CHAPTER 16 REVIEW EXERCISES

CONCEPT CHECK EXERCISES

Determine each of the following as being either true or false. If it is false, explain why.

  1. 2 [ 3 − 10 − 2 ]  =  [ 6 − 202 ] 

  2.  [ 3 − 102 ]  [ 13 − 2 − 1 ]  =  [ 510 − 4 − 2 ] 

  3. If A =  [  − 114 − 3 ]  ,  then A − 1 =  [  − 3 − 1 − 4 − 1 ] 

  4.  | 31 − 12 − 23012 |  = 3 |  − 2312 |  − 2 | 1 − 112 | 

  5.  [ 2 − 15 − 3 ]  [ 37 ]  =  [  − 1 − 6 ]  is the matrix solution of 2x − y = 35x − 3y = 7 . 

  6. After using two steps of the Gaussian elimination method for solving the system 2x + 3y = 43x + 2y = 1 ,  we would have x + 32y = 1 − 52y =  − 5 . 

PRACTICE AND APPLICATIONS

In Exercises 7–12, determine the values of the literal numbers.

  1.  [ 4aa − b ]  =  [ 85 ] 

  2.  [ x − y2x + 2z4y + z ]  =  [ 13 − 1 ] 

  3.  [ 2x3y2zx + ay + bc − z ]  =  [ 4 − 987 − 42 ] 

  4.  [ a + bjbajb − aj ]  =  [ 6j2d2cjej2 ] (j =  − 1)

  5.  [ cosπsinπ6x + yx − y ]  =  [ xyab ] 

  6.  [ lnelog100a2b2 ]  =  [ a + ba − bxy ] 

In Exercises 13–18, use the given matrices and perform the indicated operations.

A =  [ 2 − 341 − 502 − 3 ] B =  [  − 104 − 6 − 3 − 21 − 7 ] C =  [ 5 − 6280 − 2 ] 

  1. A + B

  2. 2C

  3. B − A

  4. 2A − B

  5. 2A − 3B

  6. 2(A − B)

In Exercises 19–22, perform the indicated matrix multiplications.

  1.  [ 2 − 1 − 21 ]  [ 1 − 12 − 2 ] 

  2.  [ 6 − 41020 − 43 ]  [ 7 − 164013 − 25910 ] 

  3.  [ 00.60.20.00.4 − 0.1 ]  [ 0.1 − 0.40.50.50.10.0 ] 

  4.  [ 0 − 168147 − 2 − 1 ]  [ 5 − 1715010411 − 2301 ] 

In Exercises 23–30, find the inverses of the given matrices. Check each by using a calculator.

  1.  [ 2 − 52 − 4 ] 

  2.  [  − 5 − 301050 ] 

  3.  [  − 0.8 − 0.10.4 − 0.7 ] 

  4.  [ 50 − 1242 − 80 ] 

  5.  [ 11 − 2 − 1 − 21034 ] 

  6.  [  − 1 − 12230141 ] 

  7.  [ 2 − 434 − 65 − 21 − 1 ] 

  8.  [ 31 − 4 − 31 − 2 − 603 ] 

In Exercises 31–38, solve the given systems of equations using the inverse of the coefficient matrix.

  1. 2x − 3y =  − 94x − y =  − 13

  2. 5A − 7B = 626A + 5B =  − 6

  3. 33x + 52y =  − 45045x − 62y = 1380

  4. 0.24x − 0.26y =  − 3.10.40x + 0.34y =  − 1.3

  5. 2u − 3v + 2w = 73u + v − 3w =  − 6u + 4v + w =  − 13

  6. 6x + 6y − 3z = 24x + 4y + 2z = 53x − 2y + z = 17

  7. x + 2y + 3z = 13x − 4y − 3z = 27x − 6y + 6z = 2

  8. 3x + 2y + z = 22x + 3y − 6z = 3x + 3y + 3z = 1

In Exercises 39–46, solve the given systems of equations by Gaussian elimination. For Exercises 39–44, use those that are indicated from Exercises 31–38.

