## Review Exercises for Finite MathematicsApplied to the Real World Topic: Matrix Algebra

Chapter 3 Summary
True/False Quiz
Index of Review Exercises
Everything for Calculus
Everything for Finite Math
Everything for Finite Math & Calculus
Utility: Matrix Algebra Tool

Let A=
 1 2 3 4 5 6
,   B=
 1 -1 0 1
,   C=
 -1 0 1 1 0 1
,   and  D=
 -3 -2 -1 1 2 3
.

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.
 1 A + C Select One Defined Not Defined Answer (if defined): 2 A - CT Select One Defined Not Defined Answer (if defined): 3 CT + 3D Select One Defined Not Defined Answer (if defined): 4 BA Select One Defined Not Defined Answer (if defined): 5 BAT Select One Defined Not Defined Answer (if defined): 6 BC Select One Defined Not Defined Answer (if defined): 7 CB Select One Defined Not Defined Answer (if defined): 8 B4 Select One Defined Not Defined Answer (if defined): 9 AAT Select One Defined Not Defined Answer (if defined): 10 DTD Select One Defined Not Defined Answer (if defined):

For each of the given matrices, find the inverse or determine that the matrix is singular.

11.
 1 -1 -1 1
Inverse (if defined):

12.
 1 2 3 4
Inverse (if defined):

13.
 1 2 3 1 3 2 0 1 1
Inverse (if defined):

14.
 1 2 3 1 3 2 1 1 4
Inverse (if defined):

15.
 1 2 3 4 0 1 2 3 0 0 1 2 0 0 0 1
Inverse (if defined):

16.
 1 2 3 4 2 3 3 3 0 1 2 3 0 0 1 2
Inverse (if defined):

Write each of the following systems of linear equations as a matrix equation, and solve by inverting the coefficient matrix.

 17 x + y = 6 x - y = 2 18 x + y + z = 2 x +2y + z = 6       y + z = 1 19 x - y           = 0 y - z       = 1 z - w = 0 x         + w = 3

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 23 matrix corresponding to the purchases, and let C be the 23 matrix corresponding to the costs per item. Compute the product CTQ. 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