Question 1

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 2

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 3

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

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 $16,300, and risk factors make it prudent to invest some money in all three investments. Let x denote the amount invested at 8 %, y denote the amount invested at 7%, and z denote the amount invested at 10%. Find a general description for the amounts invested at the three rates. ​

Accepted Answer

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

Question 5

Suppose the weights (in grams) and lengths (in centimeters) of three groups of laboratory animals are given by matrix A, where column 1 gives the lengths and each row corresponds to one group. ​   ​
​
If the increase in both weight and length over the next two weeks is 12% for group I, 5% for group II, and 3% for group III, then the increases in the measures during the 2 weeks can be found by computing GA, where
​   ​
​
Find the matrix that gives the new weights and measures at the end of this period by computing   , where   is the 3×3 identity matrix.
​

Accepted Answer

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

Question 6

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)

Question 7

Set up the system of equations and then solve it by using inverse matrices. Transportation The Ace Freight Company has an order for two products to be delivered to two stores of a company. The table below gives information regarding the two products.
​
 
​
​
If truck A can carry 2300 cu ft and 23,000 lb, how many of each product can it carry?
​

Accepted Answer

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

Question 8

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 9

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 10

​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. ​

Accepted Answer

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

Question 11

A man has $282,000 invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is $27,000 and the amount invested at 8% is twice that invested at 12%. How much is invested in each property?

Accepted Answer

A)  $33,840 at 12%, $28,200 at 10%, $22,560 at 8% 
B)  $88,800 at 12%, $44,400 at 10%, $148,800 at 8% 
C)  $76,800 at 12%, $68,400 at 10%, $136,800 at 8% 
D)  $55,500 at 12%, $111,000 at 10%, $115,500 at 8% 
E)  $60,000 at 12%, $102,000 at 10%, $120,000 at 8% 
A)  $33,840 at 12%, $28,200 at 10%, $22,560 at 8% 
B)  $88,800 at 12%, $44,400 at 10%, $148,800 at 8% 
C)  $76,800 at 12%, $68,400 at 10%, $136,800 at 8% 
D)  $55,500 at 12%, $111,000 at 10%, $115,500 at 8% 
E)  $60,000 at 12%, $102,000 at 10%, $120,000 at 8%

Question 12

Describe the procedure for finding the general solution of a system of three linear equations involving p, q, and r if its reduced matrix indicate has one row of zeros at the bottom. You may assume that the coefficients in the augmented matrix correspond to variables in alphabetical order from left to right. ​

Accepted Answer

A)  Use the reduced matrix to find equations for each variable p, q, and r. These equations can be used as the general solution to the system of equations. 
B)  Use the reduced matrix to find a general equation for p in terms of q and r. This equation can be used as the general solution to the system of equations. 
C)  Use the reduced matrix to solve for p and q in terms of r. These solutions will be valid for any real value of r. 
D)  Use the reduced matrix to solve for r in terms of p and q. This solution will be valid for any real value of p and q. 
E)  It is not possible to find a general solution for a system of three linear equations if its reduced matrix indicates an infinite number of solutions. 
A)  Use the reduced matrix to find equations for each variable p, q, and r. These equations can be used as the general solution to the system of equations. 
B)  Use the reduced matrix to find a general equation for p in terms of q and r. This equation can be used as the general solution to the system of equations. 
C)  Use the reduced matrix to solve for p and q in terms of r. These solutions will be valid for any real value of r. 
D)  Use the reduced matrix to solve for r in terms of p and q. This solution will be valid for any real value of p and q. 
E)  It is not possible to find a general solution for a system of three linear equations if its reduced matrix indicates an infinite number of solutions.

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

The given matrix is an augmented matrix used in the solution of a system of linear equations. What is the solution of the system?

Accepted Answer

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

Question 15

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Form a matrix for the 1995 data and a matrix for the 2002 data. ​

Accepted Answer

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

Question 16

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 17

The tables below give the numbers of some species of threatened and endangered wildlife in the United States and in foreign countries in 2003. Write a matrix A that contains the number of each of these species in the United States in 2003 and a matrix B that contains the number of each of these species outside the United States in 2003. ​

Accepted Answer

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

Question 18

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 19

Use determinants to decide whether the matrix has an inverse.
​

Accepted Answer

A) True 
 B)False

Question 20

Find AB and BA if   and   , with   . ​

Accepted Answer

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

Find the inverse matrix, if one exists.

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

Use the matrices below. Perform the indicated operations. DC

Find the inverse matrix, if one exists.

Find x, y, z and w.

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.

A man has $282,000 invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is $27,000 and the amount invested at 8% is twice that invested at 12%. How much is invested in each property?

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.

The given matrix is an augmented matrix used in the solution of a system of linear equations. What is the solution of the system?

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Form a matrix for the 1995 data and a matrix for the 2002 data.

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

Use determinants to decide whether the matrix has an inverse.

Find AB and BA if and , with .

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

Find the inverse matrix, if one exists. ​

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

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

Find the inverse matrix, if one exists.

Find x, y, z and w. ​ ​

​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. ​

A man has $282,000 invested in three properties. One earns 12%, one 10%, and one 8%. His annual income from the properties is $27,000 and the amount invested at 8% is twice that invested at 12%. How much is invested in each property?

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. ​

The given matrix is an augmented matrix used in the solution of a system of linear equations. What is the solution of the system?

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Form a matrix for the 1995 data and a matrix for the 2002 data. ​

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

Use determinants to decide whether the matrix has an inverse. ​

Find AB and BA if and , with . ​

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

Find the inverse matrix, if one exists.

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

Use the matrices below. Perform the indicated operations. DC

Find x, y, z and w.

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.

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.

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Form a matrix for the 1995 data and a matrix for the 2002 data.

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

Use determinants to decide whether the matrix has an inverse.

Find AB and BA if and , with .