Deck 3: Matrices

Question 1

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

A)
B)
C)
D)
E)

Answer 1

Question 2

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

A)
B)
C)
D)
E)

Answer 2

Question 3

Write the transpose of matrix A. ​ ​

A)
B)
C)
D)
E)

Answer 3

Question 4

Write the matrix that is the negative of matrix B. ​ ​

A) ​ ​
B) ​​
C) ​​
D) ​​
E) ​ ​

Question 5

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

A)
B)
C)
D)
E)

Question 6

What is the order of matrices A, B, C, D, E? ​
​ , , , , ​

A) Matrix A is 2×2, matrix B is 3×2, matrix C is 2×3, matrix D is 3×3, matrix E is 5×2.
B) Matrix A is 3×1, matrix B is 3×3, matrix C is 2×5, matrix D is 12×11, matrix E is 3×7.
C) Matrix A is 2×2, matrix B is 2×3, matrix C is 3×2, matrix D is 3×3, matrix E is 2×5.
D) Matrix A is 2×2, matrix B is 3×3, matrix C is 3×3, matrix D is 3×3, matrix E is 5×5.
E) Matrix A is 1×2, matrix B is 3×2, matrix C is 2×3, matrix D is 12×3, matrix E is 3×2.

Question 7

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

A) ​ ​
B)
C) ​
D) ​
E) ​

Question 8

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

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

Question 9

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

A)
B)
C)
D)
E)

Question 10

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

A)
B)
C)
D)
E)

Question 11

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

A)
B)
C)
D)
E)

Question 12

What is element ? ​ ​

A)
B)
C)
D)
E)

Question 13

Write the transpose of matrix F. ​ ​

A)
B)
C)
D)
E)

Question 14

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

A)
B)
C)
D)
E)

Question 15

If matrix M has element , what is ? ​ ​

A)
B)
C)
D)
E)

Question 16

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

A)
B)
C)
D)
E)

Question 17

Find x, y, z and w. ​ ​

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

Question 18

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

A)
B)
C)
D)
E)

Question 19

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

A)
B)
C)
D)
E)

Question 20

What is the sum of matrix M and its negative? ​ ​

A)
B)
C)
D)
E)

Question 21

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

Question 22

A poll of 3620 people revealed that of the respondents who were registered Republicans, 893 approved of the president's job performance, 257 did not, and 135 had no opinion. Of the registered Democrats, 142 approved of the president's job performance, 426 did not, and 501 had no opinion. Of those registered as Independents, 177 approved, 64 did not approve, and 801 had no opinion. Of the remaining respondents, who were not registered, 142 approved, 38 did not approve, and 44 had no opinion. Represent these data in a 3×4 matrix.

A)
B)
C)
D)
E)

Question 23

The following tables give the death rates, per 100,000 population, by age for selected years for males and females. If matrix M gives the male data and matrix F gives the female data, find matrix M - F and describe what it means. ​​
​

A) ​
M - F represents how much greater the numbers are for males in each category per year.
B) ​
M - F represents how much greater the numbers are for males in each category per year.
C) ​
M - F represents how much greater the numbers are for females in each category per year.
D) M - F represents how much greater the numbers are for females in each category per year.
E) M - F represents only the male death rates, per 100,000 population by age.

Question 24

The tables below give the numbers of some species of threatened and endangered wildlife in the United States and in foreign countries in 2003. Let A represent the matrix that contains the number of each of these species in the United States in 2003 and B represent the matrix that contains the number of each of these species outside the United States in 2003. Find the matrix B - A. What do the negative entries in matrix B - A mean? ​
​

A) ​
The negative entries tell us that that particular species is extinct.
B) ​
The negative entries tell us how many more endangered or threatened species are in foreign countries.
C) ​
The negative entries tell us that there are more endangered or threatened species in foreign countries.
D) ​
The negative entries tell us that that particular species is extinct.
E) ​
The negative entries tell us how many more endangered or threatened species are in the United States.

