Chapter 9 Matrices And Determinants 91 Matrix Solutions

Ch.9 MatricesandDeterminants

9.1 MatrixSolutionstoLinearSystems

1 WritetheAugmentedMatrixforaLinearSystem

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writetheaugmentedmatrixforthesystemofequations.

1) 9x+5y+5z =56

7x+2y+2z =19

2x+6y+6z =76

A)

95556

72219

26676

B)

97256

52619

52676

C)

56 5 5 9

19 2 2 7

76 6 6 2

D)

955

722

266

2) 9x+6z=54

 5y+8z=59

8x+5y+2z=73

A)

90654

05859

85273

B)

90854

05559

68273

C)

96054

58059

85273

D)

906

058

852

3) x–5y+z=17

 y+9z=19

 z=15

A)

1–5117

01919

00115

B)

0–5017

00919

00015

C)

15117

01919

00115

D)

1–5117

11919

11115

4) 7x+9y–8z+w=12

4y+z=11

x–y–3z=9

6x–6y+12z=–3

A)

79

–8112

041 011

1–1–309

6–612 0

–3

B)

79

–812

0416

1–1–39

6–612–3

C)

7 019

94

–1–6

–81

–312

1 000

12 11 9 –3

D)

798 112

041 011

11 309

6 6 12 0 –3

Page1

Writethesystemoflinearequationsrepresentedbytheaugmentedmatrix.Usex,y,z,and,ifnecessary,wforthe

variables.

5)

672–2

3034

730 2

A) 6x+7y+2z=–2

3x+3z=4

7x+3y=2

B) 6x–7y+2z= –2

3x+3z=–4

7x+3y=–2

C) 6x+7y+2z= –2

3x+3z=4

7x+3z=2

D) 6x+7y+2z=–2

3x+y+3z=4

7x+3y+z=2

6)

3109

–12

–1510 12

4007

–11

020

–2–8

A)

3x +y+9w =-

12

–x+5y +z=12

4x +7w =-

11

2y –2w =-

8

B)

3x +y+z+9w =-

12

–x+5y +z+w=12

4x +y+z+7w =-

11

x+2y +z–2w =-

8

C)

3x +y+9w =-

12

x+5y +z=12

4x +7w =-

11

2y +2w =-

8

D)

3x +y+9z =-

12

–x+5y +z=12

4x +7y =-

11

2x –2y =-

8

Writethesystemoflinearequationsrepresentedbytheaugmentedmatrix.Usex,y,z,and,ifnecessary,wforthe

variables.Thenuseback–substitutiontofindthesolution.

7)

17

–85

01

–7–3

001 3

A) {(–97

,

18

,

3)} B) {(5

,

–3

,

3)} C) {(5

,

3

,

2)} D) {(–187

,

–24

,

3)}

8)

1–5

215

2

01 3

26

00 1 4

A) {(–3

2,0,4)} B) {( 5

2,6,4)} C) {(3,7

2,3)} D) {(–47

2,0,4)}

9)

11

–11

–8

01

–58 0

00 1 320

00 0 1 4

A) {(–12

,

8

,

8

,

4)} B) {(–8

,

0,20

,

4)} C) {(–10

,

–4

,

16

,

3)} D) {(4

,

8

,

8

,

–12)}

Page2

2 PerformMatrixRowOperations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Performthematrixrowoperation(oroperations)andwritethenewmatrix.

1)

20 30 –60 –15

113–30

2–7421

 1

5R1

A)

46

–12 –3

113

–30

2–7421

B)

20 30 –60 –15

1

5

13

5–3

50

2–7421

C)

46

–12 –15

113–30

2–7421

D)

46

–12 –3

1

5

13

5–3

50

2

5–7

5

4

5

21

5

2)

3–51 4

–40 5–3

–12

–2–1

–5R1+R2

A)

3–51 4

–19 25 0 –23

–12

–2–1

B)

3–514

11 –25 10 17

–12

–2–1

C)

23 –5–24 19

–40 5

–3

–12

–2–1

D)

–19 25 0 –23

–405

–3

–12

–2–1

3)

11

–11 3

0–35

–50

50

–5–51

–410 2

–2

 –4R1+R3

2R1+R4

A)

11

–11 3

0–35

–50

1–4–1–9–11

–23

–24 4

B)

11

–113

0–35

–50

1–4–1–9–11

–41 0 2

–2

C)

11

–113

0–35

–50

94

–9–113

–23

–244

D)

11

–11 3

0–35

–50

1–4–1–9–3

–23

–243

Page3

3 UseMatricesandGaussianEliminationtoSolveSystems

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequationsusingmatrices.UseGaussianeliminationwithback –substitution.

1) x+y+z=5

x–y+3z =-

3

2x+y+z=10

A) {(5

,

2

,

–2)} B) {(5

,

–2

,

2)} C) {(–2

,

2

,

5)} D) {(–2

,

5

,

2)}

2) x–y+5z =-

11

3x+z=–2

x+4y+z=2

A) {(0

,

1

,

–2)} B) {(0

,

–2

,

1)} C) {(–2

,

1

,

0)} D) {(–2

,

0

,

1)}

3) 7x–y–7z =8

–2x+8z =38

2y+z=19

A) {(9

,

6

,

7)} B) {(9

,

7

,

6)} C) {(–9

,

6

,

18)} D) {(–9

,

18

,

6)}

4) 3x+5y–2w =-

13

2x+7z–w=-

1

4y+3z+3w =1

–x+2y+4z =-

5

A) {(1,–2,0,3)} B) {( 4

3,–13

20 ,0,5

2)} C) {( 3

4,–2,0,3

4)} D) {(–1,–20

13 ,0,2

5)}

5) x+y+z–w=6

2x–y+3z+4w =-

4

4x+2y–z–w=-

13

–x–2y+4z+3w =12

A) {(–4,3,5,–2)} B) {(4,–3,–5,2)} C) {(–1

4,1

3,1

5,–1

2)} D) {( 1

4,–1

3,–1

5,1

2)}

4 UseMatricesandGauss–JordanEliminationtoSolveSystems

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequationsusingmatrices.UseGauss–

J

ordanelimination.

