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By definition, A is idempotent if A2 = A, and B is nilpotent if Bm = 0 for some positive integer m. Give examples (different from 0 or I). Also give examples such that A2 = I (the unit matrix).
Find all 2 x 2 matrices A = [ajk], B = [bjk], C = [cjk] and general scalar.
Verigy (4) for the vectors in Probs. 21 and 23
Find all vectors v in R3 orthogonal to [2 0 1]T.
(c) Circle. Fine a similar formula for a circle in the plane through three given points. Find and sketch the circle through (2, 6), (6, 4), (7, 1).
(d) Sphere. Find the analog of the formula in (c) for a sphere through four given points. Find the sphere through (0, 0, 5), (4, 0, 1), (0, 4, 1), (0, 0, – 3) by this formula or byinspection.

(c) Circle. Fine a similar formula for a circle in
Calculate AB in Prob. 1 column wise. (See Example 6)
Find the value of the determinant of the n x n matrix and with main diagonal entries all 0 and all others 1. Try to find a formula for this. Try to prove it by induction. Interpret A3 and A4 as "incidence matrices" (as in Problem Set 7.1 but without the minuses) of a triangle and a tetrahedron, respectively; similarly for an "n-simplex", having n vertices and n (n – 1)/2 edges (and spanning Rn–1, n = 5, 6 . . .)
Give an application of the matrix in Prob. 3 that makes the form of its inverse obvious.
Prove the formula in Prob. 15.
Prove the formula in Prob. 17.
Same question as in Prob. 14 for the matrix in Prob. 9.
Find and Eigen basis and diagonalize, (Show thedetails).

Find and Eigen basis and diagonalize,
Find and Eigen basis and diagonalize, (Show thedetails).

Find and Eigen basis
Verify that A and  = P–1 AP have the same spectrum. Here A, Pare:

Verify that A and  = P–1 AP have the
Reduce the quadratic form to principal axes.
11.56x12 + 20.16x1 x2 + 17.44x22 = 100
Reduce the quadratic form to principal axes.
14x12 + 24x1 x2 – 4x22 = 20
Find and Eigen basis and diagonalize, (Show thedetails).

Find and Eigen basis and diagonalize, (Show the details)
Show (BA) T = – AB for A and B in Example 2, for any n X n Hermitian A and skew-Hermitian B.
Find the Eigen values and Eigen vectors of the so-called Pauli spin matrices and show that SxSy = iSz, SySx = – iSz, Sx2 = Sy2 = Sz2 =I,

Find the Eigen values and Eigen vectors of the so-called
Find the Eigen values and Eigen vectors of the so-called Pauli spin matrices and show that SxSy = iSz, SySx = – iSz, Sx2 = Sy2 = Sz2 =I,

CAS Experiment Orthogonal Matrices  (a) Prove that the product o
Prove the following statements and illustrate them with examples of your own choice. Here, λ1, ∙ ∙ ∙ λn are the (not necessarily distinct) Eigen values of a given n x n matrix A = [ajk].
(a) Trace the sum of the main diagonal entries is called the trace of A. It equals the sum of the Eigen values.
(b) “Spectral shift”. A – k1 has the Eigen values λ1 – k, ∙∙∙ λn – k and the same Eigen vectors as A.
(c) Scalar multiples, powers. kA has the Eigen values k λ1, ∙∙∙ kλn. Am (m = 1, 2, ∙∙∙) has the Eigen values λ1m, ∙∙∙ λnm. The Eigen vectors are those of A. (d) Spectral mapping theorem, the "polynomial matrix"
Has the Eigen values where j = 1, ∙∙∙ nm and the same Eigen vectors as A.
(e) Peron’s theorem, Show that a Leslie matrix L with positive l12, l13, l21, l32, has a positive Eigen value. (This is a special case of the famous Perron-Frobenius theorem in Sec. 20.7, which is difficult to prove in its general form.).


