Chapter 5

The Campus Bookstore's inventory of books is Hardcover: textbooks, 5280 ; fiction, 1680 ; nonfiction, 2320; reference, 1890 Paperback: fiction, 2810; nonfiction, 1490; reference, \(2070 ;\) textbooks, 1940 The College Bookstore's inventory of books is Hardcover: textbooks, \(6340 ;\) fiction, 2220 ; nonfiction, \(1790 ;\) reference, 1980 Paperback: fiction, 3100; nonfiction, 1720; reference, \(2710 ;\) textbooks, 2050 a. Represent Campus's inventory as a matrix \(A\). b. Represent College's inventory as a matrix \(B\). c. The two companies decide to merge, so now write a matrix \(C\) that represents the total inventory of the newly amalgamated company.

Determine the value of \(k\) such that the following system of linear equations has a solution, and then find the solution: $$\begin{array}{r}2 x+3 y=2 \\\=x+4 y=6 \\\5 x+k y=2\end{array}$$

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{array}{rr}2 x+y-2 z= & 4 \\ x+3 y-z= & -3 \\ 3 x+4 y-z= & 7\end{array}\)

Jackson Farms has allotted a certain amount of land for cultivating soybeans, corn, and wheat. Cultivating 1 acre of soybeans requires 2 labor-hours, and cultivating 1 acre of corn or wheat requires 6 labor-hours. The cost of seeds for 1 acre of soybeans is \(\$ 12\), for 1 acre of corn is \(\$ 20\), and for 1 acre of wheat is \(\$ 8\). If all resources are to be used, how many acres of each crop should be cultivated if the following hold? a. 1000 acres of land are allotted, 4400 labor-hours are available, and \(\$ 13,200\) is available for seeds. b. 1200 acres of land are allotted, 5200 labor-hours are available, and \(\$ 16,400\) is available for seeds.

The property damage claim frequencies per 100 cars in Massachusetts in the years 2000,2001 , and 2002 are \(6.88,7.05\), and \(7.18\), respectively. The corresponding claim frequencies in the United States are 4.13, \(4.09\), and \(4.06\), respectively. Express this information using \(2 \times 3\) matrix.

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned} 2 x+2 y+z=& 9 \\ x+z=& 4 \\ 4 y-3 z=& 17 \end{aligned}\)

Lawnco produces three grades of commercial fertilizers. A \(100-\mathrm{lb}\) bag of grade \(\mathrm{A}\) fertilizer contains \(18 \mathrm{lb}\) of nitrogen, \(4 \mathrm{lb}\) of phosphate, and \(5 \mathrm{lb}\) of potassium. A \(100-\mathrm{lb}\) bag of grade \(\mathrm{B}\) fertilizer contains \(20 \mathrm{lb}\) of nitrogen and \(4 \mathrm{lb}\) each of phosphate and potassium. A 100-lb bag of grade \(\mathrm{C}\) fertilizer contains \(24 \mathrm{lb}\) of nitrogen, \(3 \mathrm{lb}\) of phosphate, and \(6 \mathrm{lb}\) of potassium. How many 100 -lb bags of each of the three grades of fertilizers should Lawnco produce if a. \(26,400 \mathrm{lb}\) of nitrogen, \(4900 \mathrm{lb}\) of phosphate, and \(6200 \mathrm{lb}\) of potassium are available and all the nutrients are used? b. \(21,800 \mathrm{lb}\) of nitrogen, \(4200 \mathrm{lb}\) of phosphate, and \(5300 \mathrm{lb}\) of potassium are available and all the nutrients are used?

Ethan just returned to the United States from a Southeast Asian trip and wishes to exchange the various foreign currencies that he has accumulated for U.S. dollars. He has 1200 Thai bahts, 80,000 Indonesian rupiahs, 42 Malaysian ringgits, and 36 Singapore dollars. Suppose the foreign exchange rates are U.S. \(\$ 0.03\) for one baht, U.S. \(\$ 0.00011\) for one rupiah, U.S. \(\$ 0.294\) for one Malaysian ringgit, and U.S. \(\$ 0.656\) for one Singapore dollar. a. Write a row matrix \(A\) giving the value of the various currencies that Ethan holds. (Note: The answer is not unique.) b. Write a column matrix \(B\) giving the exchange rates for the various currencies. c. If Ethan exchanges all of his foreign currencies for U.S. dollars, how many dollars will he have?

Mortality actuarial tables in the United States were revised in 2001, the fourth time since 1858 . Based on the new life insurance mortality rates, \(1 \%\) of 60 -yr-old men, \(2.6 \%\) of 70 -yr-old men, \(7 \%\) of 80 -yr-old men, \(18.8 \%\) of 90 -yr-old men, and \(36.3 \%\) of 100 -yr-old men would die within a year. The corresponding rates for women are \(0.8 \%, 1.8 \%, 4.4 \%, 12.2 \%\), and \(27.6 \%\), respectively. Express this information using a \(2 \times 5\) matrix.