1)

8x +9y –z=87

x–5y –6z =-

56

7x +y+z=18

A) {(1

,

9

,

2)} B) {(1

,

2

,

9)} C) {(–1

,

9

,

2)} D) {(2

,

9

,

–1)}

2)

9x –y+3z =18

–6x +5y –2z =27

–7x –8y +z=-

83

A) {(2

,

9

,

3)} B) {(2

,

3

,

9)} C) {(–2

,

9

,

4)} D) {(4

,

9

,

–2)}

3) x=–4–y–z

x–y+4z =12

5x+y=-8–z

A) {(–1

,

–5

,

2)} B) {(–5

,

2

,

–1)} C) {(2

,

–5

,

–1)} D) {(2

,

–1

,

–5)}

Page4

4) 3x+5y+2w=–12

2x+6z–w=–5

–2y+3z–3w=-3

–x+2y+4z+w=–2

A) {(–1,–3,0,3)} B) {(1,–3,0,3)} C) {(–1,3,0,–3)} D) {(1,3,0,–3)}

5) x+y–z+w=–5

3x–y+3z–2w=7

–2x+2y+z–w=16

–x–2y–3z+3w=–22

A) {(–2,3,4,–2)} B) {(–2,–3,5,1

2)}

C) {(2,–3,–4,–2)} D) {( 1

2,–1

3,–1

4,–1

2)}

Writeasystemoflinearequationsinthreevariables,andthenusematricestosolvethesystem.

6) Ronattendsacocktailparty(withhisgraphingcalculatorinhispocket).Hewantstolimithisfoodintaketo

117gprotein,100gfat,and150gcarbohydrate.Accordingtothehealthconscioushostess,themarinated

mushroomcapshave3gprotein,5gfat,and9gcarbohydrate;thespicymeatballshave14gprotein,7gfat,

and15gcarbohydrate;andthedeviledeggshave13gprotein,15gfat,and6gcarbohydrate.Howmanyof

eachsnackcanheeattoobtainhisgoal?

A) 7mushrooms;5meatballs;2eggs B) 5 mushrooms;2 meatballs;7eggs

C) 2mushrooms;7meatballs;5eggs D) 8 mushrooms;6 meatballs;3eggs

7) Aceramicsworkshopmakeswreaths,trees,andsleighsforsaleatChristmas.Awreathtakes3hoursto

prepare,2hourstopaint,and10hourstofire.Atreetakes14hourstoprepare,3hourstopaint,and4hoursto

fire.Asleightakes4hourstoprepare,17hourstopaint,and7hourstofire.Iftheworkshophas104hoursfor

preptime,95hoursforpainting,and108hoursforfiring,howmanyofeachcanbemade?

A) 6wreaths;5trees;4sleighs B) 5 wreaths;4 trees;6sleighs

C) 4wreaths;6trees;5sleighs D) 7 wreaths;6 trees;5sleighs

8) Thetablebelowshowsthenumberofbirdsforthreeselectedyearsafteranendangeredspeciesprotection

programwasstarted.

x(Numberofyearsafter1980) 1  23

y(Numberofbirds) 45 72 109

Usethequadraticfunctiony=ax2+bx+ctomodelthedata.Solvethesystemoflinearequationsinvolvinga,

b,andcusingmatrices.Findtheequationthatmodelsthedata.

A) y=5x2+12x+28 B) y=6x2+24x+23 C) y=7x2–12x+31 D) y=10x2–36x+24

9) Therewereapproximately100,000vehiclessoldataparticulardealershiplastyear.Thedealertrackssalesby

agegroupformarketingpurposes.Thepercentageof36–to59–year–oldbuyersandthepercentageofbuyers

60andoldercombinedexceedsthepercentageofbuyers35andyoungerby36%.Ifthepercentageofbuyers

intheoldestgroupisdoubled,itis26%lessthanthepercentageofusersinthemiddlegroup.Findthe

percentageofbuyersineachofthethreeagegroups.

A) 32%35andyounger;54%36–59yearolds;14%60andolder

B) 34%35andyounger;51%36–59yearolds;15%60andolder

C) 26%35andyounger;56%36–59yearolds;18%60andolder

D) 14%35andyounger;54%36–59yearolds;32%60andolder

Page5

9.2 InconsistentandDependentSystemsandTheirApplications

1 ApplyGaussianEliminationtoSystemsWithoutUniqueSolutions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UseGaussianeliminationtofindthecompletesolutiontothesystemofequations,orstatethatnoneexists.