Prove the following statements and illustrate them with examples
Show that the inverse A–1 exists if and only if none of the eigenvalues λ1, ∙ ∙ ∙, λn of A is zero, and then A–1 has the eigenvalues 1/ λ1, ∙ ∙ ∙, 1/λn.
If |p| = 1 and |q| = 2, what can be said about the magnitude and direction of the resultant? Can you think of an application where this matters?
If a ray of light is reflected once in each of two mutually perpendicular mirrors, what can you say about the reflected ray?
Team Project, Geometric Applications
To increase your skill in dealing with vectors, use vectors to prove the following (see the figures)
(a) The diagonals of a parallelogram bisect each other.
(b) The line through the midpoints of adjacent sides of a parallelogram bisects one of the diagonals in the ratio 1:3.
(c) Obtain (b) from (a).
(d) The three medians of a triangle (the segments from a vertex to the midpoint of the opposite side) meet at a single point, which divides the medians in the ratio 2:1.
(e) The quadrilateral whose vertices are the midpoints of the sides of an arbitrary quadrilateral is a parallelogram.
(f) The four space diagonals of a parallelepiped meet and bisect each other.
(g) The sum of the vectors drawn from the center of a regular polygon to its vertices is the zerovectors.

Team Project, Geometric Applications  To increase your skill in
Find Eigen vectors of A, B, C in Examples 2 and 3.
Show that a consumption matrix as considered in Prob. 15 must have column sums 1 and always has the eigenvalue 1.
Determine whether the statement is true or false.
The lines 2x + 3y = 4 and 3x - 2y = 4 are perpendicular.
Determine whether a linear model might fit the data.
(a)
Determine whether a linear model might fit the data.
(a)
(b)
(c)
(d)
(b)
Determine whether a linear model might fit the data.
(a)
(b)
(c)
(d)
(c)
Determine whether a linear model might fit the data.
(a)
(b)
(c)
(d)
(d)
Determine whether a linear model might fit the data.
(a)
(b)
(c)
(d)
The table below illustrates the growth in worldwide Internet use.
Year, x Number of Internet Users Worldwide,
y (in millions)
2001, 0 ............................................... 495
2002, 1 ............................................... 677
2003, 2 ............................................... 785
2004, 3 ............................................... 914
2005, 4 ............................................... 1036
2006, 5 ............................................... 1159
2007, 6 ............................................... 1393
2008, 7 ............................................... 1611
2009, 8 ............................................... 1858
2010, 9 ............................................... 2084*
(a) Model the data with a linear function. Let the independent variable represent the number of years after 2001; that is, the data points are 10, 4952, 13, 9142, and so on. Answers may vary depending on the data points used.
(b) Using the function found in part (a), estimate the number of Internet users worldwide in 2013 and in 2018.
The table below illustrates the upward trend in America to choose cremation.
Year, x Percentage of Deaths
Followed by Cremation, y
2005, 0 .............................................. 32.3%
2006, 1 ........................................ 33.6
2007, 3 ........................................ 34.3
2008, 4 ........................................ 35.8
2009, 4 ........................................ 36.9
(a) Model the data with a linear function. Let the independent variable represent the number of years after 2005. Answers may vary depending on the data points used.
(b) Using the function found in part (a), estimate the percentage of deaths followed by cremation in 2013 and in 2016.
Data on total U.S. expenditures on pets, pet products, and related services are given in the table below. Model the data with a linear function. Then, using that function, estimate total U.S. expenditures on pets in 2005 and predict total expenditures in 2015. Answers may vary depending on the data points used.
Year, x Total U.S. Expenditures
On pets (in billions)
1991, 0 ............................................... $19.6
1994, 3 ........................................... 24.9
1997, 6 ........................................... 32.5
2000, 9 .......................................... 39.7
2003, 12 .......................................... 46.8
2006, 15 .......................................... 56.9
2009, 18 .......................................... 67.1
Data on total U.S. expenditures on pets, pet products, and
Data on the median age of the U.S. population in selected years are listed in the table below. Model the data with a linear function, estimate the median age in 1998, and predict the median age in 2020. Answers may vary depending on the data points used.
