Matrices Revisited · matrices, we can search for the additive inverses and multiplicative inverses...

The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2018 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Office of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

MODULE 10 HONORS

Matrices Revisited

ALGEBRA II

An Integrated Approach

ALGEBRA II // MODULE 10H

MATRICES REVISITED

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 10H - TABLE OF CONTENTS

MATRICES REVISITED

10.1H Row by Row . . . – A Solidify Understanding Task

Extending row reduction of matrices to systems of equations in n variables (MVP Honors

Standard)

READY, SET, GO Homework: Matrices Revisited 10.1H

10.2H . . . and Row by Column – A Solidify Understanding Task

Reviewing matrix multiplication in preparation for solving systems of equations using inverse

matrices (N.VM.6, N.VM.8)


10.3H More Arithmetic of Matrices – A Solidify Understanding Task

Examining properties of matrix addition and multiplication, including identity and inverse

properties (N.VM.8, N.VM.9)


10.4H The Determinant of a Matrix – A Develop Understanding Task

Finding the determinant of a matrix and relating it to the area of a parallelogram (N.VM.10,

N.VM.12)


10.5H Solving Systems with Matrices, Revisited – A Solidify Understanding Task

Solving a system of linear equations using the multiplicative inverse matrix (A.REI.1)



MATRICES REVISITED


10.6H All Systems Go – A Practice Understanding Task

Solving systems of linear equations using matrices (A.REI.8, A.REI.9)



MATRICES REVISITED – 10.1H


10.1H “Row by Row . . .”

A Solidify Understanding Task

Carloslikestobuysuppliesforthetwin’sbusiness,CurbsideRivalry,attheAllaDollarPaint

Storewherethepriceofeveryitemisamultipleof$1.Thismakesiteasytokeeptrackofthetotalcost

ofhispurchases.ClaritaisworriedthatitemsatAllaDollarPaintStoremightcostmore,sosheis

goingovertherecordstoseehowmuchCarlosispayingfordifferentsupplies.Unfortunately,Carlos

hasonceagainforgottentowritedownthecostofeachitemhepurchased.Instead,hehasonly

recordedwhathepurchasedandthetotalcostofalloftheitems.

CarlosandClaritaaretryingtofigureoutthecostofagallonofpaint,thecostofapaintbrush,

andthecostofarollofmaskingtapebasedonthefollowingpurchases:

Week1:Carlosbought2gallonsofpaintand1rollofmaskingtapefor$30.

Week2:Carlosbought1gallonofpaintand4brushesfor$20.

Week3:Carlosbought2brushesand1rollofmaskingtapefor$10.

1. Determinethecostofeachitemusingwhateverstrategyyouwant.Showthedetailsofyourworksothatsomeoneelsecanfollowyourstrategy.

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Youprobablyrecognizedthatthisproblemcouldberepresentedasasystemofequations.In

previousmathcoursesyouhavedevelopedseveralmethodsforsolvingsystems:graphing,

substitution,elimination,androwreductionofmatrices.

2. Whichofthemethodsyouhavedevelopedpreviouslyforsolvingsystemsofequationscouldbeappliedtothissystem?Whichmethodsseemmoreproblematic?Why?

IntheMVPAlgebraItasksToMarketwithMatricesandSolvingSystemswithMatricesyou

learnedhowtosolvesystemsofequationsinvolvingtwoequationsandtwounknownquantitiesusing

rowreductionofmatrices.(Youmaywanttoreviewthosetwotasksbeforecontinuing.)

3. Modifythe“rowreductionofmatrices”strategysoyoucanuseittosolveCarlosandClarita’ssystemofthreeequationsusingrowreduction.Whatmodificationsdidyouhavetomake,andwhy?

2




4. Decideonareasonablecontextandwriteastoryfortheinformationgiveninthefollowingmatrix:

"1 2 2 162 1 3 171 2 1 13

(

5. Solvefortheunknownsinyourstorybyusingrowreductiononthegivenmatrix.Checkyourresultsinthestorycontexttomakesuretheyarecorrect.

3

ALGEBRA II // MODULE 10

MATRICES REVISITED– 10.1H

10.1H


Needhelp?Visitwww.rsgsupport.org

READY Topic:SolvingSystemsbySubstitutionandElimination

Solveeachsystemofequationsusinganalgebraicmethod.