1) 5x+2y+z=-

11

2x–3y–z=17

7x–y=12

A) ∅B) {(0,–6,1)} C) {(–2,0,–1)} D) {(1,–5,0)}

2) 4x–y+3z =12

x+4y+6z =-

32

5x+3y+9z =20

A) ∅B) {(2,–7,–1)} C) {(8,–7,–2)} D) {(–8,–7,9)}

3) x+8y+8z =8

7x+7y+z=1

8x+15y+9z =-

9

A) ∅B) {(0,0,1)} C) {(1,–1,1)} D) {(–1,0,1)}

4) x+y+z=9

2x–3y+4z =7

x–4y+3z =-

2

A) {(–7z

5+34

5,2z

5+11

5,z)} B) {( z

5+34

5,2z

5+11

5,z)}

C) {(–7z

5+34

5,2z

5–11

5,z)} D) {( 7z

5+34

5,2z

5–11

5,z)}

5) x+y+z=7

x–y+2z =7

2x+3z =14

A) {(–3z

2+7,z

2,z)} B) {(–3z

2–7,z

2,z)} C) {(–3z

2+7,2z,z)} D) {(–3z

2–7,2z,z)}

6) x+3y+2z =11

4y+9z =-

12

x+7y+11z =-

1

A) {( 19z

4+20,–9z

4–3,z)} B) {( 19z

4+20,–9z

4+3,z)}

C) {( 19z

4+20,9z

4+3,z)} D) {(–19z

4+20,–9z

4+3,z)}

7) x+y+z+w=7

3x–2z+5w =11

–4x+3y+w=4

–x–y–z–w=6

A) ∅B) {( 3

2,1,1

3,–2)} C) {( 7

4,–1

2,5,–1

6)} D) {(–11,7

19 ,6

19 ,–4)}

Page6

8) 3x–2y+2z–w=2

4x+y+z+6w =8

–3x+2y–2z+w=5

5x+3z–2w =1

A) ∅B) {(2,0,–3

37 ,9

37 )} C) {( 1

2,0,–37

3,37

9)} D) {(1,–1

3,4

9,6)}

9) x+y+z+w=8

3x+2y+z+4w=21

4x+4y+5z+8w=30

2x+3y+6z+9w=15

A) {(–6w+3,9w+7,–4w–2,w)} B) {(5w+11,–3w–7,–3w+4,w)}

C) {(–3,16,–6,1)} D) ∅

10) x–y+z–w=10

–2x+3y+5w=–28

x+2y+8z+3w=–10

x–4y–6z–5w=30

A) {(–17w–10,–13w–16,5w+4,w)} B) {(3w–2,–8w+3,4w+9,w)}

C) {(24,10,–6,–2)} D) ∅

2 ApplyGaussianEliminationtoSystemswithMoreVariablesthanEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UseGaussianeliminationtofindthecompletesolutiontothesystemofequations,orstatethatnoneexists.

1) x+y+z=9

2x–3y+4z=7

A) {(–7

5z+34

5,2

5z+11

5,z)} B) {( 3

5z+16

5,–8

5z+29

5,z)}

C) {( 27

5,13

5,1)} D) ∅

2) x+y+z=7

x–y+2z=7

A) {(–3

2z+7,1

2z,z)} B) {(–3z+14,2z–7,z)}

C) {(4,1,2)} D) {(8,–3,2)}

3) 5x–y+z=8

7x+y+z=6

A) {(–1

6z+7

6,1

6z–13

6,z)} B) {(–z+3,4z+7,z)}

C) {( 1

6z+7

6,1

6z,z)} D) ∅

4) 3x+y+z–2w=10

2x+3y+3z+w=–5

2x+y+4z+11w=11

A) {(w+5,3w–7,–4w+2,w)} B) {(2w+3,6w–7,–10w+8,w)}

C) {(6,–4,–2,1)} D) {(7,–1,–6,2)}

Page7

5) 2x+y+2z–4w=10

x+3y+2z–11w=17

3x+y+7z–21w=0

A) {(–3w+5,2w+6,4w–3,w)} B) {(3w+5,6w+6,–4w–3,w)}

C) {(w+5,8w+4,–3w–2,w)} D) {(w–5,8w–4,–3w+2,w)}

3 SolveProblemsInvolvingSystemsWithoutUniqueSolutions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblemusingmatrices.

1) Thefigurebelowshowstheintersectionofthreeone–waystreets.Tokeeptrafficmoving,thenumberofcars

perminuteenteringanintersectionmustequalthenumberofcarsleavingthatintersection.Setupasystemof

equationsthatkeepstrafficmoving,anduseGaussianeliminationtosolvethesystem.Ifconstructionlimitsz

totcarsperminute,howmanycarsperminutemustpassthroughtheotherintersectionstokeeptraffic

moving?

A) t+8cars/minbetweenI2andI1;t+3cars/minbetweenI1andI3

B) t+1cars/minbetweenI2andI1;t+4cars/minbetweenI1andI3

C) t–2cars/minbetweenI2andI1;t+1cars/minbetweenI1andI3

D) t+2cars/minbetweenI2andI1;t–3cars/minbetweenI1andI3

2) Thenutritionalcontentperounceforthreefoodsisgiveninthetablebelow.

Fat(g/oz) Protein(g/oz) Fiber(g/oz)

FoodA241

FoodB121

FoodC816 5

Whatcombinationofthesefoodscanprovideexactly14gramsoffat,27gramsofprotein,and10gramsof

fiber?

A) Nopossiblecombinationofthesefoods B) 3ozofFoodA;5ozofFoodB;1ozofFoodC

C) 7ozofFoodA;7ozofFoodB;1ozofFoodCD)4ozofFoodA;6ozofFoodB;2ozofFoodC

Page8

3) AcompanythatmanufacturesproductsA,B,andCdoesbothassemblyandtesting.Thehoursneededto

assembleandtesteachproductareshowninthetablebelow.

