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ISBN: 9798214012933

Chapter 3: Matrices

Exercises 3.1

1. Matrix B has 3 rows.

2. Matrix E is 23 

3. A, F, and Z have the same order as G

4. 42 –35 D  =

5. 123 –101 234 F

6. Matrix D is 22  . The zero matrix of that order is 00 . 00 

7. A, C, D, F, G, and Z are square.

8. 1130 4211 3201 B

9. 23 1 a =

10. 24 1 b =

11. 134 020 213 TA

12. 112 203 314 T F

13. 000 +(–A)000 000 A

14. If 3 0, j a = then j = 2 since the only element that is 0 in the 3rd row is in the 2nd column.

15. 543295 + 132547 CD

16. 1+10+2-2+3221 +=3–12+01+1222

21. Because the orders are different, D – G is undefined.

22. Because the orders are different, B + F is undefined.

23. 11303390 33421112633 32019603 B

24. 42168 44 351220 D

25. 2012842816 42 486101018 CD  +=+=

26. 40241262818 83 81691511 CD

27. Because the orders are different, 23AB is undefined.

28. 4812633 434043012 812161530 101115 1016 23916 FG

29. 10310 0023 21421 x yz w

Chapter 3: Matrices

By setting corresponding elements equal we have x = 3, y = 2, z = 3, w = 4.

30. 01041 331 0210 x yyy zw

By setting corresponding elements equal we have w = 2, x = 4, y = 1, and z = 1.

31. 321247 441131 xxxz yywy

2x – 4 = x or x = 4, z = 3, y = 1, w + 1 = 4 or w = 3.

32. 3211 4454 xyxxw zyxy +−−  = 

Setting

33. 3342144 33681815 xyxyy yzyz

Setting

0.5394.432.011.83 0.1356.335.122.43

The matrix DC measure, the increase in the leakage to the oceans from various regions from 2020 to 2050. The negative entries tell us that the leakage to the oceans has decreased from 2020 to 2050 from the region.

36. a. 1341.3148.793.3 73.869.469.2 1.562.163.05 19.641.820.9

1211.5192.4165.5 80.879.078.4 1.881.681.95 8.312.510.2 A

b. 129.843.772.2 79.69.2 0.320.481.1 11.329.310.7

c. The negative entries in row 3 indicate that the infant mortality rates of all three countries are projected to be reduced, which is a positive change for the countries.

Chapter 3: Matrices

37. a. 11041.78978.46461 8739.89159.66877.3 CapitalExpendituresGrossOperatingCosts= 9798.19086.76448.4 9696.68926.76109.5

b. Air pollution (R4, C1)

41381341815636136

41778237116796607

38. a.

41877834716496987

41477935016577314

MF represents the projected differences of male and female populations for different ages and years.

b. The female population is projected to be greater than the male population for those over age 64 in all years 2025 – 2040.

39. a. 8255801560 810650350 AB

40. Table form.

Republicans Democrats Independents Not Registered 843 257 135 92 Approve 426 451 127 64 Disapprove 751 92

A 34  matrix representing this information is 84325713592 42645112764. 751923844

41. a. M F 78.682.9

16.5518.419.3 5.998.611.59 12.7212.9613.2

This represents the U.S. balance of trade for various energy types for selected years.

c. The life expectancy difference is greatest in 2020.

This represents the average monthly U.S. consumption of various energy types for selected years.

Chapter 3: Matrices

This represents the total U.S. production of various energy types for selected years.

44. a.

120,000180,00078,000117,000 0.6575,00045,00048,75029,250 105,000174,00068,250113,100

24,000216,000 15,00054,000 21,000208,800

45. a.

(This is 20% of the first column and 120% of the second.)

92.4016842021 84168840

1.05117.602941890 632948442 84042021 A

96.8017644022 88176880

1.10123.203081980 663088844 88044022 A

50,00052,00024,000

46. a.

0.8030,000104,00010,000

16,00040,000240,000

40,00041,60019,200

24,00083,2008000

12,80032,000192,000

b. They paid 100% of the debts due in 30 days and 50% of the debts due in 60 days.