1.! − 2$ = 63! − $ = 13

2.2! − 3$ = 9! + 5$ = −28 3.

! − $ = 04! + $ = 9

4.2! + 3$ = 2−4! − 6$ = −14

5.7! − $ = 14! + 7$ = −48

6.8! + 5$ = 9−! − 5$ = 5

7.Doanyofthesystemsinproblems1-6representparallellines?

Ifso,howdoyouknow?

8.Doanyofthesystemsinproblems1-6representperpendicularlines?

Ifso,howdoyouknow?

READY, SET, GO! Name PeriodDate

4



10.1H



SET Topic:Solvingmatricesusingrowreduction

9.Createamatrixtomatcheachstepinthesolvingofthesystemofequationsgiven.Also,writea

descriptionofwhathappenedtotheequationandthematrixbetweensteps.

SystemofEquations Description Matrix

GivenSystem 03! + 2$ = 40! − 7$ = −2 13 2

1 −7 240−23 ê −345 → 45 ê

Step1 0 3! + 2$ = 40−3! + 21$ = 6 ê 1 2

−3 2406 3 ê ê

Step2 0 3! + 2$ = 400! + 23$ = 46 ê 1 0 2403

ê ê

Step3 03! + 2$ = 400! + $ = 2 ê 1 2 3

ê ê

Step4 03! + 0$ = 360! + $ = 2 ê 1 2 3

ê ê

Step5 0! + 0$ = 120! + $ = 2 1 2 3

5



10.1H



GO Topic:Reviewinghistograms

10.Flipacoin5times.Recordthenumberoftimesthecoinlandswithheadsup.Repeatthis

process20times,eitherbyhandorbysimulationusingtechnology,eachtimerecordingyour

resultsinthetablebelow.(Everytimeyouperformthesimulation,countthenumberofheadsyouhaveandrecordtheresultintheTallycolumnbelow.Forexample,ifyouflipthecoin5timesandget3heads,putatallymarkbythe3headsor60%row.)http://www.rossmanchance.com/applets/CoinTossing/CoinToss.html

11.Createahistogramofyourresults.Describetheshapeofthehistogram(Shape,Center,Spread)

#Heads %Heads Tally0 0%

1 20%

2 40%

3 60%

4 80%

5 100%

6



10.1H



12.Flipacoin20times.Recordthenumberoftimesheadslandssideup.Repeatthisprocess20

timeseitherbyhandorbysimulationusingtechnology.

http://www.rossmanchance.com/applets/CoinTossing/CoinToss.html

Recordyourresultsinthetablebelow.

13.Createahistogramofyourresultsbelow.Describetheshapeofthehistogram(Shape,Center,

Spread)

#Heads

%Heads

Frequency #Heads

%Heads

Frequency

0 0% 11 55%

1 5% 12 60%

2 10% 13 65%

3 15% 14 70%

4 20% 15 75%

5 25% 16 80%

6 30% 17 85%

7 35% 18 90%

8 40% 19 95%

9 45% 20 100%

10 50%

7



10.1H



14.Comparetheshapecenterandspreadofeachdistribution.Whatdoyounotice?

15.Ifyourepeatedthisprocesswith500flipsinsteadof5or20,predictwhatwouldhappentothe

shape,spread,andcenterofthenewhistogram.

8




10.2H “. . . and Row by Column”


Intheprevioustaskyouchoseacontextandwroteastoryfortheinformationgiveninthe

followingmatrix:

"1 2 2 162 1 3 171 2 1 13

(

Yourstorycouldalsohavebeenrepresentedbythefollowingsystemofequations:) + 2+ + 2, = 162) + + + 3, = 17) + 2+ + , = 13

Usingrowreductiononthematrix,youfoundthatthesolutiontothissystemofequationsis:

x=2,y=4,z=3.Inalatertaskinthismoduleyouwilllearnanothermethodforsolvinglinear

systemsusingmatrices.Thisnewmethodwillusematrixmultiplication,solet’sreviewthat

operation.

Wecanverifythatx=2,y=4,z=3isasolutiontothesystemofequationsgivenaboveusing

matrixmultiplication.