Year, x Total U.S. Expenditures
1970, 0 ............................ 28.0
1980, 10 ......................... 30.0
1990, 20 ......................... 32.8
2000, 30 ......................... 35.3
2008, 38 .......................... 36.8
2011, 41 .......................... 36.9*
The net sales data in several years for Nike are given in the table at right. Model the data with a linear function, and predict the net sales in 2015. Answers may vary depending on the data points used.
Year, x Net Sales (in billions)
2005, 0 .......................... $13.7
2006, 1 ........................... 15.0
2007, 2 ........................... 16.3
2008, 3 ........................... 18.6
2009, 4 ........................... 19.2
2010, 5 ........................... 19.0
Data on average credit-card debt per U.S. household are given in the table below. Model the data with a linear function, and estimate the average debt in 2007 and in 2014. Answers may vary depending on the data points used.
Year, x Credit-Card Debt
Per Household, y
1992, 0 .............................................. $ 3,803
1996, 0 ........................................ 6,912
2000, 8 ........................................ 8,308
2004, 12 ........................................ 9,577
2008, 16 ........................................ 10,691
2011, 19 ........................................ 14,750
(a) Use a graphing calculator to fit a regression line to the data in Exercise 61.
(b) Estimate the number of Internet users worldwide in 2013 and compare the value with the result found in Exercise 61.
In Exercise 61
Year, x Number of Internet Users Worldwide,
y (in millions)
2001, 0 ............................................... 495
2002, 1 ............................................... 677
2003, 2 ............................................... 785
2004, 3 ............................................... 914
2005, 4 ............................................... 1036
2006, 5 ............................................... 1159
2007, 6 ............................................... 1393
2008, 7 ............................................... 1611
2009, 8 ............................................... 1858
2010, 9 ............................................... 2084*
(c) Find the correlation coefficient for the regression line and determine whether the line fits the data closely.
(a) Use a graphing calculator to fit a regression line to the data in Exercise 62.
In Exercise 62
Year, x Percentage of Deaths
Followed by Cremation, y
2005, 0 .............................................. 32.3%
2006, 1 ........................................ 33.6
2007, 3 ........................................ 34.3
2008, 4 ........................................ 35.8
2009, 4 ........................................ 36.9
(b) Estimate the percentage of deaths followed by cremation in 2013 and compares the result with the estimate found with the model in Exercise 62.
(c) Find the correlation coefficient for the regression line and determine whether the line fits the data closely.
(a) Use a graphing calculator to fit a regression line to the data in Exercise 63.
In Exercise 63
Year, x Total U.S. Expenditures
On pets (in billions)
1991, 0 ............................................... $19.6
1994, 3 ........................................... 24.9
1997, 6 ........................................... 32.5
2000, 9 .......................................... 39.7
2003, 12 .......................................... 46.8
2006, 15 .......................................... 56.9
2009, 18 .......................................... 67.1
(b) Estimate total U.S. expenditures on pets in 2015 and compare the value with the result found in Exercise 63.
(c) Find the correlation coefficient for the regression line and determine whether the line fits the data closely.
Write a slope-intercept equation for a line with the given characteristics.
(a) m = 2/9, y-intercept (0, 4)
(b) m = - 3/8, y-intercept (0, 5)
(c) m = -4, y-intercept (0, -7)
(a) Use a graphing calculator to fit a regression line to the data in Exercise 66.
In Exercise 66
Year, x Credit-Card Debt
Per Household, y
1992, 0 .............................................. $ 3,803
1996, 0 ........................................ 6,912
2000, 8 ........................................ 8,308
2004, 12 ........................................ 9,577
2008, 16 ........................................ 10,691
2011, 19 ........................................ 14,750
(b) Estimate average credit-card debt per U.S. house hold in 2014 and compare the result with the estimate found with the model in Exercise 66.
(c) Find the correlation coefficient for the regression line and determine whether the line fits the data closely.