Hoursneeded

weeklytoassemble

Hoursneeded

weeklytotest

ProductA14

ProductB15

ProductC210

Thecompanyhasexactly24hoursperweekavailableforassemblyand107hoursperweekavailablefor

testing.IfthecompanymustproducetunitsofProductCthisweek,howmanyunitsofProductsAandBcan

theyproduce?

A) 13ofProductA;–2t+11ofProductBB)13t

ofProductA;2t+11ofProductB

C) t+13ofProductA;t+11ofProductBD)13ofProductA;11ofProductB

9.3 MatrixOperationsandTheirApplications

1 UseMatrixNotation

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Givetheorderofthematrix,andidentifythegivenelementofthematrix.

1)

1–711 7

05

–13 2 ;a12

A) 2×4;–7B)4×2; –7C)2×4; 0D)4×2; 0

2)

–8–56107

10 –e6

–15 π

31112 13 6

1

311 1 10 13

;a34

A) 4×5;13 B) 5×4; 1 C) 20; 6D)4×4; 12

2 UnderstandWhatisMeantbyEqualMatrices

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findvaluesforthevariablessothatthematricesareequal.

1)

x

2

=1

y

A) x=1;y=2B)x=2;y=1C)x=1;y=1D)x=2;y=2

2)

16

–7–6

=xy

–7z

A) x=1;y=6;z=–6B)x=1;y=6;z= –7

C) x=1;y=–7;z=–6D)x=6;y=1;z= –6

Page9

3)

x+3y+4

7–5

=7–4

7z

A) x=4;y=–8;z=–5B)x= –4;y=8;z=5

C) x=7;y=–4;z=–5D)x=4;y= –5;z=7

4)

xy+9

3z 4

=515

24 4

A) x=5;y=6;z=8B)x=5;y=15;z=24 C) x=15;y=4;z=5D)x=4;y=24;z=72

3 AddandSubtractMatrices

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1)

LetA=–11

25

andB=62

32 .FindA+B.

A)

53

57

B)

34

37

C)

51

21

D)

20

2)

LetA=–10

41

andB=–14

31 .FindA–B.

A)

0–4

10

B)

–24

72

C)

04

–10

D)

–3

3)

LetA=

–15

04

7–4

andB=

72

17 4

42

.FindA–B.

A)

–83

–17 0

3–6

B)

16

78

11 –1

C)

13

70

3–2

D)

3–4

70

–36

4)

LetA=

–3–6

–9–7

–34

andB=

–69

6–7

–2–6

.FindA+B.

A)

–93

–3–14

–5–2

B)

3–15

–15 0

–1–5

C)

–93

3–7

–52

D)

–9–7

–3–14

–5–2

Page10

5)

LetA=

3

–2

–4

andB=

–5

1

5

.FindA+B.

A)

–2

–1

1

B)

[–2–11]

C)

3–5

–21

–45

D)

2

1

2

6)

LetA=

4–42

8–6–1

133

andB=

341

10–2

–24–3

.FindA+B.

A)

703

9–6–3

–170

B)

983

7–6–3

–170

C)

983

9–61

170

D)

703

7–61

170

7)

LetA=

281

46–3

349

andB=

–1–83

304

–47–4

.FindA–B.

A)

316–2

16

–7

7–313

B)

30

–2

16

–7

1–35

C)

3–16 –2

16

–7

7313

D)

104

761

–1115

4 PerformScalarMultiplication

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1)

LetA=–35

02 .Find4A.

A)

–12 20

08

B)

–12 20

02

C)

–12 5

02

D)

19

46

2) LetB=[–144–3].Find–3B.

A) 3–12 –12 9 B) 344–3C) –31212–9D) –322–5

3)

LetA=23

26

andB=04

–16 .Find3A+B.

A)

613

524

B)

621

336

C)

613

112

D)

67

512

Page11

4)

LetA=

1

–3

2

andB=

–1

3

–2

.FindA–3B.

A)

4

–12

8

B)

–2

6

–4

C)

–4

12

–8

D)

4

–6

4

5) LetA=[–42]andB=[10].Find3A+4B.

A) –86 B) –12 4 C) –74 D) –12

6)

LetA=

75

–2

–885

3–5–2

andB=

–7–2–4

–2–1–7

6–6–1

.Find–4A–3B.

A)

–7–14 20

38 –29 1

–30 38 11

B)

–35 –22 4

30 –33 –27

–614 7

C)

03

–6

–10 7 –2

9–11 –3

D)

0–10 9

37

–11

–6–2–3

5 SolveMatrixEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

SolvethematrixequationforX.

1)

LetA=1–5

–2–5

andB=–22

1–4;X+A=B

A)

X=–37

31

B)

X=7–3

13

C)

X=31

–37

D)

X=13

7–3

2)

LetA=5–3

4–5

andB=4–2

51;X+A=B

A)

X=–11

16

B)

X=1–1

61

C)

X=16

–11

D)

X=6–1

11

3)

LetA=

–18

–28

–2–1

andB=

45

89

48

;X–B=A

A)

313

617

27

B)

–53

–10 –1

–6–4

C)

313

–68

2–7

D)

38

617

27

Page12

4)

LetA=

2–2

–80

7–2

andB=

70

0–6

2–2

;4X+A=B

A)

X=

5

4 1

2

2–3

2

–5

40

B)

X=

–5

4–1

2

–23

2

5

40

C)

X=

52

8–6

–50

D)

X=

–52

–86

50

5)

LetA=

–25

0–5

7–8

andB=

7–6

25

06

;B–X=3A

A)

X=

13 –21

220

–21 30

B)

X=

19

–2–10

21 –18

C)

X=

13 –21

–220

–21 30

D)

X=

19

–2–10

7–18

6)

LetA=

41

–3

300

1–43

andB=

–1–4–3

011

304

;3B–3A=X

A)

X=

–15 –15 0

–933

6123

B)

X=

–933

6123

–15 –15 0

C)

X=

–15 –15 0

–911

6123

D)

X=

–911

6123

–15 –15 0

6 MultiplyMatrices

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

FindtheproductAB,ifpossible.