Solve the system: $$\begin{array}{l}\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=-1 \\\\\frac{2}{x}+\frac{3}{y}+\frac{2}{z}=3 \\ \frac{2}{x}+\frac{1}{y}+\frac{2}{z}=-7\end{array}$$

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned} 2 x+3 y-2 z=& 10 \\ 3 x-2 y+2 z=& 0 \\ 4 x-y+3 z=&-1 \end{aligned}\)

A private investment club has a certain amount of money earmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high risk, medium risk, and low risk. Management estimates that high-risk stocks will have a rate of return of \(15 \%\) year; medium-risk stocks, \(10 \% /\) year; and low-risk stocks, \(6 \% /\) year. The members have decided that the investment in low-risk stocks should be equal to the sum of the investments in the stocks of the other two categories. Determine how much the club should invest in each type of stock in each of the following scenarios. (In all cases, assume that the entire sum available for investment is invested.) a. The club has \(\$ 200,000\) to invest, and the investment goal is to have a return of \(\$ 20,000 /\) year on the total investment. b. The club has $$\$ 220,000$$ to invest, and the investment goal is to have a return of $$\$ 22,000$$ year on the total investment. c. The club has $$\$ 240,000$$ to invest, and the investment goal is to have a return of $$\$ 22,000 /$$ year on the total investment.

Kaitlin and her friend Emma returned to the United States from a tour of four cities: Oslo, Stockholm, Copenhagen, and Saint Petersburg. They now wish to exchange the various foreign currencies that they have accumulated for U.S. dollars. Kaitlin has 82 Norwegian krones, 68 Swedish krones, 62 Danish krones, and 1200 Russian rubles. Emma has 64 Norwegian krones, 74 Swedish krones, 44 Danish krones, and 1600 Russian rubles. Suppose the exchange rates are U.S. \(\$ 0.1651\) for one Norwegian krone, U.S. \(\$ 0.1462\) for one Swedish krone, U.S. \(\$ 0.1811\) for one Danish krone, and U.S. \(\$ 0.0387\) for one Russian ruble. a. Write a \(2 \times 4\) matrix \(A\) giving the values of the various foreign currencies held by Kaitlin and Emma. (Note: The answer is not unique.) b. Write a column matrix \(B\) giving the exchange rate for the various currencies. c. If both Kaitlin and Emma exchange all their foreign currencies for U.S. dollars, how many dollars will each have?

Figures for life expectancy at birth of Massachusetts residents in 2002 are \(81.0,76.1\), and \(82.2 \mathrm{yr}\) for white, black, and Hispanic women, respectively, and \(76.0 .69 .9\), and \(75.9\) years for white, black, and Hispanic men, respectively. Express this information using a \(2 \times 3\) matrix and a \(3 \times 2\) matrix.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. A system of linear equations having fewer equations than variables has no solution, a unique solution, or infinitely many solutions.

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned}-x_{2}+x_{3} &=2 \\ 4 x_{1}-3 x_{2}+2 x_{3} &=16 \\ 3 x_{1}+2 x_{2}+x_{3} &=11 \end{aligned}\)

The Carver Foundation funds three nonprofit organizations engaged in alternate-energy research activities. From past data, the proportion of funds spent by each organization in research on solar energy, energy from harnessing the wind, and energy from the motion of ocean tides is given in the accompanying table. $$\begin{array}{lccc}\hline &{\text { Proportion of Money Spent }} \\ & \text { Solar } & \text { Wind } & \text { Tides } \\\\\hline \text { Organization I } & 0.6 & 0.3 & 0.1 \\ \hline \text { Organization II } & 0.4 & 0.3 & 0.3 \\\\\hline \text { Organization III } & 0.2 & 0.6 & 0.2 \\ \hline\end{array}$$ Find the amount awarded to each organization if the total amount spent by all three organizations on solar, wind, and tidal research is a. $$\$ 9.2$$ million, $$\$ 9.6$$ million, and $$\$ 5.2$$ million, respectively. b. $$\$ 8.2$$ million, $$\$ 7.2$$ million, and $$\$ 3.6$$ million, respectively.

The market share of motorcycles in the United States in 2001 follows: Honda \(27.9 \%\), Harley-Davidson \(21.9 \%\), Yamaha \(19.2 \%\), Suzuki \(11.0 \%\), Kawasaki \(9.1 \%\), and others \(10.9 \%\). The corresponding figures for 2002 are \(27.6 \%, 23.3 \%, 18.2 \%\), \(10.5 \%, 8.8 \%\), and \(11.6 \%\), respectively. Express this information using a \(2 \times 6\) matrix. What is the sum of all the elements in the first row? In the second row? Is this expected? Which company gained the most market share between 2001 and \(2002 ?\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. A system of linear equations having more equations than variables has no solution, a unique solution, or infinitely many solutions.

