Let A =
,
B =
,
C =
,
and D =
.
For each of the following, determine whether the expression is defined, and if it is, evaluate it.
If the expression is not defined, choose "Not Defined" and click on "Check"
If the expression is defined, choose "Defined" and enter the result in the space provided. Entries in each row should be seperated by commas, and each row should be on a new line. Then press "Check"
"Peek" will show the correct answer, correctly formatted.
Press here to have a small pop-up window showing the matrices A, B, C and D.
For each of the given matrices, find the inverse or determine that the matrix is singular.
Write each of the following systems of linear equations as a matrix equation, and solve by inverting the coefficient matrix.
20. Airport Delays The following table shows the average departure delay per flight at the four busiest airports in the US.
Airport Atlanta Chicago O'Hare Dallas-Fort Worth Los Angeles Int'l.
Avg. Delay (minutes) 8.8 7.5 6.4 5.5
On a certain day, 100 flights depart from Atlanta, 80 from O'Hare, 70 from Dallas, and 60 from Los Angeles. Use matrix algebra to compute the total delay on all departing flights from these airports.
Figures are for the first half of 1999. Sources: CI World Traffic Report; FAA/The New York Times, October 13, 1999, p. C1.
21. More Airport Delays Referring to the departure delay figures for the first half of 1999 in the preceding exercise, suppose that, Atlanta cuts the average departure delay by 2 minutes every six months, O'Hare by 1.5 minutes, Dallas by 1.5, whereas departure delays in Los Angeles increase by 1.5 minutes every six months. On a certain day at the end of 2000, 100 flights depart from Atlanta, 80 from O'Hare, 70 from Dallas, and 60 from Los Angeles. Use matrix algebra to compute the total delay on all departing flights from these airports.
22. Spending The Logan Primadonna Co. manufactures ballet tutus and tights, and is in intense competition with McCormack Theatrics and Justino Pirouette Inc. Unfortunately, the quality of the work at all three companies leaves something to be desired, and many ballet companies have been switching brands every season in the vain hope of finding tights and tutus that will stand up to the technical demands of the dancers for more than a day or two. Last ballet season, the Avante-Garde Ballet company purchased items from all three companies as shown in the following table.
Purchases
Logan Primadona McCormack Theatrics Justino Pirouette
Tutus 20 10 20
Tights (Pairs) 20 30 10
The cost of these items are given in the following chart.
Cost Per Item ($)
Logan Primadona McCormack Theatrics Justino Pirouette
Tutus 30 40 50
Tights 20 20 15
(a) Let Q be the 2 3 matrix corresponding to the purchases, and let C be the 2 3 matrix corresponding to the costs per item. Compute the product CT Q. What do the diagonal entries of the product represent?
(b) With Q and C as above, compute the product CQT . What do the diagonal entries of the product represent?
23. Jones Beach Jones Beach has a jetty on its westernmost edge, a swimming area is on its easternmost edge, and a volleyball play area in between. One sunny day, you notice that one in six of the people in the volleyball area stroll to the jetty every hour, while two in six stroll over to the swim area. Due to the bad jellyfish infestation, half the beach goers at the swim area meander over to the volleyball area every hour, while the rest of them stay put. Nobody leaves the jetty. The above data can be summarized in the following table.
From To Jetty Swimming Area Volleyball Area
Jetty 1 0 0
Swimming Area 0 1/2 1/2
Volleyball Area 1/6 1/3 1/2
Use matrix multiplication to answer the following questions.
(a) At noon, there was no one at the jetty, 300 people in the swimming area, and 600 people in the volleyball area. How crowded was the volleyball area by 1 pm.?
people
(b) How crowded was the swimming area by 1 pm.?
at the swimming area
24. Germany Input-Output Table Germany Input-Output Table* Two of the sectors of the former West German economy are (1) iron and steel, and (2) road vehicles. In 1980, the input-output table involving these two sectors was as follows (all figures are in millions of deutschmarks).
From To 1 2
1 63,080 4,956
2 73 24,475
Total Output 112,120 143,955
Determine how these two sectors would react to an increase in demand for steel (Sector 1) of DM10,000 million, and how they would react to an increase in demand for vehicles (Sector 2) of DM10,000 million.
Source: Input-Output Tabellen 1980, Statistisches Bundesamt, Wiesbaden, Federal Republic of Germany.
Last Updated:February, 2000
Copyright © 2000 StefanWaner and Steven R. Costenoble