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3 Cramer’s Rule, Volume, and Linear Transformations

3 Cramer’s Rule, Volume, and Linear Transformations

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178



CHAPTER 3



Determinants



SOLUTION Viewthesystemas Ax D b. Usingthenotationintroducedabove,

Ä

Ä

Ä

3 2

6

2

3 6

AD

;

A1 .b/ D

;

A 2 .b / D

5 4

8 4

5 8

Since det A D 2, thesystemhasauniquesolution. ByCramer’srule,

det A1 .b/

24 C 16

D

D 20

det A

2

det A2 .b/

24 C 30

x2 D

D

D 27

det A

2



x1 D



Application to Engineering

A numberofimportantengineeringproblems, particularlyinelectricalengineeringand

controltheory, canbeanalyzedby Laplacetransforms. Thisapproachconvertsan

appropriatesystemoflineardifferentialequationsintoasystemoflinearalgebraic

equationswhosecoefficientsinvolveaparameter. Thenextexampleillustratesthetype

ofalgebraicsystemthatmayarise.



EXAMPLE 2 Considerthefollowingsysteminwhich s isanunspecifiedparameter.

Determinethevaluesof s forwhichthesystemhasauniquesolution, anduseCramer’s

ruletodescribethesolution.

3sx1 2x2 D 4

6x1 C sx2 D 1



SOLUTION Viewthesystemas Ax D b. Then

Ä

Ä

3s

2

4

2

AD

; A1 .b/ D

;

6

s

1

s

Since



det A D 3s 2



A2 .b/ D



12 D 3.s C 2/.s







3s

6



4

1



2/



thesystemhasauniquesolutionpreciselywhen s Ô ˙2. Forsuchan s , thesolutionis

.x1 ; x2 /, where



x1 D

x2 D



det A1 .b/

4s C 2

D

det A

3.s C 2/.s 2/



3s C 24

sC8

det A2 .b/

D

D

det A

3.s C 2/.s 2/

.s C 2/.s 2/



A Formula for A–1

Cramer’sruleleadseasilytoageneralformulafortheinverseofan n

The j thcolumnof A 1 isavector x thatsatisfies



n matrix A.



Ax D ej

where ej isthe j thcolumnoftheidentitymatrix, andthe i thentryof x isthe .i; j /-entry

of A 1 . ByCramer’srule,

˚

«

det Ai .ej /

.i; j /-entryof A 1 D xi D

det A



(2)



3.3



Cramer's Rule, Volume, and Linear Transformations



179



Recallthat Aj i denotesthesubmatrixof A formedbydeletingrow j andcolumn i . A

cofactorexpansiondowncolumn i of Ai .ej / showsthat

det Ai .ej / D . 1/i Cj det Aj i D Cj i



(3)



where Cj i isacofactorof A. By(2), the .i; j /-entryof A 1 isthecofactor Cj i divided

by det A. [Notethatthesubscriptson Cj i arethereverseof .i; j /.] Thus

2

3

C11

C21

Cn1

C22

Cn2 7

1 6

6 C12

7

(4)

A 1D

6 ::

::

:: 7

det A 4 :

:

:5



C1n



C2n



Cnn



Thematrixofcofactorsontherightsideof(4)iscalledthe adjugate (or classicaladjoint)of A, denotedby adj A. (Theterm adjoint alsohasanothermeaningin

advancedtextsonlineartransformations.) Thenexttheoremsimplyrestates(4).



THEOREM 8



An Inverse Formula

Let A beaninvertible n



n matrix. Then

1

adj A

A 1D

det A

2



2

EXAMPLE 3 Findtheinverseofthematrix A D 4 1

1

SOLUTION Theninecofactorsare

ˇ

ˇ

ˇ

ˇ 1

ˇ1

1 ˇˇ

ˇ

C11 D Cˇˇ

D

2;

C

D

12

ˇ

ˇ1

4 2

ˇ

ˇ

ˇ

ˇ1

ˇ2

3 ˇˇ

ˇ

C21 D ˇˇ

D

14;

C

D

C

22

ˇ

ˇ1

4 2

ˇ

ˇ

ˇ

ˇ 1

ˇ2

3 ˇˇ

C31 D Cˇˇ

D 4;

C32 D ˇˇ

ˇ

1 1

1



ˇ

1 ˇˇ

D 3;



ˇ

3 ˇˇ

D 7;



ˇ

3 ˇˇ

D 1;





1

1

4



3

3

1 5.

