matrices revisited · matrices, we can search for the additive inverses and multiplicative inverses...
TRANSCRIPT
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)
READY, SET, GO Homework: Matrices Revisited 10.2H
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)
READY, SET, GO Homework: Matrices Revisited 10.3H
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)
READY, SET, GO Homework: Matrices Revisited 10.4H
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)
READY, SET, GO Homework: Matrices Revisited 10.5H
ALGEBRA II // MODULE 10H
MATRICES REVISITED
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
10.6H All Systems Go – A Practice Understanding Task
Solving systems of linear equations using matrices (A.REI.8, A.REI.9)
READY, SET, GO Homework: Matrices Revisited 10.6H
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.1H
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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$10.
1. Determinethecostofeachitemusingwhateverstrategyyouwant.Showthedetailsofyourworksothatsomeoneelsecanfollowyourstrategy.
CC
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ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.1H
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Youprobablyrecognizedthatthisproblemcouldberepresentedasasystemofequations.In
previousmathcoursesyouhavedevelopedseveralmethodsforsolvingsystems:graphing,
substitution,elimination,androwreductionofmatrices.
2. Whichofthemethodsyouhavedevelopedpreviouslyforsolvingsystemsofequationscouldbeappliedtothissystem?Whichmethodsseemmoreproblematic?Why?
IntheMVPAlgebraItasksToMarketwithMatricesandSolvingSystemswithMatricesyou
learnedhowtosolvesystemsofequationsinvolvingtwoequationsandtwounknownquantitiesusing
rowreductionofmatrices.(Youmaywanttoreviewthosetwotasksbeforecontinuing.)
3. Modifythe“rowreductionofmatrices”strategysoyoucanuseittosolveCarlosandClarita’ssystemofthreeequationsusingrowreduction.Whatmodificationsdidyouhavetomake,andwhy?
2
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MATRICES REVISITED – 10.1H
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4. Decideonareasonablecontextandwriteastoryfortheinformationgiveninthefollowingmatrix:
"1 2 2 162 1 3 171 2 1 13
(
5. Solvefortheunknownsinyourstorybyusingrowreductiononthegivenmatrix.Checkyourresultsinthestorycontexttomakesuretheyarecorrect.
3
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10.1H
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READY Topic:SolvingSystemsbySubstitutionandElimination
Solveeachsystemofequationsusinganalgebraicmethod.
1.! − 2$ = 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
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10.1H
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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
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MATRICES REVISITED– 10.1H
10.1H
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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%
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MATRICES REVISITED– 10.1H
10.1H
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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
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MATRICES REVISITED– 10.1H
10.1H
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14.Comparetheshapecenterandspreadofeachdistribution.Whatdoyounotice?
15.Ifyourepeatedthisprocesswith500flipsinsteadof5or20,predictwhatwouldhappentothe
shape,spread,andcenterofthenewhistogram.
8
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.2H
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10.2H “. . . and Row by Column”
A Solidify Understanding Task
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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ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.2H
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2. Inacontext,eachentryinamatrixrepresentstwopiecesofinformation,dependingontherowandcolumninwhichitislocated.Organizethefollowinginformationintoamatrix,andlabeltherowsandcolumns.
• Week1:Claritapainted6curbsidelogosandCarlospainted4drivewaymascots• Week2:Claritapainted8curbsidelogosandCarlospainted3drivewaymascots• Week3:Claritapainted5curbsidelogosandCarlospainted6drivewaymascots
3. CarlosandClaritacharge$8foracurbsidelogoand$20foradrivewaymascot.Usingthisadditionalinformation,showhowyoucanusematrixmultiplicationtodeterminehowmuchrevenueCarlosandClaritacollectedduringweeks1,2and3.
10
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MATRICES REVISITED– 10.2H
10.2H
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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
• Day2–53ordersoffriedchicken,60ordersoffish,and125ordersoffrenchfries
• Day3–76ordersoffriedchicken,82ordersoffish,and198ordersoffrenchfries
• Day4–84ordersoffriedchicken,68ordersoffish,and147ordersoffrenchfries
• Day5–91ordersoffriedchicken,88ordersoffish,and203ordersoffrenchfries
READY, SET, GO! Name PeriodDate
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MATRICES REVISITED– 10.2H
10.2H
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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
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.3H
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10.3H More Arithmetic of Matrices
A Solidify Understanding Task
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
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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?