A person who is exercising should not exceed his or her maximum heart rate, which is determined on the basis of that person's sex, age, and resting heart rate. The table below relates resting heart rate and maximum heart rate for a 20-year-old man.
Resting Heart Rate, Maximum Heart Rate,
H (in beats per minute) M (in beats per minute)
50 .................................................... 166
60 .................................................... 168
70 .................................................... 170
80 .................................................... 172
(a) Use a graphing calculator to model the data with a linear function.
(b) Estimate the maximum heart rate if the resting heart rate is 40, 65, 76, and 84.
(c) What is the correlation coefficient? How confident are you about using the regression line to estimate function values?
A math instructor asked her students to keep track of how much time each spent studying a chapter on functions in her algebra-trigonometry course. She collected the information together with test scores from that chapter's test. The data are listed in the table below.
Study Time, x (in hours) Test Grade, y (in percent)
23 ............................................. 81%
15 ............................................. 85
17 ............................................. 80
9 .............................................. 75
21 ............................................. 86
13 ............................................. 80
16 ............................................. 85
11 ............................................. 93
(a) Use a graphing calculator to model the data with a linear function.
(b) Predict a student's score if he or she studies 24 hr, 6 hr, and 18 hr.
(c) What is the correlation coefficient? How confident are you about using the regression line to predict function values?
Find the slope of the line containing the given points.
(a) (2, -8) and (-5, -1)
(b) (5, 7) and (5, -7)
Find an equation for a circle satisfying the given conditions.
(a) Center (-7, -1), radius of length 9 / 5
(b) Center (0, 3), diameter of length 5
Find k so that the line containing the points (-3, k) and (4, 8) is parallel to the line containing the points (5, 3) and (1, -6).
Using the figure below, find the road grade and an equation giving the height y as a function of the horizontal distance x.
Using the figure below, find the road grade and an
Find an equation of the line passing through the point (4, 5) and perpendicular to the line passing through the points (-1, 3) and (2, 9).
Use this graph for Exercised 1 and 2.
Use this graph for Exercised 1 and 2.
(a) Find the
(a) Find the coordinates of points A, B, C, D, E, and F.
(b) Find the coordinates of points G, H, I, J, K, and L.
Dannon recently replaced its 8-oz cup of yogurt with a 6-oz cup and reduced the suggested retail price from 89 cents to 71 cents. Was the price per ounce reduced by the same percent as the size of the cup? If not, find the price difference per ounce in terms of a percent.
One week 10 copies of the novel The Last Song by Nicholas Sparks were sold for every 7.9 copies of David Baldacci's Deliver Us from Evil that were sold. If a total of 10,919 copies of the two books were sold, how many copies of each were sold?
A 150-lb person who runs at 6 mph for 1 hr burns about 720 calories. The same person, walking at 4 mph for 90 min, burns about 480 calories. Suppose a 150-lb person runs at 6 mph for 75 min. How far would the person have to walk at 4 mph in order to burn the same number of calories used running?
In the 2009-2010 school year in the United States, there were 128,000 students from China. This number is 22% more than the number of students from India. How many foreign students were from India?
The Toyota Prius gets 44 miles per gallon (mpg) overall. The Hummer H2 gets 11 mpg less than one-half of the milesper- gallon rate for the Toyota Prius. Find the milesper- gallon rate for the Hummer H2.
It is estimated that 710,000 metric tons of olive oil were consumed in Italy in 2009-2010. This is 60,000 metric tons more than 2.5 times the amount consumed in the United States during the same time period. Find the amount of olive oil consumed in the United States in 2009-2010.
It is estimated that 710,000 metric tons of olive oil
The average salary of a landscape architect for the federal government is $80,830 per year. This is about 38.5% higher than the yearly salary of a private-sector landscape architect. Find the salary of a private-sector landscape architect.