1. Usethefollowingtoreviewhowmatrixmultiplicationworks.Explainhowthenumbersinthematrixontherightsideoftheequationwereobtainedastheproductofthetwomatricesontheleftsideoftheequation.

"1 2 22 1 31 2 1

( ∙ "243( = "

161713(

Myexplanation:

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2. Inacontext,eachentryinamatrixrepresentstwopiecesofinformation,dependingontherowandcolumninwhichitislocated.Organizethefollowinginformationintoamatrix,andlabeltherowsandcolumns.

• Week1:Claritapainted6curbsidelogosandCarlospainted4drivewaymascots• Week2:Claritapainted8curbsidelogosandCarlospainted3drivewaymascots• Week3:Claritapainted5curbsidelogosandCarlospainted6drivewaymascots

3. CarlosandClaritacharge$8foracurbsidelogoand$20foradrivewaymascot.Usingthisadditionalinformation,showhowyoucanusematrixmultiplicationtodeterminehowmuchrevenueCarlosandClaritacollectedduringweeks1,2and3.

10



10.2H



READY Topic:Addingmatrices

Addthegivenmatrices.Ifthematricescan’tbeadded,giveareasonwhynot.

1.! 5 −223 11' + !

10 8−4 19' = 2./

99 5598 6478 12

2 + ! 1 45 222 88 0' =

3./70 14 62−31 39 8566 13 −7

2 + /−70 −14 −6231 −39 −85−66 −13 7

2 = 4./6−8102 + /

−108−6

2 =

SET Topic:Multiplyingmatrices5.Recallfromtoday’slessonthatinacontext,eachentryinamatrixrepresentstwopiecesof

information,dependingontherowandcolumninwhichitislocated.Organizethe

followinginformationintoamatrix,andlabeltherowsandcolumns.

ThefollowingnumberofitemsweresoldduringthelunchrushatFriedFreddy’sCafé:

• Day1–65ordersoffriedchicken,62ordersoffish,and145ordersoffrenchfries






11



10.2H



5./1 1 22 2 31 3 4

2 ∙ /155102 = /

11111111111111111

2

6./10 20 2020 14 511 15 2

2 ∙ /5282 = /

11111111111111111

2

7./4 3 0.52 1 35 4 0.25

2 ∙ /2530202 = /

11111111111111111

2

8./5 2 44 4 48 2 5

2 ∙ /915252 = /

11111111111111111

2

GO

Topic:Findingprobabilitiesfromatwo-waytable

Thefollowingdatarepresentsarandomsampleofboysandgirlsandhowmanyprefercatsordogs.Usetheinformationtoanswerthequestionsbelow.

Cats Dogs TotalBoys 32 68 100

Girls 41 11 52

Total 73 79 152

9.5(7) = 10.5(9) = 11.5(:) = 12.5(;) =

13.5(:|9) = 14.5(:=>7) = 15.5(;|7) = 16.5(7 ∩ ;) =

17.Ifthisisarandomsamplefromaschool,whattotalpercentofboysinthisschooldoyouthink

wouldpreferdogs?

18.Whatpercentofstudentsattheschoolwouldprefercats?

19.Ifyousampledadifferent152students,wouldyougetthesamepercentages?Explain.

20.Whatwouldhappentoyourpercentagesifyouusedalargersamplesize?

12




10.3H More Arithmetic of Matrices


Inthistaskyouwillhaveanopportunitytoexaminesomeofthepropertiesofmatrix

additionandmatrixmultiplication.Wewillrestrictthisworktosquare2´2matrices.

Thetablebelowdefinesandillustratesseveralpropertiesofadditionandmultiplicationfor

realnumbersandasksyoutodetermineifthesesamepropertiesholdformatrixadditionand

matrixmultiplication.Whilethechartasksforasingleexampleforeachproperty,youshould

experimentwithmatricesuntilyouareconvincedthatthepropertyholdsoryouhavefounda

counter-exampletoshowthatthepropertydoesnothold.Canyoubaseyourjustificationonmore

thatjusttryingoutseveralexamples?