€
a ⋅1= 1⋅ a = a
€
1a
€
a ⋅ 1a
= 1
€
3 14 2"
# $
%
& ' +
a bc d"
# $
%
& ' =
3 14 2"
# $
%
& '
14
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.3H
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
TheMultiplicativeIdentityMatrix
Findvaluesfora,b,canddsothatthematrixbelowthatcontainsthesevariablesplaysthe
roleof1,orthemultiplicativeidentitymatrix,forthefollowingmatrixmultiplication.Willthis
samematrixworkasthemultiplicativeidentityforall2´2matrices?
Nowthatwehaveidentifiedtheadditiveidentityandmultiplicativeidentityfor2´2
matrices,wecansearchfortheadditiveinversesandmultiplicativeinversesofmatrices.
FindinganAdditiveInverseMatrix
Findvaluesfora,b,canddsothatthematrixbelowthatcontainsthesevariablesplaysthe
roleoftheadditiveinverseofthefirstmatrix.Willthissameprocessworkforfindingtheadditive
inverseofall2´2matrices?
€
3 14 2"
# $
%
& ' ⋅
a bc d"
# $
%
& ' =
3 14 2"
# $
%
& '
€
3 14 2"
# $
%
& ' +
a bc d"
# $
%
& ' =
0 00 0"
# $
%
& '
15
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.3H
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FindingaMultiplicativeInverseMatrix
Findvaluesfora,b,canddsothatthematrixbelowthatcontainsthesevariablesplaysthe
roleofthemultiplicativeinverseofthefirstmatrix.Willthissameprocessworkforfindingthe
multiplicativeinverseofall2´2matrices?
€
3 14 2"
# $
%
& ' ⋅
a bc d"
# $
%
& ' =
1 00 1"
# $
%
& '
16
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MATRICES REVISITED– 10.3H
10.3H
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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.
€
A =5 23 1"
# $
%
& '
€
B =4 23 2"
# $
%
& '
READY, SET, GO! Name PeriodDate
17
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MATRICES REVISITED– 10.3H
10.3H
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5.IsABCDarectangle?
Provideconvincingevidenceforyouranswer.
6.IsABCDarhombus? Provideconvincingevidenceforyouranswer.
7.IsABCDasquare? Provideconvincingevidenceforyouranswer.
18
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.4H
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10.4H The Determinant of a Matrix
A Solidify Understanding Task
Intheprevioustaskwelearnedhowtofindthemultiplicativeinverseofamatrix.Usethatprocess
tofindthemultiplicativeinverseofthefollowingtwomatrices.
1.
2.
3. Wereyouabletofindthemultiplicativeinverseforbothmatrices?
Thereisanumberassociatedwitheverysquarematrixcalledthedeterminant.Ifthe
determinantisnotequaltozero,thenthematrixhasamultiplicativeinverse.
Fora2´2matrixthedeterminantcanbefoundusingthefollowingrule:(note:thevertical
lines,ratherthanthesquarebrackets,whichareusedtoindicatethatwearefindingthe
determinantofthematrix)
4 Usingthisrule,findthedeterminantofthetwomatricesgiveninproblems1and2above.
€
5 16 2"
# $
%
& '
€
6 23 1"
# $
%
& '
€
a bc d
= ad − bc
http://en.wikipedia.org/wiki/
File:Area_parallellogram
_as_determinant.svg
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ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.4H
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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
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MATRICES REVISITED – 10.4H
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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
€
a bc d"
# $
%
& ' ⋅
M1 M2
M3 M4
"
# $
%
& ' =
1 00 1"
# $
%
& '
€
a bc d"
# $
%
& '
−1
=1
ad − bcd −b−c a"
# $
%
& '
21
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MATRICES REVISITED– 10.4H
10.4H
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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.
€
5x − 3y = 32x + y = 10# $ %
READY, SET, GO! Name PeriodDate
22
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MATRICES REVISITED– 10.4H
10.4H
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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.
€
8 −24 1#
$ %
&
' (
€
3 26 4"
# $
%
& '
€
4 23 1"
# $
%
& '
23
ALGEBRA II // MODULE 10
MATRICES REVISITED– 10.4H
10.4H
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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
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.5H
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10.5H Solving Systems with
Matrices, Revisited
A Solidify Understanding Task
Whenyousolvelinearequations,youusemanyofthepropertiesofoperationsthatwere
revisitedinthetaskMoreArithmeticofMatrices.