2



ˇ

ˇ1

C13 D Cˇˇ

1

ˇ

ˇ2

C23 D ˇˇ

1

ˇ

ˇ2

C33 D Cˇˇ

1



ˇ

1 ˇˇ

D5



ˇ

1 ˇˇ

D 7



ˇ

1 ˇˇ

D 3





Theadjugatematrixisthe transpose ofthematrixofcofactors. [Forinstance, C12 goes

inthe .2; 1/ position.] Thus

2

3

2 14 4

7 15

adj A D 4 3

5

7 3

Wecouldcompute det A directly, butthefollowingcomputationprovidesacheckon

thecalculationsabove and produces det A:

3

2

32

3 2

14

0

0

2 14

4

2 1 3

0 5 D 14I

7

1 54 1 1 1 5 D 4 0 14

.adj A/ A D 4 3

0

0 14

5

7

3

1 4

2

Since .adj A/A D 14I , Theorem8showsthat det A D 14 and

2

3 2

3

2 14 4

1=7

1

2=7

1

4 3

7 1 5 D 4 3=14

1=2 1=14 5

A 1D

14

5

7 3

5=14

1=2

3=14



180



CHAPTER 3



Determinants



NUMERICAL NOTES

Theorem8isusefulmainlyfortheoreticalcalculations. Theformulafor A 1

permitsonetodeducepropertiesoftheinversewithoutactuallycalculatingit.

Exceptforspecialcases, thealgorithminSection 2.2givesamuchbetterwayto

compute A 1 , iftheinverseisreallyneeded.

Cramer’sruleisalsoatheoreticaltool. Itcanbeusedtostudyhowsensitive

thesolutionof Ax D b istochangesinanentryin b orin A (perhapsdue

toexperimentalerrorwhenacquiringtheentriesfor b or A). When A isa

3 3 matrixwith complex entries, Cramer’sruleissometimesselectedforhand

computationbecauserowreductionof Œ A b  withcomplexarithmeticcanbe

messy, andthedeterminantsarefairlyeasytocompute. Foralarger n n matrix

(realorcomplex), Cramer’sruleishopelesslyinefficient. Computingjust one

determinanttakesaboutasmuchworkassolving Ax D b byrowreduction.



Determinants as Area or Volume

Inthenextapplication, weverifythegeometricinterpretationofdeterminantsdescribed

inthechapterintroduction. Althoughageneraldiscussionoflengthanddistancein Rn

willnotbegivenuntilChapter 6, weassumeherethattheusualEuclideanconceptsof

length, area, andvolumearealreadyunderstoodfor R2 and R3 .



THEOREM 9



SG



PROOF The theoremisobviouslytrueforany 2 2 diagonalmatrix:

ˇ Ä

ˇ

ˇ

0 ˇˇ

areaof

ˇ det a

D jad j D

ˇ

0 d ˇ

rectangle



A Geometric Proof

3–12



y

⎡0 ⎡

⎢d ⎢

⎣ ⎣



⎡ a⎡

⎢ ⎢

⎣ 0⎣

FIGURE 1



Area D jad j.



If A isa 2 2 matrix, theareaoftheparallelogramdeterminedbythecolumnsof

A is jdet Aj. If A isa 3 3 matrix, thevolumeoftheparallelepipeddetermined

bythecolumnsof A is jdet Aj.



x



SeeFig.1. Itwillsufficetoshowthatany 2 2 matrix A D Œ a1 a2  canbetransformedintoadiagonalmatrixinawaythatchangesneithertheareaoftheassociated

parallelogramnor jdet Aj. FromSection 3.2, weknowthattheabsolutevalueofthe

determinantisunchangedwhentwocolumnsareinterchangedoramultipleofone

columnisaddedtoanother. Anditiseasytoseethatsuchoperationssufficetotransform

A intoadiagonalmatrix. Columninterchangesdonotchangetheparallelogramatall.

Soitsufficestoprovethefollowingsimplegeometricobservationthatappliestovectors

in R2 or R3 :

Let a1 and a2 be nonzero vectors. Then for any scalar c , the area of the

parallelogram determined by a1 and a2 equals the area of the parallelogram

determinedby a1 and a2 C c a1 .

Toprovethisstatement, wemayassumethat a2 isnotamultipleof a1 , forotherwise thetwoparallelogramswouldbedegenerateandhavezeroarea. If L istheline

through 0 and a1 , then a2 C L isthelinethrough a2 parallelto L, and a2 C c a1 ison

thisline. SeeFig. 2. Thepoints a2 and a2 C c a1 havethesameperpendiculardistance

to L. HencethetwoparallelogramsinFig. 2havethesamearea, sincetheysharethe

basefrom 0 to a1 . Thiscompletestheprooffor R2 .