  7. 2x + 3y − z = 10x − 2y + 6z =  − 65x + 4y + 4z = 14

  8. x − 3y + 4z − 2t = 62x + y − 2z + 3t = 73x − 9y + 12z − 6t = 12

In Exercises 47–50, solve the systems of equations by determinants. As indicated, use the systems from Exercises 35–38.

In Exercises 51–58, solve the given systems of equations by using the coefficient matrix. Use a calculator to perform the necessary matrix operations and display the results and the check.

  1. 3x − 2y + z = 62x + 3z = 34x − y + 5z = 6

  2. 7n + p + 2r = 34n − 2p + 4r =  − 22n + 3p − 6r = 3

  3. 2x − 3y + z − t =  − 84x + 3z + 2t =  − 32y − 3z − t = 12x − y − z + t = 3

  4. 6x + 4y − 4z − 4t = 05y + 3z + 4t = 36y − 3z + 4t = 96x − y + 2z − 2t =  − 3

  5. 3x − y + 6z − 2t = 82x + 5y + z + 2t = 74x − 3y + 8z + 3t =  − 173x + 5y − 3z + t = 8

  6. A + B + 2C − 3D = 153A + 3B − 8C − 2D = 96A − 4B + 6C + D =  − 62A + 2B − 4C − 2D = 8

  7. 4r − s + 8t − 2u + 4v =  − 13r + 2s − 4t + 3u − v = 43r + 3s + 2t + 5u + 6v = 136r − s + 2t − 2u + v = 0r − 2s + 4t − 3u + 3v = 1

  8. 2v + 3w + 2x − 2y + 5z =  − 17v + 8w + 3x + y − 4z = 3v − 2w − 4x − 4y − 8z =  − 93v − w + 7x + 5y − 3z =  − 184v + 5w + x + 3y − 6z = 7

In Exercises 59–62, use matrices A and B.

A =  [ 1034 ] B =  [ 010001100 ] 

  1. Find A2 , A3 ,  and A4 . 

  2. Show that (A2)2 = A4 . 

  3. Show that B3 = I . 

  4. Show that B4 = B . 

In Exercises 63–66, evaluate the given determinants by expansion by minors.

  1.  | 7231 − 50 − 64 − 3 | 

  2.  | 4 − 5 − 1616 − 24 − 3 | 

  3.  | 140 − 33125 − 2 − 2 − 41 − 163 − 4 | 

  4.  |  − 266 − 11 − 2 − 525 − 443 − 31 − 2 − 3 | 

In Exercises 67–70, use the determinants for Exercises 63–66, and evaluate each using a calculator.

In Exercises 71 and 72, use the matrix N.

N =  [ 0 − 110 ] 

  1. Show that N − 1 =  − N . 

  2. Show that N2 =  − I . 

In Exercises 73–76, solve the given problems.

  1. For any real number n, show that  [ n1 + n1 − n − n ] 2 = I . 

  2. For the matrix N =  [ 1111 ]  ,  find (a) N2 ,  (b) N3 ,  (c) N4 .  What is N20 ?  Explain.

  3. If B − 1 =  [ 1234 ]  and AB =  [  − 1302 ]  ,  find A.

  4. For A =  [ 1234 ]  , B =  [ 2 − 335 ]  , C =  [ 0124 ]  ,  verify the associative law of multiplication, A(BC) = (AB)C . 

In Exercises 77–80, use matrices A and B.

A =  [ 1 − 203 ]  B =  [  − 312 − 1 ] 

  1. Show that (A + B)(A − B) ≠ A2 − B2 . 

  2. Show that (A + B)2 ≠ A2 + 2AB + B2 . 

  3. Show that the inverse of 2A is A − 1 / 2.

  4. Show that the inverse of B/2 is 2B − 1 . 

In Exercises 81–84, solve the given systems of equations by use of matrices as in Section 16.4.

  1. Two electric resistors, R1 and R2 ,  are tested with currents and voltages such that the following equations are found:

    2R1 + 3R2 = 263R1 + 2R2 = 24

    Find the resistances R1 and R2 (in Ω).

  2. A company produces two products, each of which is processed in two departments. Considering the worker time available, the numbers x and y of each product produced each week can be found by solving the system of equations

    4.0x + 2.5y = 12003.2x + 4.0y = 1200

    Find x and y.