47. a. 0110 1011 0001 0110

b. 0111 1011 0101 1110

c. 0221 2022 0102 1220 AB

Sum in Row 1 = 5, Row 2 = 6, Row 3 = 3, Row 4 = 5. Person 2 is the most active.

48. a. 01010 00001 01001 10101 01000 A

b. 00101 00000 01001 01001 00000 B

c. 01111 00001 02002 11102 01000 AB  

Sum in Row 1 = 4, Row 2 = 1, Row 3 = 4, Row 4 = 5, Row 5 = 1. Player 4 is the top-ranked player.

49. a. 503030458075 3622446258106 AB ++  +== ++ 

b. 80967552 ++ 588110637 ABC ++  =  ++  176127 139143  =  

c. 50–4030–26104 –36–2944–4272 AD  ==  

d. 30–4045–261019 –= 22–2962–42720 BD  =   Shortage; take from inventory.

50. 1 2 3 4 5 6

WHN1232315262034

WHN2242316282133

WHN3181715181722

Chapter 3: Matrices

52. Subtract 0.01 from each entry in column 2 and column 5.

Subtract 0.02 from each entry in column 7.

1.000.940.950.950.940.850.830.850.85

0.850.990.950.950.940.950.830.850.85

0.850.841.000.950.940.950.930.850.85

0.850.840.851.000.940.950.930.950.85

New0.850.840.850.850.990.950.930.950.95

0.950.840.850. E = 850.841.000.930.950.95

0.950.940.850.850.840.850.980.950.95

0.950.940.950.850.840.850.831.000.95

0.950.940.950.950.840.850.830.851.00

53. For each row: (C1 + C2 + C3 + C4) ÷ 4 = average efficiency

W1: 0.9625W2: 0.9375W3: 0.9125

W4: 0.8875W5: 0.85W6: 0.875

W7: 0.90W8: 0.925W9: 0.95

W5 is least efficient. He works best at center 5.

54. Work Centers 123456789 [0.880.900.920.940.960.930.910.890.87]

Number 5; Number 9

Exercises 3.2

7. 42104 35510 41002160 32505120 14216 28512 DE

Chapter 3: Matrices

8. 531013 122134 5603591512 14021638 11343 5255

9. 10211301061043000027532 3214211383342920021149113 403320140940612000031310123 AB

10. EC is undefined.

11. BA is undefined.

12. 143 1013122103040133003 266 2134310219082346204 1050 011 TFB

13. 1130 104101210830000413934 4211 510540520151001097161 3201 EB

16. 1021510850095 321013041520717 40340401220001620 TAE

Chapter 3: Matrices

19. 3 53535353253156 121212125234 5328211402110521161126 1277281421144235

20. 2 F is undefined.

No,

CDE

22. 5342104292510415425116 () 12355101012510701240

CDE

5342104531421615425116 () 12355101228512701240

They are equal.

23. No. See problems 5 and 7

24. Yes. In general, (kA)(B) = k(AB).

25. 25412122528440442101255880 143582120212323318942670 1327113115142242216628441

−−−−+++−−   =−=−−−+++−=−   −−−−−++−−−+− 

26. 121254221583462100 58214310822532620244010 711–313214113354492833661811

−−++−+−−+−   =−=−++−+−−+−=   −−−−+−−++−++− 

27. 304448120121201224024000 1711124734738146000 304336120121201224024000

−+−+−−++   =−−−=−+−+−+−=   −−+−+−−++ 

28. 4483041242402801643282820 112171316070418275 336304931802101232462115

−+−++−−−− 

Chapter 3: Matrices

30.

31. AI = A See Example 6.

32. IA = A

33. Z(CI) = ZC = Z (ZC)I = ZI = Z Product of zero matrix and A is the zero matrix.

34. IFZ = Z

35. Problem 27 shows that a product can be “zero” and neither of the factors is zero.

36. Yes, it is true that multiplication by the zero matrix gives a result of a zero matrix, provided the multiplication is defined.