Property ExamplewithRealNumbers ExamplewithMatrices

AssociativePropertyof

Addition

(a+b)+c=a+(b+c)

AssociativePropertyof

Multiplication

(ab)c=a(bc)

CommutativePropertyof

Addition

a+b=b+a

http://commons.wikimedia.org/wiki/File:Matriz_A_por_B.png

13




CommutativePropertyof

Multiplication

ab=ba

DistributivePropertyof

MultiplicationOverAddition

a(b+c)=ab+ac

Inadditiontothepropertieslistedinthetableabove,additionandmultiplicationofreal

numbersincludepropertiesrelatedtothenumbers0and1.Forexample,thenumber0isreferred

toastheadditiveidentitybecausea+0=0+a=a,andthenumber1isreferredtoasthe

multiplicativeidentitysince .Oncetheadditiveandmultiplicativeidentitieshave

beenidentified,wecanthendefineadditiveinversesaand–asincea+-a=0,andmultiplicative

inversesaand since .Todecideifthesepropertiesholdformatrixoperations,wewill

needtodetermineifthereisamatrixthatplaystheroleof0formatrixaddition,andifthereisa

matrixthatplaystheroleof1formatrixmultiplication.

TheAdditiveIdentityMatrix

Findvaluesfora,b,canddsothatthematrixbelowthatcontainsthesevariablesplaysthe

roleof0,ortheadditiveidentitymatrix,forthefollowingmatrixaddition.Willthissamematrix

workastheadditiveidentityforall2´2matrices?

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a ⋅1= 1⋅ a = a

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1a

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a ⋅ 1a

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TheMultiplicativeIdentityMatrix


roleof1,orthemultiplicativeidentitymatrix,forthefollowingmatrixmultiplication.Willthis

samematrixworkasthemultiplicativeidentityforall2´2matrices?

Nowthatwehaveidentifiedtheadditiveidentityandmultiplicativeidentityfor2´2

matrices,wecansearchfortheadditiveinversesandmultiplicativeinversesofmatrices.

FindinganAdditiveInverseMatrix


roleoftheadditiveinverseofthefirstmatrix.Willthissameprocessworkforfindingtheadditive

inverseofall2´2matrices?

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FindingaMultiplicativeInverseMatrix


roleofthemultiplicativeinverseofthefirstmatrix.Willthissameprocessworkforfindingthe

multiplicativeinverseofall2´2matrices?

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10.3H



READY Topic: SET Topic::Findingtheadditiveandmultiplicativeinverseofamatrix

2.Given:Matrix

a.FindtheadditiveinverseofmatrixAb.FindthemultiplicativeinverseofmatrixA

3.Given:Matrix

a.FindtheadditiveinverseofmatrixB. b.FindthemultiplicativeinverseofmatrixB

GO Topic:Parallellines,perpendicularlines,andlengthfromacoordinategeometryperspective

Giventhefourpoints:A(2,1),B(5,2),C(4,5),andD(1,4)

4.IsABCDaparallelogram?

Provideconvincingevidenceforyouranswer.

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10.3H



5.IsABCDarectangle?

Provideconvincingevidenceforyouranswer.

6.IsABCDarhombus? Provideconvincingevidenceforyouranswer.

7.IsABCDasquare? Provideconvincingevidenceforyouranswer.

18




10.4H The Determinant of a Matrix


Intheprevioustaskwelearnedhowtofindthemultiplicativeinverseofamatrix.Usethatprocess

tofindthemultiplicativeinverseofthefollowingtwomatrices.

1.

2.

3. Wereyouabletofindthemultiplicativeinverseforbothmatrices?

Thereisanumberassociatedwitheverysquarematrixcalledthedeterminant.Ifthe

determinantisnotequaltozero,thenthematrixhasamultiplicativeinverse.

Fora2´2matrixthedeterminantcanbefoundusingthefollowingrule:(note:thevertical

lines,ratherthanthesquarebrackets,whichareusedtoindicatethatwearefindingthe

determinantofthematrix)

4 Usingthisrule,findthedeterminantofthetwomatricesgiveninproblems1and2above.

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= ad − bc

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File:Area_parallellogram

_as_determinant.svg

19




Theabsolutevalueofthedeterminantofa2´2matrixcanbevisualizedastheareaofa

parallelogram,constructedasfollows:

• Drawonesideoftheparallelogramwithendpointsat(0,0)and(a,c).• Drawasecondsideoftheparallelogramwithendpointsat(0,0)and(b,d).• Locatethefourthvertexthatcompletestheparallelogram.