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{array}{rr}2 x+4 y-6 z= & 38 \\ x+2 y+3 z= & 7 \\ 3 x-4 y+4 z= & -19\end{array}\)

Find the value(s) of \(k\) such that $$A=\left[\begin{array}{ll} 1 & 2 \\\k & 3\end{array}\right]$$ has an inverse. What is the inverse of \(A\) ?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are matrices of the same size and \(c\) is a scalar, then \(c(A+B)=c A+c B\).

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned} x_{1}-2 x_{2}+x_{3} &=6 \\ 2 x_{1}+x_{2}-3 x_{3} &=-3 \\\ x_{1}-3 x_{2}+3 x_{3} &=10 \end{aligned}\)

Find the value(s) of \(k\) such that $$A=\left[\begin{array}{rrr}1 & 0 & 1 \\\\-2 & 1 & k \\\\-1 & 2 & k^{2}\end{array}\right]$$ has an inverse.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are matrices of the same size, then \(A-B=\) \(A+(-1) B\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a square matrix with inverse \(A^{-1}\) and \(c\) is a nonzer real number, then $$(c A)^{-1}=\left(\frac{1}{c}\right) A^{-1}$$

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned} 2 x &+3 z=&-1 \\ 3 x-2 y+z &=9 \\ x+y+4 z &=4 \end{aligned}\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a matrix, then \(\left(A^{T}\right)^{T}=A\).

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned} 2 x_{1}-x_{2}+3 x_{3} &=-4 \\ x_{1}-2 x_{2}+x_{3} &=-1 \\\ x_{1}-5 x_{2}+2 x_{3} &=-3 \end{aligned}\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A^{-1}\) does not exist, then the system \(A X=B\) of \(n\) linear equations in \(n\) unknowns does not have a unique solution.

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{array}{rr}x_{1}-x_{2}+3 x_{3}= & 14 \\ x_{1}+x_{2}+x_{3}= & 6 \\ -2 x_{1}-x_{2}+x_{3}= & -4\end{array}\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Let $$A=\left[\begin{array}{ll}a & b \\\c & d\end{array}\right]$$ a. Find \(A^{-1}\) if it exists. b. Find a necessary condition for \(A\) to be non singular. c. Verify that \(A A^{-1}=A^{-1} A=I\).

Refer to Example 6 in this section. Suppose Ace Novelty received an order from another amusement park for 1200 Pink Panthers, 1800 Giant Pandas, and 1400 Big Birds. The quantity of each type of stuffed animal to be produced at each plant is shown in the following production matrix: Each Panther requires \(1.3 \mathrm{yd}^{2}\) of plush, \(20 \mathrm{ft}^{3}\) of stuffing, and 12 pieces of trim. Assume the materials required to produce the other two stuffed animals and the unit cost for each type of material are as given in Example 6 . a. How much of each type of material must be purchased for each plant? b. What is the total cost of materials that will be incurred at each plant? c. What is the total cost of materials incurred by Ace Noyelty in filling the order?

Solve the system of linear equations using the Gauss-Jordan elimination method. \(2 x_{1}-x_{2}-x_{3}=0\) \(3 x_{1}+2 x_{2}+x_{3}=7\) \(x_{1}+2 x_{2}+2 x_{3}=5\)

Correspond to those in Exercises \(15-27\), Section \(5.1 .\) Use the results of your previous work to help you solve these problems. The Johnson Farm has 500 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn and wheat (including seeds and labor) is $$\$ 42$$ and $$\$ 30$$ per acre, respectively. Jacob Johnson has \(\$ 18,600\) available for cultivating these crops. If he wishes to use all the allotted land and his entire budget for cultivating these two crops, how many acres of each crop should he plant?

The total output of loudspeaker systems of the Acrosonic Company at their three production facilities for May and June is given by the matrices \(A\) and \(B\), respectively, where The unit production costs and selling prices for these loudspeakers are given by matrices \(C\) and \(D\), respectively, where Compute the following matrices and explain the meaning of the entries in each matrix. a. \(A C\) b. \(A D\) c. \(B C\) d. \(B D\) e. \((A+B) C\) f. \((A+B) D\) g. \(A(D-C)\) h. \(B(D-C)\) i. \((A+B)(D-C)\)

Correspond to those in Exercises \(15-27\), Section \(5.1 .\) Use the results of your previous work to help you solve these problems. Michael Perez has a total of $$\$ 2000$$ on deposit with two savings institutions. One pays interest at the rate of $$6 \% /$$ year, whereas the other pays interest at the rate of $$8 \%$$ year. If Michael earned a total of $$\$ 144$$ in interest during a single year, how much does he have on deposit in each institution?