3.3



Cramer's Rule, Volume, and Linear Transformations

a2



a 2 + c a1



181



a2 + L



L

a1



0



c a1



FIGURE 2 Twoparallelogramsofequalarea.

z

⎡0 ⎡

⎢0 ⎢

⎢c ⎢

⎣ ⎣



x



⎡a⎡

⎢0⎢

⎢0⎢

⎣ ⎣



⎡ 0⎡

⎢ b⎢

⎢ ⎢

⎣ 0⎣



y



Theprooffor R3 issimilar. Thetheoremisobviouslytruefora 3 3 diagonal

matrix. SeeFig. 3. Andany 3 3 matrix A canbetransformedintoadiagonalmatrix

usingcolumnoperationsthatdonotchange jdet Aj. (Thinkaboutdoingrowoperations

on AT .) Soitsufficestoshowthattheseoperationsdonotaffectthevolumeofthe

parallelepipeddeterminedbythecolumnsof A.

A parallelepipedisshowninFig. 4asashadedboxwithtwoslopingsides. Its

volumeistheareaofthebaseintheplane Span fa1 ; a3 g timesthealtitudeof a2 above

Span fa1 ; a3 g. Anyvector a2 C c a1 hasthesamealtitudebecause a2 C c a1 liesinthe

plane a2 C Span fa1 ; a3 g, whichisparallelto Span fa1 ; a3 g. Hencethevolumeofthe

parallelepipedisunchangedwhen Œ a1 a2 a3  ischangedto Œ a1 a2 C c a1 a3 .

Thusacolumnreplacementoperationdoesnotaffectthevolumeoftheparallelepiped.

Sincecolumninterchangeshavenoeffectonthevolume, theproofiscomplete.



FIGURE 3



Volume D jabcj.



}

,a3

a1

{

n



a



a2



+



Sp



a2



0



}

,a3

a1

{

n



a3



n{



a

Sp



}

,a3



a1



a2



+



a3



a

Sp

a 2 + ca 1 a 2



}

,a3

a1

{

n

pa



S



a1



a1



0



FIGURE 4 Twoparallelepipedsofequalvolume.



EXAMPLE 4 Calculatetheareaoftheparallelogramdeterminedbythepoints

. 2; 2/, .0; 3/, .4; 1/, and .6; 4/. SeeFig. 5(a).



SOLUTION Firsttranslatetheparallelogramtoonehavingtheoriginasavertex. For

example, subtractthevertex . 2; 2/ fromeachofthefourvertices. Thenewparallelogramhasthesamearea, anditsverticesare .0; 0/, .2; 5/, .6; 1/, and .8; 6/. See

x2



x2



x1



(a)



x1



(b)



FIGURE 5 Translatingaparallelogramdoesnotchangeits



area.



182



CHAPTER 3



Determinants



Fig. 5(b). Thisparallelogramisdeterminedbythecolumnsof

Ä

2 6

AD

5 1

Since jdet Aj D j 28j, theareaoftheparallelogramis28.



Linear Transformations

Determinantscanbeusedtodescribeanimportantgeometricpropertyoflineartransformationsintheplaneandin R3 . If T isalineartransformationand S isasetinthe

domainof T , let T .S/ denotethesetofimagesofpointsin S . Weareinterestedinhow

thearea(orvolume)of T .S / compareswiththearea(orvolume)oftheoriginalset S .

Forconvenience, when S isaregionboundedbyaparallelogram, wealsoreferto S as

aparallelogram.



THEOREM 10



Let T W R2 ! R2 bethelineartransformationdeterminedbya 2

S isaparallelogramin R2 , then



2 matrix A. If



fareaof T .S /g D jdet Aj fareaof Sg



(5)



fvolumeof T .S /g D jdet Aj fvolumeof Sg



(6)



If T isdeterminedbya 3



3 matrix A, andif S isaparallelepipedin R3 , then



PROOF Considerthe 2 2 case, with A D Œ a1 a2 . A parallelogramattheoriginin

R2 determinedbyvectors b1 and b2 hastheform

S D fs1 b1 C s2 b2 W 0 Ä s1 Ä 1; 0 Ä s2 Ä 1g

Theimageof S under T consistsofpointsoftheform



T .s1 b1 C s2 b2 / D s1 T .b1 / C s2 T .b2 /

D s1 Ab1 C s2 Ab2

where 0 Ä s1 Ä 1, 0 Ä s2 Ä 1. Itfollowsthat T .S / istheparallelogramdetermined

bythecolumnsofthematrix Œ Ab1 Ab2 . Thismatrixcanbewrittenas AB , where

B D Œ b1 b2 . ByTheorem9andtheproducttheoremfordeterminants,



fareaof T .S /g D jdet ABj D jdet Aj jdet Bj

D jdet Aj fareaof Sg



(7)



Anarbitraryparallelogramhastheform p C S , where p isavectorand S isaparallelogramattheorigin, asabove. Itiseasytoseethat T transforms p C S into T .p/ C T .S /.