1)

A=–13

32 ,B=–20

–13

A)

–19

–86

B)

20

–36

C)

2–6

–23

D)

9–1

6–8

2)

A=–13

14 ,B=0–26

1–32

A)

3–70

4–14 14

B) ABisnotdefined. C)

34

–7–14

014

D)

0–618

1–12 8

Page13

3)

A=3–21

04

–1,B=30

–21

A) ABisnotdefined. B)

9–63

–68

–3

C)

9–6

–68

3–3

D)

90

04

4)

A=13–1

30 5 ,B=

30

–11

05

A)

0–2

925

B) ABisnotdefined. C)

3–30

0025

D)

–20

25 9

5)

A=–628 ,B=

6

0

–3

A) –60 B) 282 C) –36 0 –24 D)

–36

0

–24

6)

A=–391

–231 ,B=

4

–1

5

A)

–16

–6

B) ABisnotdefined. C) –16 –6D)

–391

–231

4–15

7)

A=[–833],B=

1–3–4

2–8–3

7–9–9

A) 19 –27 –4B)

19

–27

–4

C)

–8303

1–3–4

2–8–3

7–9–9

D)

–8–9–12

–16 –24 –9

–56 –27 –27

Page14

8)

A=

346

7–91

–9–4–4

,B=

317

–2–22

8–13

A)

49 –11 47

47 24 34

–51 3–83

B)

49 47 –51

–11 24 3

47 34 –83

C)

346

7–91

–9–4–4

317

–2–22

8–13

D)

9442

–14 18 2

–72 4 –12

9)

A=3–21

04

–1,B=50

–22

A) ABisnotdefined. B)

15 –10 5

–612–4

C)

15 –6

–10 12

5–4

D)

15 0

08

10)

A=–47

–8

1–22,B=

–8

2

–9

A)

118

–30

B) ABisnotdefined. C) 118 –30 D)

–47

–8

1–22

–82

–9

Page15

7 ModelAppliedSituationswithMatrixOperations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

The⊥shapeinthefigurebelowisshownusing9pixelsina3×3grid.Thecolorlevelsaregiventotherightofthe

figure.Usethematrix

131

333

thatrepresentsadigitalphotographofthe⊥shapetosolvetheproblem.

1) Adjustthecontrastbychangingtheblacktodarkgreyandthelightgreytowhite.Usematrixadditionto

accomplishthis.

A)

131

333

+

–1–1–1

=

020

222

B)

131

333

+

0–10

–1–1–1

=

121

222

C)

131

333

+

111

=

242

444

D)

131

333

+

0–10

–1–1–1

=

020

222

2) Adjustthecontrastbychangingtheblacktolightgreyandthelightgreytoblack.Usematrixadditionto

accomplishthis.

A)

131

333

+

2–22

–2–2–2

=

313

111

B)

131

333

+

–2–2–2

=

313

111

C)

131

333

+

1–11

–1–1–1

=

313

111

D)

131

333

+

222

=

313

111

Page16

3) Adjustthecontrastbychangingtheblacktowhiteandthelightgreytodarkgrey.Usematrixadditionto

accomplishthis.

A)

131

333

+

1–31

–3–3–3

=

202

000

B)

131

333

+

–13–1

333

=

202

000

C)

131

333

+

–1–1–1

=

020

222

D)

131

333

+

0–30

–3–3–3

=

101

000

4) Adjustthecontrastbyleavingtheblackaloneandchangingthelightgreytodarkgrey.Usematrixadditionto

accomplishthis.

A)

131

333

+

101

000

=

232

333

B)

131

333

+

111

=

232

333

C)

131

333

+

0–10

–1–1–1

=

121

322

D)

131

333

+

1–11

–1–1–1

=

121

322

5) Usingthesamecolorlevelsfromtheinstructions,writea3×3matrixAthatrepresentstheletterLindarkgrey

onawhitebackground.Thenfinda3×3matrixBsothatA+BlightensonlytheletterLfromdarkgreyto

lightgrey.

A)

A=

200

222

;B=

–100

–1–1–1

B)

A=

211

222

;B=

100

111

C)

A=

100

111

;B=

–100

–1–1–1

D)

A=

311

333

;B=

–2–1–1

–2–2–2

Page17

Solvetheproblemusingmatrices.

6) StateUniversityhasaCollegeofArts&Sciences,aCollegeofBusiness,andaCollegeofEngineering.The

percentageofstudentsineachcategoryaregivenbythefollowingmatrix.

Freshman Sophomore Junior Senior

Arts&Sciences

Business

Engineering

60% 50% 40% 70%

20% 40% 30% 10%

20% 10% 30% 20%

Thestudentpopulationisdistributedbyclassandageasgiveninthefollowingmatrix.

Female Male

Freshman

Sophomore

Junior

Senior

430 730

550 750

860 620

630 480

HowmanyfemalestudentsareintheCollegeofBusiness?HowmanymalestudentsareintheCollegeofArts

&Sciences?

A) 627students;1397students B) 525 students;680students

C) 680students;503students D) 1318 students;627students

7) Thefinalgradeforanalgebracourseisdeterminedbygradesonthemidtermandfinalexam.Thegradesfor

fourstudentsandtwopossiblegradingsystemsaremodeledbythefollowingmatrices.