(Notethattheelementsinthecolumnsofthematrixareusedtodefinetheendpointsofthevectors

thatformtwosidesoftheparallelogram.)

5. Usethefollowingdiagramtoshowthattheareaoftheparallelogramisgivenbyad–bc.

6. Drawtheparallelogramswhoseareasrepresentthedeterminantsofthetwomatriceslistedinquestions1and2above.Howdoesazerodeterminantshowupinthesediagrams?

20




7. Createamatrixforwhichthedeterminantwillbenegative.Drawtheparallelogram

associatedwiththedeterminantofyourmatrixandfindtheareaoftheparallelogram.

Thedeterminantcanbeusedtoprovideanalternativemethodforfindingtheinverseof2´2

matrix.

8. Usetheprocessyouusedpreviouslytofindtheinverseofageneric2´2matrixwhoseelementsaregivenbythevariablesa,b,candd.Fornow,wewillrefertotheelementsoftheinversematrixasM1,M2,M3andM4asillustratedinthefollowingmatrixequation.FindexpressionsforM1,M2,M3andM4intermsoftheelementsofthefirstmatrix,a,b,candd.

M1=M2=M3=M4=Useyourworkabovetoexplainthisstrategyforfindingtheinverseofa2´2matrix:(note:the-1superscriptisusedtoindicatethatwearefindingthemultiplicativeinverseofthematrix)

wheread–bcisthedeterminantofthematrix

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M3 M4

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10.4H



READY Topic:Solvingsystemsofequationsusingrowreduction

Giventhesystemofequations

1.Zacstartedsolvingthisproblembywriting !5 −3 32 1 10( → !1 −5 −17

2 1 10 (

DescribewhatZacdidtogetfromthematrixonthelefttothematrixontheright.

2.Leastartedsolvingthisproblembywriting!5 −3 32 1 10( → +

5 −3 31 ,

- 5.

DescribewhatLeadidtogetfromthematrixonthelefttothematrixontheright.

3.UsingeitherZac’sorLea’sfirststep,continuesolvingthesystemusingrowreduction.Showeach

matrixalongwithnotationindicatinghowyougotfromonematrixtoanother.Besuretocheck

yoursolution.

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5x − 3y = 32x + y = 10# $ %


22



10.4H



SET Topic:Findingthedeterminantofa2X2matrix

4.Usethedeterminantofeach2´2matrixtodecidewhichmatriceshavemultiplicativeinverses,

andwhichdonot.

a. b. c.

5.Findthemultiplicativeinverseofeachofthematricesin4,providedtheinversematrixexists.

a. b. c.

6.Generallymatrixmultiplicationisnotcommutative.Thatis,ifAandBarematrices,typically

/ ∙ 1 ≠ 1 ∙ /.However,multiplicationofinversematricesiscommutative.Testthisoutbyshowingthatthepairsofinversematricesyoufoundinquestion7givethesameresultwhen

multipliedineitherorder.

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10.4H



GO Topic:Parallelandperpendicularlines

Determineifthefollowingpairsoflinesareparallel,perpendicularorneither.Explainhow

youarrivedatyouranswer.

7. 3x+2y=7and6x+4y=9

8. and

9. and4x+3y=3

10.Writetheequationofalinethatisparallelto andhasay-interceptat(0,4).

€

y =23x − 5

€

y = −23x + 7

€

y =34x − 2

€

y =45x − 2

24




10.5H Solving Systems with

Matrices, Revisited


Whenyousolvelinearequations,youusemanyofthepropertiesofoperationsthatwere

revisitedinthetaskMoreArithmeticofMatrices.

1. Solvethefollowingequationforxandlistthepropertiesofoperationsthatyouuseduringtheequationsolvingprocess.

Thelistofpropertiesyouusedtosolvethisequationprobablyincludedtheuseofa

multiplicativeinverseandthemultiplicativeidentityproperty.Ifyoudidn’tspecificallylistthose

properties,gobackandidentifywheretheymightshowupintheequationsolvingprocessforthis

particularequation.

Systemsoflinearequationscanberepresentedwithmatrixequationsthatcanbesolved

usingthesamepropertiesthatareusedtosolvetheaboveequation.First,weneedtorecognize

howamatrixequationcanrepresentasystemoflinearequations.