1. Solvethefollowingequationforxandlistthepropertiesofoperationsthatyouuseduringtheequationsolvingprocess.
Thelistofpropertiesyouusedtosolvethisequationprobablyincludedtheuseofa
multiplicativeinverseandthemultiplicativeidentityproperty.Ifyoudidn’tspecificallylistthose
properties,gobackandidentifywheretheymightshowupintheequationsolvingprocessforthis
particularequation.
Systemsoflinearequationscanberepresentedwithmatrixequationsthatcanbesolved
usingthesamepropertiesthatareusedtosolvetheaboveequation.First,weneedtorecognize
howamatrixequationcanrepresentasystemoflinearequations.
2. Writethelinearsystemofequationsthatisrepresentedbythefollowingmatrixequation.(Thinkabouttheprocedureformultiplyingmatricesyoudevelopedinprevioustasks.)
€
23x = 8
€
3 52 4"
# $
%
& ' ⋅
xy"
# $ %
& ' =
−14"
# $
%
& '
25
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.5H
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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" # $
€
23
€
32
€
2 33 4"
# $
%
& '
€
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
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.5H
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Applythepropertyofthemultiplicativeidentityontheleftsideoftheequation
6. Whatdoesthelastlineinthetableinquestion5tellyouaboutthesystemofequationsinquestion3?
7. Usetheprocessyouhavejustexaminedtosolvethefollowingsystemoflinearequations.
€
x = 12
€
3x + 5y = −12x + 4y = 4# $ %
27
ALGEBRA II // MODULE 10
MATRICES REVISITED– 10.5H
10.5H
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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
" # $
READY, SET, GO! Name PeriodDate
28
ALGEBRA II // MODULE 10
MATRICES REVISITED– 10.5H
10.5H
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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
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.6H
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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.
€
1 0 00 1 00 0 1
"
#
$ $ $
%
&
' ' '
⋅
2 0 11 4 00 2 1
"
#
$ $ $
%
&
' ' '
CCBYNASA
https://flic.kr/p/9y3p6p
30
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.6H
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
ALGEBRA II // MODULE 10H
MATRICES REVISITED – 10.6H
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
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MATRICES REVISITED– 10.6H
10.6H
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READY Topic:ReviewingrationalexponentsandmethodsforsolvingquadraticsWriteeachexponentialexpressioninradicalform.1. 2. 3.
10#$ %
&' 3)
&#
4. 5. 6.
6$+ 7
'# -
.'
Writeeachradicalexpressioninexponentialform.
7. 8. 9.
√30 1√723 4' 5%#10. 11. 12.
5)'6 1√)7 48 √)9:
Explaineachstrategyforsolvingquadraticequationsandexplainthecircumstancesinwhichthestrategyismostefficient.13.Graphing 14.Factoring 15.Completingthesquare 16.Whatotherstrategiesdoyouknowforsolvingquadraticequations?Whenwouldyouusethem?
READY, SET, GO! Name PeriodDate
33
ALGEBRA II // MODULE 10
MATRICES REVISITED– 10.6H
10.6H
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SET Topic:Solvingsystemswiththreeunknowns.Solvethesystemofequationsusingmatrices.Createamatrixequationforthesystemofequationsthatcanbeusedtofindthesolution.Thenfindtheinversematrixanduseittosolvethesystem.
17.;2% − 4? + A = 05% − 4? − 5A = 124% + 4? + A = 24
18.;% + 2? + 5A = −15% + ? − 4A = 12% − 6? + 4A = −12
19.;4D + E − 2F = 5
−3D − 3E − 4F = −164D − 4E + 4F = −4
20.;−6% − 4? + A = −20−3% − ? − 3A = −8−5% + 3? + 6A = −4
GO Topic:SolvingquadraticequationsSolveeachoftheequationsbelowusinganappropriateandefficientmethod.21. 22. 23.
%$ − 5% = −6 3%$ − 5 = 0 5%$ − 10 = 0
24. 25. 26.%$ + 1% − 30 = 0 %$ + 2% = 48 %$ − 3% = 0
34