Correspond to those in Exercises \(15-27\), Section \(5.1 .\) Use the results of your previous work to help you solve these problems. The Coffee Shoppe sells a coffee blend made from two coffees, one costing $$\$ 5 / \mathrm{lb}$$ and the other costing $$\$ 6 / \mathrm{lb}$$. If the blended coffee sells for $$\$ 5.60 / \mathrm{lb}$$, find how much of each coffee is used to obtain the desired blend. Assume that the weight of the blended coffee is $$100 \mathrm{lb}$$.

Correspond to those in Exercises \(15-27\), Section \(5.1 .\) Use the results of your previous work to help you solve these problems. Cantwell Associates, a real estate developer, is planning to build a new apartment complex consisting of one-bedroom units and two- and three-bedroom townhouses. A total of 192 units is planned, and the number of family units (two- and three-bedroom townhouses) will equal the number of one-bedroom units. If the number of one-bedroom units will be 3 times the number of threebedroom units, find how many units of each type will be in the complex.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A, B\), and \(C\) are matrices and \(A(B+C)\) is defined, then \(B\) must have the same size as \(C\) and the number of columns of \(A\) must be equal to the number of rows of \(B\).

Correspond to those in Exercises \(15-27\), Section \(5.1 .\) Use the results of your previous work to help you solve these problems. The annual returns on Sid Carrington's three investments amounted to $$\$ 21,600: 6 \%$$ on a savings account, \(8 \%\) on mutual funds, and \(12 \%\) on bonds. The amount of Sid's investment in bonds was twice the amount of his investment in the savings account, and the interest earned from his investment in bonds was equal to the dividends he received from his investment in mutual funds. Find how much money he placed in each type of investment.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a \(2 \times 4\) matrix and \(B\) is a matrix such that \(A B A\) is defined, then the size of \(B\) must be \(4 \times 2\).

Correspond to those in Exercises \(15-27\), Section \(5.1 .\) Use the results of your previous work to help you solve these problems. The management of Hartman Rent-A-Car has allocated $$\$ 1.5$$ million to buy a fleet of new automobiles consisting of compact, intermediate-size, and full- size cars. Compacts cost $$\$ 12,000$$ each, intermediatesize cars cost $$\$ 18,000$$ each, and full-size cars cost $$\$ 24,000$$ each. If Hartman purchases twice as many compacts as intermediate-size cars and the total number of cars to be purchased is 100 , determine how many cars of each type will be purchased. (Assume that the entire budget will be used.)

Correspond to those in Exercises \(15-27\), Section \(5.1 .\) Use the results of your previous work to help you solve these problems. The management of a private investment club has a fund of $$\$ 200,000$$ earmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high risk, medium risk, and low risk. Management estimates that high-risk stocks will have a rate of return of $$15 \% /$$ year; medium- risk stocks, $$10 \% /$$ year; and low-risk stocks, $$6 \% /$$ year. The investment in low-risk stocks is to be twice the sum of the investments in stocks of the other two categories. If the investment goal is to have an average rate of return of $$9 \% /$$ year on the total investment, determine how much the club should invest in each type of stock. (Assume that all of the money available for investment is invested.)

For the opening night at the Opera House, a total of 1000 tickets were sold. Front orchestra seats cost $$\$ 80$$ apiece, rear orchestra seats cost $$\$ 60$$ apiece, and front balcony seats cost $$\$ 50$$ apiece. The combined number of tickets sold for the front orchestra and rear orchestra exceeded twice the number of front balcony tickets sold by 400\. The total receipts for the performance were $$\$ 62,800$$. Determine how many tickets of each type were sold.

An executive of Trident Communications recently traveled to London, Paris, and Rome. He paid 180, 230, and 160 per night for lodging in London, Paris, and Rome, respectively, and his hotel bills totaled 2660. He spent 110, 120 and 90 per day for his meals in London, Paris, and Rome, respectively, and his expenses for meals totaled 1520 . If he spent as many days in London as he did in Paris and Rome combined, how many days did he stay in each city?

Joan and Dick spent 2 wk (14 nights) touring four cities on the East Coast- Boston, New York, Philadelphia, and Washington. They paid \(\$ 120, \$ 200, \$ 80\), and \(\$ 100\) per night for lodging in each city, respectively, and their total hotel bill came to \(\$ 2020\). The number of days they spent in New York was the same as the total number of days they spent in Boston and Washington, and the couple spent 3 times as many days in New York as they did in Philadelphia. How many days did Joan and Dick stay in each city?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. When simple interest is used, the accumulated amount is a linear function of \(t\).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If the augmented matrix corresponding to a system of three linear equations in three variables has a row of the form \(\left[\begin{array}{lll}0 & 0 & 0 & a\end{array}\right]\), where \(a\) is a nonzero number, then the system has no solution.