  3. A beam is supported as shown in Fig. 16.23. Find the force F and the tension T by solving the following system of equations:

    Forces on a beam.

    Fig. 16.23

    0.500F = 0.866T0.866F + 0.500T = 350

  4. To find the electric currents (in A) indicated in Fig. 16.24, it is necessary to solve the following equations.

    A 4 V circuit with parallel current I sub Ay, I sub B, and I sub C with 5 Ohms, 2 Ohms, and 1 Ohms, respectively.

    Fig. 16.24

    IA + IB + IC = 05IA − 2IB =  − 42IB − IC = 0

    Find IA , IB ,  and IC . 

In Exercises 85–88, solve the systems of equations in Exercises 81–84, by Gaussian elimination.

In Exercises 89–92, solve the given problems by setting up the necessary equations and solving them by any appropriate method of this chapter.

  1. A crime suspect passes an intersection in a car traveling at 110 mi/h. The police pass the intersection 3.0 min later in a car traveling at 135 mi/h. How long is it before the police overtake the suspect?

  2. A contractor needs a backhoe and a generator for two different jobs. Renting the backhoe for 5.0 h and the generator for 6.0 h costs $425 for one job. On the other job, renting the backhoe for 2.0 h and the generator for 8.0 h costs $310. What are the hourly charges for the backhoe and the generator?

  3. By mass, three alloys have the following percentages of lead, zinc, and copper.

    		 Lead Zinc Copper
    Alloy A 60% 30% 10%
    Alloy B 40% 30% 30%
    Alloy C 30% 70%

    How many grams of each of alloys A, B, and C must be mixed to get 100 g of an alloy that is 44% lead, 38% zinc, and 18% copper?

  4. On a 750-mi trip from Salt Lake City to San Francisco that took a total of 5.5 h, a person took a limousine to the airport, then a plane, and finally a car to reach the final destination. The limousine took as long as the final car trip and the time for connections. The limousine averaged 55 mi/h, the plane averaged 400 mi/h, and the car averaged 40 mi/h. The plane traveled four times as far as the limousine and car combined. How long did each part of the trip and the connections take?

In Exercises 93–96, perform the indicated matrix operations.

  1. An automobile maker has two assembly plants at which cars with either 4, 6, or 8 cylinders and with either standard or automatic transmission are assembled. The annual production at the first plant of cars with the number of cylinders–transmission type (standard, automatic) is as follows:

    4: 12,000, 15,000; 6: 24,000, 8000; 8: 4000, 30,000

    At the second plant the annual production is

    4: 15,000, 20,000; 6: 12,000, 3000; 8: 2000, 22,000

    Set up matrices for this production, and by matrix addition find the matrix for the total production by the number of cylinders and type of transmission.

  2. Set up a matrix representing the information given in Exercise 91. A given shipment contains 500 g of alloy A, 800 g of alloy B, and 700 g of alloy C. Set up a matrix for this information. By multiplying these matrices, obtain a matrix that gives the total weight of lead, zinc, and copper in the shipment.

  3. The matrix equation

    ( [ R1 − R2 − R2R1 ]  + R2 [ 1001 ] ) [ i1i2 ]  =  [ 60 ] 

    may be used to represent the system of equations relating the currents and resistances of the circuit in Fig. 16.25. Find this system of equations by performing the indicated matrix operations.

    A 6 V circuit with R sub 1, R sub 2, and R sub 1 running in parallel. Currents i sub 2 and i sub 1 run perpendicular from R sub 1 to R sub 2, and from R sub 2 to R sub 1.

    Fig. 16.25

  4. A person prepared a meal of the following items, each having the given number of grams of protein, carbohydrates, and fat, respectively. Beef stew: 25, 21, 22; coleslaw: 3, 10, 10; (light) ice cream: 7, 25, 6. If the calorie count of each gram of protein, carbohydrate, and fat is 4.1 Cal/g, 3.9 Cal/g, and 8.9 Cal/g, respectively, find the total calorie count of each item by matrix multiplication. (1 Cal = 1 kcal . )

  5. A hardware company has 60 different retail stores in which 1500 different products are sold. Write a paragraph or two explaining why matrices provide an efficient method of inventory control, and what matrix operations in this chapter would be of use.

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