Question 25

Find x, y, z and w. ​ ​

A) and
B) and
C) and ​
D) and
E) and

Question 26

Find x, y, z, and w . ​ ​

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

Question 27

The following tables give the death rates, per 100,000 population, by age for selected years for males and females. If matrix M gives the male data and matrix F gives the female data, use matrix operations to find the death rate per 100,000 for all people in the age categories given and for the given years. ​
​

A) ​
B) ​
C) ​
D)
E)

Question 28

​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 total U.S. trade with these groupings by finding A + B. ​

A)
B)
C)
D) ​
E)

Question 29

The following tables give the capital expenditures and gross operating costs of manufacturing establishments for pollution abatement, in millions of dollars. Which type of pollution abatement was most expensive in 2005? ​

A) ​Solid Contained Waste
B) ​Water
C) ​Air

Question 30

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Use a matrix operation to find the change from 1995 to 2002 in rank for each breed and in registration numbers for each breed. ​

A)
B)
C)
D)
E)

Question 31

From the data in the following table, make a matrix A that gives the value (in billions of dollars) of U.S. imports from the various country groupings in the years 2000 and 2002. ​

A)
B)
C)
D)
E)

Question 32

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. ​

A) ​ ,
B) ​ ,
C) ​ ,
D) ​ ,
E) ​ ,

Question 33

​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. ​

A)
B)
C)
D) ​
E)

Question 34

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Which breed had the greatest improvement in rank?

A)
B)
C)
D)
E)

Question 35

The following tables give the rank and number of registered dogs for the top ten breeds for 1995 and their 2002 data. Which breed had the largest increase in the number of registrations? ​

A) ​ ​
B)
C)
D)
E)

Question 36

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. ​

A) ​
B) ​
C) ​
D)
E)

Question 37

Solve for x, y and z. ​ ​

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

Question 38

The tables below give the numbers of some species of threatened and endangered wildlife in the United States and in foreign countries in 2003. Find a matrix with the total number of these species. Assume that U.S. and foreign species are different. ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 39

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. ​ ​ ; ​

A)
B) ​
C) ​
D)
E) ​

Question 40

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. ​ ​ ; ​

A)
B) ​
C) ​
D)
E) ​

Question 41

When a firm buys another company, the company frequently has some outstanding debt that the purchaser must pay. Consider three companies, A, B, and C, that are purchased by Maxx Industries. The following table gives the amount of each company's debt, classified by the number of days remaining until the debt must be paid. What payment plan did Maxx Industries use if the outstanding debt is given by the following matrix?

A) ​They paid 30% of the debts due in 30 days, 100% of the debts due in 60 days, and 100% of the debts due in more than 60 days.
B) ​They paid 30% of the debts due in 30 days, 30% of the debts due in 60 days, and 100% of the debts due in more than 60 days.
C) ​They paid 100% of the debts due in 30 days, 100% of the debts due in 60 days, and 30% of the debts due in more than 60 days.
D) They paid 100% of the debts due in 30 days, 30% of the debts due in 60 days, and 100% of the debts due in more than 60 days.
E) They paid 100% of the debts due in 30 days, 30% of the debts due in 60 days, and 30% of the debts due in more than 60 days.

Question 42

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

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

Question 43

Ace, Baker, and Champ are being purchased by ALCO, Inc., and their outstanding debts must be paid by the purchaser. The matrix below gives the amounts of debt and the terms for the companies being purchased. Suppose ALCO decides to pay 80% of all debts due in 30 days and to increase the debts due in 60 days by 20%. Write a matrix that gives the debts after these transactions are made. ​
​ ​

A)
B)
C)
D)
E)

Question 44

​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. ​​

Let matrix A represent the matrix with the following elements: .
Let matrix B represent the matrix with the following elements: .
The player i is the top-ranked player if the sum of row i in the matrix A + B is the largest. What is the number of this player?
​

A) ​player 3
B) ​player 2
C) ​player 5
D) ​player 1
E) ​player 4

Question 45

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

A)
B)
C)
D)
E)

Question 46

​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) ​
B) ​
C) ​
D)
E)