2. Writethelinearsystemofequationsthatisrepresentedbythefollowingmatrixequation.(Thinkabouttheprocedureformultiplyingmatricesyoudevelopedinprevioustasks.)

€

23x = 8

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# $ %

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# $

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3. Usingtherelationshipsyounoticedinquestion3,writethematrixequationthatrepresentsthefollowingsystemofequations.

4. Therationalnumbers and aremultiplicativeinverses.Whatisthemultiplicative

inverseofthematrix ?Note:Theinversematrixisusuallydenotedby .

5. Thefollowingtableliststhestepsyoumayhaveusedtosolve andasksyoutoapply

thosesamestepstothematrixequationyouwroteinquestion4.Completethetableusing

thesesamesteps.

Originalequation

Multiplybothsidesoftheequationbythemultiplicativeinverse

Theproductofmultiplicativeinversesisthemultiplicativeidentityontheleftsideoftheequation

Performtheindicatedmultiplicationontherightsideoftheequation

€

2x + 3y = 143x + 4y = 20" # $

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23

€

32

€

2 33 4"

# $

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€

2 33 4"

# $

%

& '

−1

€

23x = 8

€

23x = 8

€

2 33 4"

# $

%

& ' ⋅

xy"

# $ %

& ' =

1420"

# $

%

& '

€

32⋅23x =

32⋅8

€

1⋅ x =32⋅8

€

1⋅ x = 12

26




Applythepropertyofthemultiplicativeidentityontheleftsideoftheequation

6. Whatdoesthelastlineinthetableinquestion5tellyouaboutthesystemofequationsinquestion3?

7. Usetheprocessyouhavejustexaminedtosolvethefollowingsystemoflinearequations.

€

x = 12

€

3x + 5y = −12x + 4y = 4# $ %

27



10.5H



READY Topic:ReflectionsandRotations1.Thefollowingthreepointsformtheverticesofatriangle:(3,2),(6,1),(4,3)

a.Plotthesethreepointsonthecoordinategridandthenconnectthemtoformatriangle.

b.Reflecttheoriginaltriangleoverthey-axisandrecordthecoordinatesoftheverticeshere:

c.Reflecttheoriginaltriangleoverthex-axisandrecordthecoordinatesoftheverticeshere:

d.Rotatetheoriginaltriangle90°counter-clockwiseabouttheoriginandrecordthecoordinatesoftheverticeshere:

e.Rotatetheoriginaltriangle180°abouttheoriginandrecordthecoordinatesoftheverticeshere. SET Topic:SolvingSystemsUsingInverseMatricesTwoofthefollowingsystemshaveuniquesolutions(thatis,thelinesintersectatasinglepoint).2.Usethedeterminantofa2´2matrixtodecidewhichsystemshaveuniquesolutions,andwhichonedoesnot.

a. b. c.

€

8x − 2y = −24x + y = 5

# $ %

€

3x + 2y = 76x + 4y = −5# $ %

€

4x + 2y = 03x + y = 2

" # $


28



10.5H



3.Foreachofthesystemsin#2whichhaveauniquesolution,findthesolutiontothesystembysolvingamatrixequationusinganinversematrix.a. b. c.

GO Topic:ReviewingthePropertiesofArithmeticMatcheachexampleontheleftwiththenameofapropertyofarithmeticontheright.Notallanswerswillbeused.

_____4.2(x+3y)=2x+6y

_____5.(2x+3y)+4y=2x+(3y+4y)

_____6.2x+3y=3y+2x

_____7.2(3y)=(2×3)y=6y

_____8.

_____9.x+-x=0

_____10.xy=yx

a.multiplicativeinverses

b.additiveinverses

c.multiplicativeidentity

d.additiveidentity

e.commutativepropertyofaddition

f.commutativepropertyofmultiplication

g.associativepropertyofaddition

h.associativepropertyofmultiplication

i.distributivepropertyofadditionovermultiplication

€

23⋅32x = 1x

29




10.6H All Systems Go!

A Practice Understanding Task

Thismodulebeganwiththefollowingproblem:

CarlosandClaritaaretryingtofigureoutthecostofagallonofpaint,thecostofa

paintbrush,andthecostofarollofmaskingtapebasedonthefollowingpurchases:

Week1:Carlosbought2gallonsofpaintand1rollofmaskingtapefor$30.