Midterm Final

Student1

Student2

Student3

Student4



73 79

44 62

78 82

98 96

System

1

System

2

Midterm

Final

0.4 0.5

0.6 0.5

FindthefinalcoursescoreforStudent3forbothgradingSystem1andSystem2.

A) System1:80.4;System2:80 B) System1:72.2;System2:87.8

C) System1:76.6;System2:76 D) System1:48.6;System2:53

9.4 MultiplicativeInversesofMatricesandMatrixEquations

1 FindtheMultiplicativeInverseofaSquareMatrix

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

FindtheproductsABandBAtodeterminewhetherBisthemultiplicativeinverseofA.

1)

A=53

32 ,B=2–3

–35

A) B=A–1B) B≠A–1

Page18

2)

A=10 1

–10 ,B=01

–110

A) B≠A–1B) B=A–1

3)

A=–24

4–4,B=

1

2

1

4

1

2

1

4

A) B≠A–1B) B=A–1

4)

A= –51

–71 ,B=

1

2–1

2

7

2–5

2

A) B=A–1B) B≠A–1

5)

A=

2–10

–11

–2

10

–1

,B=

1–12

–3–24

–111

A) B≠A–1B) B=A–1

6)

A=–5–1

60

,B=

01

6

–15

6

A) B≠A–1B) B=A–1

7)

A=

100

110

111

,B=

100

–110

0–11

A) B=A–1B) B≠A–1

8)

A=

100–1

2100

111–2

0001

,B=

1001

–210–2

1–113

0001

A) B=A–1B) B≠A–1

9)

A=

2011

11–1–1

–1–210

0011

,B=

200–2

–2103

–2215

2–2–1–3

A) B≠A–1B) B=A–1

Page19

Findtheinverseofthematrix,ifpossible.

10)

A=–1–2

–6–4

A)

1

2–1

4

–3

4

1

8

B)

1

2

1

4

3

4

1

8

C)

–3

4

1

8

1

2–1

4

D)

1

8–1

4

–3

4

1

2

11)

A=0–4

4–2

A)

–1

8

1

4

–1

40

B)

–1

8–1

4

1

40

C)

–1

40

–1

8

1

4

D)

01

4

–1

4–1

8

12)

A=–10

–4–4

A)

–10

1–1

4

B)

–10

–1–1

4

C) Noinverse D)

–1

40

1–1

13)

A=3–1

42

A) Noinverse B)

–3–1

42

C)

31

42

D)

1

3–1

1

4–1

2

14)

A=2–5

06

A)

1

2

5

12

01

6

B)

1

2–5

12

01

6

C)

01

6

1

2

5

12

D)

1

6

5

12

01

2

Page20

15)

A=–43

–40

A)

0–1

4

1

3–1

3

B)

01

4

–1

3–1

3

C)

1

3–1

3

0–1

4

D)

–1

3–1

4

1

30

16)

A=

100

–110

111

A)

100

110

–2–11

B)

111

011

001

C)

1–11

01

–1

001

D)

–100

–1–10

–1–1–1

17)

A=

100

410

0–91

A)

100

–410

–36 9 1

B)

10 0

41 0

00–9

C)

1–936

011

001

D)

100

–9–10

36 4 1

18)

A=

1–100

01

–60

0 017

0 001

A)

1 1 6–42

0 1 6–42

0 0 1–7

00 01

B)

1–742

–42

016

–6

001 1

000 1

C)

10 00

–71 00

42 6 1 0

–42 –611

D)

1000

1100

6610

–7–42 –71

Page21

2 UseInversestoSolveMatrixEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

WritethelinearsystemasamatrixequationintheformAX=B,whereAisthecoefficientmatrixandBistheconstant

matrix.

1) 5x+7y=80

–2x–2y=–28

A)

57

–2–2

x

y

=80

–28

B)

80 7

–28 –2

x

y

=5

–2

C)

5–2

7–2

x

y

=80

–28

D)

57

–2–2

x

y

=–28

80

2) 6x+4y =38

4y =–4

A)

64

0 4

x

y

=38

–4

B)

64

4–4

x

y

=38

0

C)

40

6 4

x

y

=–4

4

D)

38 4

–4 0

x

y

=6

4

3) 6x–2y+8z=86

8x–2y–2z=50

6x–2y+9z=91

A)

6–28

8–2–2

6–29



x

y

z

=

86

50

91

B)

686

–2–2–2

8–29



x

y

z

=

86

50

91

C)

86 8 –2

50 –2–2

91 9 –2



x

y

z

=

6

8

6

D)

6–2

8–2

6–2



x

y

z

=

8

–2

9

4) 9x+4z =17

 5y+4z=18

–2x+8y+4z=22

A)

904

054

–284



x

y

z

=

17

18

22

B)

90

–2

058

444



x

y

z

=

17

18

22

C)

940

540

–284



x

y

z

=

17

18

22

D)

90

05

–28



x

y

z

=

4

4

4

Writethematrixequationasasystemoflinearequationswithoutmatrices.