The average depth of the Pacific Ocean is 14,040 ft. This is 8890 ft less than the sum of the average depths of the Atlantic Ocean and the Indian Ocean. The average depth of the Indian Ocean is 272 ft less than four-fifths of the average depth of the Atlantic Ocean. Find the average depth of the Indian Ocean.
The average depth of the Pacific Ocean is 14,040 ft.
Of each dollar spent on textbooks at college bookstores, 22.3 cents goes to the college store for profit, store operations, and personnel. On average, a college student at a four-year college spends $667 per year for textbooks. How much of this expenditure goes to the college store?
The total curb weight of a Toyota Tundra truck, a Ford Mustang car, and a Smart For Two car is 11,150 lb. The weight of a Mustang is 5 lb less than the weight of two Smart For Two cars. The Tundra weighs 2135 lb more than the Mustang. What is the curb weight of each vehicle?
The total curb weight of a Toyota Tundra truck, a
The points (7, 13) and (-3, -11) are at the ends of a diameter.
In a recent week, the television networks CBS, ABC, and NBC together averaged a total of 29.1 million viewers. CBS had 1.7 million more viewers than ABC, and NBC had 1.7 million fewer viewers than ABC.
Nielsen Media Research surveys TV-watching habits and provides a list of the 20 most-watched TV programs each week. Each rating point in the survey represents 1,102,000 households. One week "60 Minutes" had a rating of 11.0. How many households did this represent?
Kendal borrowed money from her father at 5% simple interest to help pay her tuition at Wellington Community College. At the end of 1 year, she owed a total of $1365 in principal and interest. How much did she borrow?
Ryan, a consumer electronics salesperson, earns a base salary of $1500 per month and a commission of 8% on the amount of sales he makes. One month Ryan received a paycheck for $2284. Find the amount of his sales for the month.
Juliet has a choice between receiving a monthly salary of $1800 from Furniture by Design or a base salary of $1600 and a 4% commission on the amount of furniture she sells during the month. For what amount of sales will the two choices be equal?
Soledad worked 48 hr one week and earned a $442 paycheck. She earns time and a half (1.5 times her regular hourly wage) for the number of hours she works in excess of 40. What is Soledad's regular hourly wage?
Soledad worked 48 hr one week and earned a $442
City Cabs charges a $1.75 pickup fee and $1.50 per mile traveled. Diego's fare for a cross-town cab ride is $19.75. How far did he travel in the cab?
In triangle ABC, angle B is five times as large as angle A. The measure of angle C is 2o less than that of angle A. Find the measures of the angles.
In triangle ABC, angle B is twice as large as angle A. Angle C measures 20o more than angle A. Find the measures of the angles.
Morgan's Seeds has a rectangular test plot with a perimeter of 322 m. The length is 25 m more than the width. Find the dimensions of the plot.
Morgan's Seeds has a rectangular test plot with a perimeter
The points (-9, 4), (-2, 5), (-8, -3), and (-1, -2) are vertices of an inscribed square.
The children at Tiny Tots Day Care planted a rectangular vegetable garden with a perimeter of 39 m. The length is twice the width. Find the dimensions of the garden.
The width of the soccer field recommended for players under the age of 12 is 35 yd less than the length. The perimeter of the field is 330 yd. find the dimensions of the field.
Marissa is designing a poster to promote the Talbot Street Art Fair. The width of the poster will be two-thirds of its height, and its perimeter will be 100 in. Find the dimensions of the poster.
Water accounts for 55% of a woman's weight (ga.water.usgs.gov/edu). Lily weighs 135 lb. How much of her body weight is water?
Water accounts for 60% of a man's weight (ga.water.usgs.gov/edu). Jake weighs 186 lb. How much of his body weight is water?
A Central Railway freight train leaves a station and travels due north at a speed of 60 mph. One hour later, an Amtrak passenger train leaves the same station and travels due north on a parallel track at a speed of 80 mph. How long will it take the passenger train to overtake the freight train?
A private airplane leaves Midway Airport and flies due east at a speed of 180 km/h. Two hours later, a jet leaves Midway and flies due east at a speed of 900 km/h. How far from the airport will the jet overtake the private plane?