Week2:Carlosbought1gallonofpaintand4brushesfor$20.

Week3:Carlosbought2brushesand1rollofmaskingtapefor$10.

IntheprevioussequenceoftasksMoreArithmeticofMatrices,SolvingSystemswithMatrices,

RevisitedandTheDeterminantofaMatrixyoulearnedhowtosolvesystemsusingmultiplicationof

matrices.Inthistask,wearegoingtoextendthisstrategytoincludesystemswithmorethattwo

equationsandtwovariables.

1. Multiplythefollowpairsofmatrices:

a.

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30




b.

2. Whatpropertyisillustratedbythemultiplicationinquestion1a?3. Whatpropertyisillustratedbythemultiplicationinquestion1b?4. Rewritethefollowingsystemofequations,whichrepresentsCarlosandClarita’sproblem,

asamatrixequationintheformAX=BwhereA,XandBareallmatrices.

5. Solveyourmatrixequationbyusingmultiplicationofmatrices.Showthedetailsofyourworksothatsomeoneelsecanfollowit.

€

0.4 0.2 −0.4−0.1 0.2 0.10.2 −0.4 0.8

#

$

% % %

&

'

( ( (

⋅

2 0 11 4 00 2 1

#

$

% % %

&

'

( ( (

€

2g + 0b +1t = 301g + 4b + 0t = 200g + 2b +1t = 10

31




Youwereabletosolvethisequationusingmatrixmultiplicationbecauseyouweregiventhe

inverseofmatrixA.Unlike2´2matrices,wheretheinversematrixcaneasilybefoundbyhand

usingthemethodsdescribedinMoreArithmeticofMatrices,theinversesofann´nmatrixin

generalcanbedifficulttofindbyhand.Insuchcases,wewillusetechnologytofindtheinverse

matrixsothatthismethodcanbeappliedtoalllinearsystemsinvolvingnequationsandn

unknownquantities.Hereisoneonlineresourceyoumightuse:https://matrixcalc.org/en/

6. Solvethefollowingsystemusingamatrixequationandinversematrices.Althoughyoumay

usetechnologytofindtheinversematrix,makesureyourecordallofyourworkinthespacebelow,includingyourinversematrix.

! + 2$ + 2% = 162! + $ + 3% = 17! + 2$ + % = 13

32



10.6H



READY Topic:ReviewingrationalexponentsandmethodsforsolvingquadraticsWriteeachexponentialexpressioninradicalform.1. 2. 3.

10#$ %

&' 3)

&#

4. 5. 6.

6$+ 7

'# -

.'

Writeeachradicalexpressioninexponentialform.

7. 8. 9.

√30 1√723 4' 5%#10. 11. 12.

5)'6 1√)7 48 √)9:

Explaineachstrategyforsolvingquadraticequationsandexplainthecircumstancesinwhichthestrategyismostefficient.13.Graphing 14.Factoring 15.Completingthesquare 16.Whatotherstrategiesdoyouknowforsolvingquadraticequations?Whenwouldyouusethem?


33



10.6H



SET Topic:Solvingsystemswiththreeunknowns.Solvethesystemofequationsusingmatrices.Createamatrixequationforthesystemofequationsthatcanbeusedtofindthesolution.Thenfindtheinversematrixanduseittosolvethesystem.

17.;2% − 4? + A = 05% − 4? − 5A = 124% + 4? + A = 24

18.;% + 2? + 5A = −15% + ? − 4A = 12% − 6? + 4A = −12

19.;4D + E − 2F = 5

−3D − 3E − 4F = −164D − 4E + 4F = −4

20.;−6% − 4? + A = −20−3% − ? − 3A = −8−5% + 3? + 6A = −4

GO Topic:SolvingquadraticequationsSolveeachoftheequationsbelowusinganappropriateandefficientmethod.21. 22. 23.

%$ − 5% = −6 3%$ − 5 = 0 5%$ − 10 = 0

24. 25. 26.%$ + 1% − 30 = 0 %$ + 2% = 48 %$ − 3% = 0

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Matrices Revisited · matrices, we can search for the additive inverses and multiplicative inverses...

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Transcript of Matrices Revisited · matrices, we can search for the additive inverses and multiplicative inverses...