5)

79

11 9

x

y

=19

20

A) 7x+9y=19

11x+9y=20

B) 7x+9y= –19

11x+9y=–20

C) 9x+7y=19

11x+9y=20

D) 7x+9y=19

9x+11y=20

Page22

6)

5–2

24

x

y

=–4

–6

A) 5x–2y=–4

2x+4y=–6

B) 5x–2y=4

2x+4y=6

C) –2x+5y= –4

2x+4y=–6

D) 5x–2y= –4

4x+2y=–6

7)

884

506

850



x

y

z

=

–2

4

2

A) 8x+8y+4z =-

2

5x+6z =4

8x+5y =2

B) 8x–8y+4z =-

2

5x+6z =-

4

8x+5y =-

2

C) 8x+8y+4z =-

2

5x+6z =4

8x+5z =2

D) 8x+8y+4z =-

2

5x+6y =4

8x+5z =2

Solvethesystemusingtheinversethatisgivenforthecoefficientmatrix.

8)

x+2y +3z =5

x+y+z=11

2x +2y +z=3

Theinverseof

123

111

221

is

–14–1

1–52

02–1

.

A) {(36

,

–44

,

19)} B) {(6

,

–29

,

24)} C) {(20

,

–22

,

3)} D) {(52

,

66

,

25)}

9)

x+2y +3z =-

6

x+y+z=11

x–2z =-

10

Theinverseof

123

111

10–2

is

–24–1

3–52

–12–1

.

A) {(66

,

–93

,

38)} B) {(55

,

–99

,

38)} C) {(–6

,

0,0)} D) {(22

,

17

,

6)}

10)

x+2y +3z =6

x+y+z=12

–x+y+2z =9

Theinverseof

123

111

–112

is

1–1–1

–352

2–3–1

.

A) {(–15

,

60

,

–33)} B) {(–12

,

27

,

9)} C) {(6

,

48

,

–18)} D) {(27

,

96

,

57)}

3 EncodeandDecodeMessages

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Encodeordecodethegivenmessage,asrequested,numberingthelettersofthealphabet1through26intheirusual

order.

1) UsethecodingmatrixA=37

25

toencodethemessageLIFE.

A) 99 53

69 37 B) 78 62

54 43 C) 54 28

129 67 D) –3–5

33

2) UsethecodingmatrixA=–1–3

25

toencodethemessageCARE.

A) –6–33

11 61 B) –57 –16

96 27 C) 18 105

–7–4D) –1–8

–4–29

Page23

3) UsethecodingmatrixA=

10–2

123

111

toencodethemessageCOME_HERE.

A)

–23 –11 –5

72 29 56

31 13 28

B)

35 5

15 0 18

13 8 5

C)

19 27 –21

–14 –30 39

811 –13

D)

–7–21 3

28 69 44

13 33 26

4) UsethecodingmatrixA=1–4

–29

anditsinverseA–1=94

21

todecodethecryptogram–7–8

16 21

.

A) ABLE B) ACTS C) ARMS D) ALAS

5) UsethecodingmatrixA=21

53

anditsinverseA–1=3–1

–52

todecodethecryptogram96

25 17

.

A) BEAD B) CARE C) DARE D) CURB

6) UsethecodingmatrixA=

111

–112

123

anditsinverseA–1=

–1–11

52–3

–3–12

todecodethecryptogram

37 16 35

38 20 4

82 40 60

.

A) GOOD_LUCK B) STAY_CALM C) LOOK_DOWN D) HELP_THEM

9.5 DeterminantsandCramerʹsRule

1 EvaluateaSecond–OrderDeterminant

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Evaluatethedeterminant.

1)

77

94

A) –35 B) 91 C) 35 D) 13

2)

4–7

3–9

A) –15 B) –57 C) 15 D) –1

3)

32

–13

A) 11 B) 7 C) –11 D) 9

4)

1

5–1

8

85

A) 2 B) 0 C) –2D)

89

40

Page24

5)

1

8

1

6

–8

9

2

5

A) 107

540 B) –53

540 C) 1

216 D) –8

45

2 SolveaSystemofLinearEquationsinTwoVariablesUsingCramerʹsRule

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UseCramerʹsruletosolvethesystem.

1) 2x+6y=22

5x+y=-15

A) {(–4

,

5)} B) {(5

,

–4)} C) {(–5

,

–4)} D) {(–4

,

–5)}

2) 3x+4y =51

2x+4y =46

A) {(5

,

9)} B) {(9

,

5)} C) {(–9

,

5)} D) {(5

,

–9)}

3) 3x+3y=33

2x–3y=-

3

A) {(6

,

5)} B) {(5

,

6)} C) {(–5

,

6)} D) {(–6

,

–5)}

4) 3x+2y =2

6x+5y =1

A) {( 8

3,–3)} B) {(–3,8

3)} C) {( 3

8,–1

3)} D) {( 4

9,5

9)}

5) 6x=–3y–18

4x=–y–14

A) {(–4

,

2)} B) {(2

,

–4)} C) {(–2

,

–4)} D) {(–4

,

–2)}

6) 2x=39–3y

5y=69–4x

A) {(6

,

9)} B) {(9

,

6)} C) {(–9

,

6)} D) {(6

,

–9)}

3 EvaluateaThird–OrderDeterminant

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Evaluatethedeterminant.

1)

800

352

723

A) 88 B) 152 C) –88 D) 93

2)

5–55

–304

20

–1

A) –25 B) –55 C) 25 D) 55

Page25

3)

125

213

125

A) 0 B) 62 C) 1 D) –12

4)

316

445

626

A) –48 B) 348 C) 48 D) –108

Solvetheproblem.

5) Theareaofatrianglewithvertices(x1

,

y1),(x2

,

y2),and(x3

,

y3)is

Area=±1

2

x1y11

x2y21

x3y31

,

wherethesymbol±indicatesthattheappropriatesignshouldbechosentoyieldapositivearea.Usethis

formulatofindtheareaofatrianglewhoseverticesare(5,10),(7,–2),and(–3,–4).