37. a. We will use 11 and .ABBA adbcadbc

abdbadbcabbaadbc AB cdcacddcbcadadbc

dbabadbcbdbdadbc BA cacdacacbcadadbc

b. For B to exist, we must have

38. 1013 2134 F

a. yes

c. No, because

39. 12122222 34216442

Thesevaluesarethesolution.

Chapter 3: Matrices

40. 31026204 22124419 11212222

These values are the solution.

41. 11211225 40124015 21112215

Thesevaluesarethesolution.

42. 102220240 310161077 121122153 2 1020.

So,1 isnotasolutionof3107 11213 xx yy zz

The graphing calculator should show 1’s on the main diagonal and 0’s everywhere else (1 may be 0.9999 or 0.998 and 0 may be 0.002 or -0.002).

Cengage

accessible website, in whole or in part.

Chapter 3: Matrices

45. 0.88036,00042,00031,68036,960 00.8550,00056,00042,50047,600

The product is the price the dealer pays for the cars.

46. Revenue

So, the revenue is as follows:

Atlanta: $11,910,000

Chicago $16,040,000

New York: $ 5,920,000

47. a. 89.5293.9798.15 33.8635.4736.93 6.666.726.74

b. 7868.817968.667959.97 2976.293007.862995.02 585.41569.86546.61

c. The 1-1 entry is the trillions of BTUs used in single-family households in 2025. The 2-3 entry is the millions of BTUs predicted to be used in multi-family households in 2035.

d. Using AB as stored in a calculator,     11111,430.5111,546.3811,501.60. AB  Left to right, each entry gives the millions of BTUs of delivered energy consumed by all households in 2025, 2030, and 2035.

0.1620.0479,371,700438,134,000

0.1190.0866,884,150801,692,000 57,850,0000

0.0660.0653,818,100605,930,000 09,322,000,000

0.2980.58217,239,3005,425,404,000

0.2020.21511,685,7002,004,

After 5 years, M has 55% and S has 45% of the population.

b. 10 years: ( ) 2PDDPD = ; 15 years: 3PD

c. 60% in M and 40% in S. The population proportions are stable.

Chapter 3: Matrices

50.

b. ( )

51. Wasteblanknotblankwantblanknot 231192052714152027231142027141520

5923195142023142715124275268205343124250 261255 6111202715271201420149334327249417149304 316310

So 124, 149, 275, 334, 268, 327, 205, 249, 343, 417, 124, 149, 250, 304, 261, 316, 255, 310 is the message sent.

52. Toblankbeblankorblanknotblanktoblankbe 20152725271518271415202720152725

59202751527152715223515326823726125531531855 6111522718142020275285184327288316310382 38767

Thus, 235, 285, 153, 184, 268, 327, 237, 288, 261, 316, 255, 310, 315, 382, 318, 387, 55, 67 is sent.

53. a. 3 21 454 33 1 454 2230

Houston needs 23,300 gallons of black crude and 18,700 gallons of gold. Gulfport needs 32,250 gallons of black and 27,750 gallons of gold.

b. 

54. a.

1200150015002000

8.505.5012.0022,80024,25023,05037,250 1200100010001500 6.004.0015.0019,50020,50019,00033,000 5005004001000

1200150015002000

b. 1

8.505.5012.001 1200100010001500

CDT represents the total costs for Labor and Materials for the entire year.

Chapter 3: Matrices

0.78.510.21.15.63.6

0.50.26.11.30.21.0

2.20.48.81.21.24.8

55. a.

56. 0.78.510.21.15.63.61.2000000 0.50.26.11.30.21.001.030000

2.20.48.81.21.24.8001.05000

251.863.481.635.254.3144.20001.2000

30.01.01.01.01.01.000001.050

1. 122 1217 31203 4221

2. 1217 3120 4221

3. 1342 2021 1211

4. 1223 1204 0111

5. x = 2, 1 2, y = z = –5