Question 47

Multiply the matrices. ​ ​

A)
B)
C)
D) ​
E)

Question 48

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

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

Question 49

Multiply the matrices. ​ ​

A)
B)
C)
D)
E)

Question 50

The figure that follows depicts the mean (average) flow time for a job when the critical-ratio rule is used to dispatch workers at several machines required to complete the job. In the figure, WHN 1, WHN 2, and WHN 3 represent three different rules for determining when a worker should be transferred to another machine, and Where represents six rules for determining the machine to which the worker is transferred. Construct a 3×6 matrix A to represent these data, with entries rounded to the nearest integer. ​ ​

A)
B)
C)
D)
E)

Question 51

Ace, Baker, and Champ are being purchased by ALCO, Inc., and their outstanding debts must be paid by the purchaser. The matrix below gives the amounts of debt and the terms for the companies being purchased. If ALCO pays 35% of the amount owed on each debt, write the matrix giving the remaining debts. ​
​ ​

A)
B)
C)
D)
E)

Question 52

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 53

The matrix that follows represents work efficiency for workers at nine different work centers. Suppose that the work centers changes in rules cause efficiency to decrease by 0.04 at work centers 2 and 5 and to decrease by 0.01 at work center 7. Write the matrix that describes the new efficiencies.​ ​
​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 54

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

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

Question 55

​When a firm buys another company, the company frequently has some outstanding debt that the purchaser must pay. Consider three companies, A, B, and C, that are purchased by Maxx Industries. The following table gives the amount of each company's debt, classified by the number of days remaining until the debt must be paid. Suppose Maxx Industries pays 20% of the amount owed on each loan. Write the matrix that would give the remaining debt in each category. ​

A) ​
B) ​
C) ​
D)
E)

Question 56

Use a computer spreadsheet to find the average efficiency for each work center over the first five workers. Which work center is the most efficient? ​
​

A) Number 4
B) Number 3
C) Number 8
D) Number 5
E) Number 1

Question 57

​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) ​
B) ​
C) ​
D)
E)

Question 58

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

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

Question 59

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

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

Question 60

Multiply the matrices. ​

A)
B)
C)
D) ​
E)

Question 61

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

A)
B)
C)
D)
E)

Question 62

Use matrix F to find F², if possible. ​ ​

A)
B)
C)
D)
E) F² is undefined.

Question 63

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

A)
B)
C)
D)
E) A² is undefined.

Question 64

Use matrix A to find A³, if possible. ​ ​

A)
B)
C)
D)
E) A³ is undefined.

Question 65

For , are FFT and F^TF defined?

Question 66

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.
​ , ,

Question 67

For , what size is each product FFT and F^TF? ​​

A) FFT is 2×2; F^TF is 2×2
B) FFT is 2×2; F^TF is 4×4
C) FFT is 4×2; F^TF is 4×4
D) FFT is 4×4; F^TF is 2×2
E) FFT is 2×4; F^TF is 4×2

Question 68

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

A)
B)
C)
D)
E)

Question 69

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 70

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

A)
B)
C)
D)
E)

Question 71

Is it true for matrices (as it is for real numbers) that multiplication by a zero matrix only sometimes gives is a result of a zero matrix provided the multiplication is defined?

Question 72

Find AB and BA if and , with . ​

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

Question 73

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 74

For , can FF^T = F^TF? Explain. ​

A) No, because FF^Tand F^TF have the same dimensions but different values.
B) Yes, because FF^T and F^TF have the same dimensions.
C) Yes, because FF^T and F^TF have the same values.
D) No, because FF^T and F^TF have different dimensions.
E) Yes, because FF^T and F^TF have the same dimensions and the same values.

Question 75

(CD)·E and C·(DE) are equal.
​

Question 76

Use the matrices to find , if possible. ​ ​

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

Question 77

Use the matrices below. Perform the indicated operations. ​
I·F·Z ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 78

CD = DC.
​

Question 79

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

A)
B)
C)
D)
E)

Question 80

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

A)
B)
C)
D)
E)