A) 62 B) 124 C) 14 D) 28

6) Determinantsareusedtoshowthatthreepointslieonthesameline(arecollinear).If

x1y11

x2y21

x3y31

=0,

thenthepoints(x1,y1),(x2,y2),and(x3,y3)arecollinear.Ifthedeterminantdoesnotequal0,thenthepoints

arenotcollinear.Arethepoints(1,–9),(0,–6),and(3,–15)collinear?

A) Yes B) No

7) Determinantsareusedtoshowthatthreepointslieonthesameline(arecollinear).If

x1y11

x2y21

x3y31

=0,

thenthepoints(x1,y1),(x2,y2),and(x3,y3)arecollinear.Ifthedeterminantdoesnotequal0,thenthepoints

arenotcollinear.Arethepoints(2,7),(0,8),and(6,8)collinear?

A) No B) Yes

8) Theequationofalinepassingthroughtwodistinctpoints(x1

,

y1)and(x2

,

y2)isgivenby

xy1

x2y21

x3y31

=0.Usethedeterminanttowriteanequationforthelinepassingthrough(–7,–9)and(9,1).

Expressthelineʹsequationinstandardform.

A) –10x+16y+74=0B)

–9x+9y–7=0C)10x–16y+74 =0D)1x–7y–81 =0

Page26

4 SolveaSystemofLinearEquationsinThreeVariablesUsingCramerʹsRule

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UseCramerʹsruletosolvethesystem.

1) 6x–3y–z=–17

x–6y+5z=–10

4x+y+z=15

A) {(1

,

6

,

5)} B) {(6

,

5

,

6)} C) {(1

,

–6

,

–5)} D) {(2

,

4

,

5)}

2) 5x–6y+4z=–6

–2x+7y–5z=12

8x–8y+8z=16

A) {(2

,

8

,

8)} B) {(8

,

8

,

8)} C) {(2

,

–8

,

–8)} D) {(3

,

6

,

8)}

3) 3x+3z=36

5x+8y–8z=3

–6x–6y=–48

A) {(7

,

1

,

5)} B) {(1

,

5

,

1)} C) {(7

,

–1

,

–5)} D) {(8

,

–1

,

5)}

4) 5x+5y–z=67

x–2y+2z=5

3x+y+z=35

A) {(9

,

5

,

3)} B) {(5

,

3

,

5)} C) {(9

,

–5

,

–3)} D) {(10

,

3

,

3)}

5) 2x+2y+4z=60

2y+3z=35

3x–4z=–12

A) {(8

,

4

,

9)} B) {(4

,

9

,

4)} C) {(8

,

–4

,

–9)} D) {(9

,

2

,

9)}

5 UseDeterminantstoIdentifyInconsistentSystemsandSystemswithDependentEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UseCramerʹsruletodetermineifthesystemisinconsistentsystemorcontainsdependentequations.

1) 3x +y=10

6x +2y =20

A) systemcontainsdependentequations B) systemisinconsistent

2) 2x –5y =-

29

4x –10y =-

31

A) systemisinconsistent B) systemcontainsdependentequations

3) 3x +y=10

3x +y=25

A) systemisinconsistent B) systemcontainsdependentequations

4) 4x–y+2z=1

3x+5y–z=0

–6x–10y+2z=0

A) systemcontainsdependentequations B) systemisinconsistent

Page27

5) x+z=1

2x–2y=–2

y+z=4

A) systemisinconsistent B) systemcontainsdependentequations

6 EvaluateHigher–OrderDeterminants

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Evaluatethedeterminant.

1)

0002

9581

4499

9147

A) –264 B) 264 C) –18 D) –8

2)

0992

0178

6948

2892

A) –1696 B) 58 C) 64 D) 0

3)

8498

8456

6306

6479

A) –108 B) 14 C) 54 D) 25

Page28

Ch.9 MatricesandDeterminants

AnswerKey

9.1 MatrixSolutionstoLinearSystems

1 WritetheAugmentedMatrixforaLinearSystem

2 PerformMatrixRowOperations

3 UseMatricesandGaussianEliminationtoSolveSystems

4 UseMatricesandGauss–JordanEliminationtoSolveSystems

9.2 InconsistentandDependentSystemsandTheirApplications

1 ApplyGaussianEliminationtoSystemsWithoutUniqueSolutions

2 ApplyGaussianEliminationtoSystemswithMoreVariablesthanEquations

3 SolveProblemsInvolvingSystemsWithoutUniqueSolutions

Page29

9.3 MatrixOperationsandTheirApplications

1 UseMatrixNotation

2) A

2 UnderstandWhatisMeantbyEqualMatrices

3 AddandSubtractMatrices

4 PerformScalarMultiplication

5 SolveMatrixEquations

6) A

6 MultiplyMatrices

7 ModelAppliedSituationswithMatrixOperations

Page30

9.4 MultiplicativeInversesofMatricesandMatrixEquations

1 FindtheMultiplicativeInverseofaSquareMatrix

2 UseInversestoSolveMatrixEquations

3 EncodeandDecodeMessages

9.5 DeterminantsandCramerʹsRule

1 EvaluateaSecond–OrderDeterminant

2 SolveaSystemofLinearEquationsinTwoVariablesUsingCramerʹsRule

3 EvaluateaThird–OrderDeterminant

Page31

4 SolveaSystemofLinearEquationsinThreeVariablesUsingCramerʹsRule

5) A

5 UseDeterminantstoIdentifyInconsistentSystemsandSystemswithDependentEquations

6 EvaluateHigher–OrderDeterminants

Page32