.
Find a general solution that will determine the number of bags of each type of food that should be ordered.
​
​

Accepted Answer

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

Question 7

Suppose that the economy of a small nation has an electronics industry, a steel industry, and an auto industry, with the following technology matrix. ​
​   ​
If the nation wishes to have surpluses of 104 units of electronics production, 271 units of steel production, and 200 automobiles, find the gross production of each industry.
​

Accepted Answer

A) electronics = 1254, steel = 1262, autos = 722 
B) electronics = 1252, steel = 1262, autos = 724 
C) electronics = 1254, steel = 1269, autos = 722 
D) electronics = 1254, steel = 1269, autos = 724 
E) electronics = 1252, steel = 1262, autos = 722 
A) electronics = 1254, steel = 1262, autos = 722 
B) electronics = 1252, steel = 1262, autos = 724 
C) electronics = 1254, steel = 1269, autos = 722 
D) electronics = 1254, steel = 1269, autos = 724 
E) electronics = 1252, steel = 1262, autos = 722

Question 8

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

Accepted Answer

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

Question 9

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. ​
​Pollution Abatement Capital Expenditures   ​Pollution Abatement Gross Operating Costs   ​

Accepted Answer

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

Question 10

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

Question 11

Use matrix A to find A2, if possible. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E) A2 is undefined. 
A)    
B)    
C)    
D)    
E) A2 is undefined.

Question 12

Use technology to find the product AB of the following matrices. ​     ​

Accepted Answer

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

Question 13

If A is a   matrix and B is its inverse, what does the product AB equal? ​

Accepted Answer

A)   
Zero matrix 
B)   
Identity matrix 
C)   
Identity matrix 
D)   
Zero matrix 
E) matrix A 
A)   
Zero matrix 
B)   
Identity matrix 
C)   
Identity matrix 
D)   
Zero matrix 
E) matrix A

Question 14

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 change in sales by type (wholesale and retail) and city (Chicago, Atlanta, and Memphis) from 2004 to 2005. ​   ​     ;   ​

Accepted Answer

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

Question 15

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 16

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 17

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 18

The following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of 1 unit of product. ​   ​
For each 100 units of manufactured products produced, how many units of fuels are required?
​

Accepted Answer

A) 33 
B) 2.9 
C) 1.1 
D) 29 
E) 11 
A) 33 
B) 2.9 
C) 1.1 
D) 29 
E) 11

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, if a solution exists.

Accepted Answer

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

Question 20

Multiplication by a matrix can be used to encode messages (and multiplication by its inverse can be used to decode messages). Given the code     , use matrix A to encode the message "The die is cast."

Accepted Answer

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

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

A man has invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is and the amount invested at 8% is twice that invested at 12%. What is the annual income from each property?

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

In order to rank the five members of its chess team for play against another school, the coach draws the following diagram. An arrow from 1 to 2 means player 1 has defeated player 2. Construct matrix A with elements .

Use the matrices below. Perform the indicated operations. DC

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.

Use matrix A to find A², if possible.

Use technology to find the product AB of the following matrices.

If A is a matrix and B is its inverse, what does the product AB equal?

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

Find x, y, z, and w .

The following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of 1 unit of product. For each 100 units of manufactured products produced, how many units of fuels are required?

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

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

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

A man has invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is and the amount invested at 8% is twice that invested at 12%. What is the annual income from each property? ​

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

In order to rank the five members of its chess team for play against another school, the coach draws the following diagram. An arrow from 1 to 2 means player 1 has defeated player 2. ​ Construct matrix A with elements . ​

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

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.

Use matrix A to find A2, if possible. ​ ​

Use technology to find the product AB of the following matrices. ​ ​

If A is a matrix and B is its inverse, what does the product AB equal? ​

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

Find x, y, z, and w . ​ ​

The following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of 1 unit of product. ​ ​ For each 100 units of manufactured products produced, how many units of fuels are required? ​

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

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

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

A man has invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is and the amount invested at 8% is twice that invested at 12%. What is the annual income from each property?

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

In order to rank the five members of its chess team for play against another school, the coach draws the following diagram. An arrow from 1 to 2 means player 1 has defeated player 2. Construct matrix A with elements .

Use the matrices below. Perform the indicated operations. DC

Use matrix A to find A², if possible.

Use technology to find the product AB of the following matrices.

If A is a matrix and B is its inverse, what does the product AB equal?

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

Find x, y, z, and w .

The following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of 1 unit of product. For each 100 units of manufactured products produced, how many units of fuels are required?