A kayak moves at a rate of 12 mph in still water. If the river's current flows at a rate of 4 mph, how long does it take the boat to travel 36 mi upstream?
A kayak moves at a rate of 12 mph in
Angelo's kayak travels 14 km/h in still water. If the river's current flows at a rate of 2 km/h, how long will it take him to travel 20 km downstream?
An airplane that travels 450 mph in still air encounters a 30-mph headwind. How long will it take the plane to travel 1050 mi into the wind?
Center (-2, 3), tangent (touching at one point) to the y-axis
Center (-2, 3), tangent (touching at one point) to the
An airplane that can travel 375 mph in still air is flying with a 25-mph tailwind. How long will it take the plane to travel 700 mi with the wind?
Erica invested a total of $5000, part at 3% simple interest and part at 4% simple interest. At the end of 1 year, the investments had earned $176 interest. How much was invested at each rate?
Dimitri's two student loans total $9000. One loan is at 5% simple interest and the other is at 6% simple interest. At the end of 1 year, Dimitri owes $492 in interest. What is the amount of each loan?
In February 2011, facebook.com had 134,078,221 unique visitors. This number of visitors was 51,604,956 less than the total number of visitors to YouTube.com and amazon.com. The number who visited YouTube.com was 42,826,225 more than the number who visited amazon.com. Find the number of visitors to YouTube.com and to amazon.com.
Together, one 8-oz serving of plain nonfat yogurt and one 1-oz serving of Swiss cheese contain 676 mg of calcium. The yogurt contains 4 mg more than twice the calcium in the cheese. Find the calcium content of each food.
The elevations of the 31 NFL stadiums range from 3 ft at MetLife Stadium in East Rutherford, New Jersey, to 5210 ft at Sports Authority Field at Mile High in Denver, Colorado. The elevation of Sports Authority Field at Mile High is 247 ft higher than seven times the elevation of Lucas Oil Stadium in Indianapolis, Indiana. What is the elevation of Lucas Oil Stadium?
The elevations of the 31 NFL stadiums range from 3
There is a total of 1525 public libraries in New York and Wisconsin. There are 151 more libraries in New York than twice the number in Wisconsin. Find the number of public libraries in New York and in Wisconsin.
There is a total of 1525 public libraries in New
In the Dominican Republic, factory-bottled water is the primary source of drinking water for 67% of the urban population. In 2009, the population of the Dominican Republic was 9,650,054, of which 66.8% was urban. For how many in the urban population of the Dominican Republic was bottled water the primary source of drinking water?
In the Dominican Republic, factory-bottled water is the primary source
A volcano that is currently about one-half mile below the surface of the Pacific Ocean near the Big Island of Hawaii will eventually become a new Hawaiian island, Loihi. The volcano will break the surface of the ocean in about 50,000 years. On average, how many inches does the volcano rise in a year?
Find the zero of the linear function.
(a) f(x) = x + 5
(b) f(x) = 5x + 20
(c) f(x) = -2x + 11
Center (4, -5) tangent to the x-axis
Use the given graph to find each of the following: (a) the x-intercept and (b) the zero of the function.
(a)
Use the given graph to find each of the following:
(b)
Use the given graph to find each of the following:
(c)
Use the given graph to find each of the following:
Write a slope-intercept equation for the line containing the point (-1, 4) and parallel to the line 3x + 4y = 7.
Write an equation of the line containing the points (-5, 4) and (3, -2).
Find the distance between (2, 2) and (-3, - 10).
Find the midpoint of the segment with end points (- 1 /2, 2 / 5), and (- 3 / 2, 3 / 5).
Given that f(x) = x / x - 3, find f(-3), f(0), and f(3).
Find the slope and the y-intercept of the line with the equation 7x - y = 1 / 2.
State whether each of the following is a linear function.
(a) f(x) = 7 - 3 / 2 x
(b) f (x) = 3 / 2x + 5
(c) f(x) = x2 + 1
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