(SeeExercise26.) Sincetranslationdoesnotaffecttheareaofaset,



fareaof T .p C S/g D fareaof T .p/ C T .S /g

D fareaof T .S /g

D j det Aj fareaof Sg

D j det Aj fareaof p C Sg



Translation

Byequation(7)

Translation



Thisshowsthat(5)holdsforallparallelogramsin R2 . Theproofof(6)forthe 3

caseisanalogous.



3



3.3



Cramer's Rule, Volume, and Linear Transformations



183



WhenweattempttogeneralizeTheorem10toaregionin R2 or R3 thatisnot

boundedbystraightlinesorplanes, wemustfacetheproblemofhowtodefineand

computeitsareaorvolume. Thisisaquestionstudiedincalculus, andweshallonly

outlinethebasicideafor R2 . If R isaplanarregionthathasafinitearea, then R can

beapproximatedbyagridofsmallsquaresthatlieinside R. Bymakingthesquares

sufficientlysmall, theareaof R maybeapproximatedascloselyasdesiredbythesum

oftheareasofthesmallsquares. SeeFig. 6.



0



0



FIGURE 6 Approximatingaplanarregionbyaunionofsquares.



Theapproximationimprovesasthegridbecomesfiner.



If T isalineartransformationassociatedwitha 2 2 matrix A, thentheimageof

aplanarregion R under T isapproximatedbytheimagesofthesmallsquaresinside R.

TheproofofTheorem 10showsthateachsuchimageisaparallelogramwhoseareais

jdet Aj timestheareaofthesquare. If R0 istheunionofthesquaresinside R, thenthe

areaof T .R0 / is jdet Aj timestheareaof R0 . SeeFig. 7. Also, theareaof T .R0 / isclose

totheareaof T .R/. Anargumentinvolvingalimitingprocessmaybegiventojustify

thefollowinggeneralizationofTheorem 10.



T



0



R'



0



T(R')



FIGURE 7 Approximating T .R/ byaunionofparallelograms.



TheconclusionsofTheorem10holdwhenever S isaregionin R2 withfinitearea

oraregionin R3 withfinitevolume.



EXAMPLE 5 Let a and b bepositivenumbers. Findtheareaoftheregion E

boundedbytheellipsewhoseequationis



x12

x22

C

D1

a2

b2



184



Determinants



CHAPTER 3



SOLUTION Weclaimthat E istheimageoftheunitdisk D underthelineartransforÄ

Ä

Ä

a

0

u1

x1

mation T determinedbythematrix A D

, becauseif u D

,xD

,

0 b

u2

x2

and x D Au, then

x1

x2

u1 D

and u2 D

a

b



u2



D



u1



1



Itfollowsthat u isintheunitdisk, with u21 C u22 Ä 1, ifandonlyif x isin E , with

.x1 =a/2 C .x2 =b/2 Ä 1. BythegeneralizationofTheorem 10,



T

x2

b



fareaofellipseg D fareaof T .D/g

D jdet Aj fareaof Dg



E

x1



a



.1/2 D ab



D ab



PRACTICE PROBLEM



Ä

Ä

1

5

Let S betheparallelogramdeterminedbythevectors b1 D

and b2 D

, and

3

1

Ä

1 :1

let A D

. Computetheareaoftheimageof S underthemapping x 7! Ax.

0 2



3.3 EXERCISES

UseCramer’sruletocomputethesolutionsofthesystemsin

Exercises1–6.

1. 5x1 C 7x2 D 3



2. 4x1 C x2 D 6



2x1 C 4x2 D 1



3.



3x1



2x2 D



5x1 C 6x2 D

5.



2x1 C x2

3x1



4.



5



5x1 C 3x2 D

3x1



D



7



x2 C 2x3 D



3



C x3 D



8



x2 D



5

4



3x1 C x2 C 3x3 D



2



2x3 D



2



InExercises7–10, determinethevaluesoftheparameter s for

whichthesystemhasauniquesolution, anddescribethesolution.

7. 6sx1 C 4x2 D



5



9x1 C 2sx2 D



9. sx1



2sx2 D



3x1 C 6sx2 D



2

1

4



8. 3sx1



5x2 D 3



9x1 C 5sx2 D 2

10. 2sx1 C



0

1

3



3

0

05

2



2



1

16. 4 0

0



2

3

0



3

4

15

3



18. Supposethatalltheentriesin A areintegersand det A D 1.

Explainwhyalltheentriesin A 1 areintegers.



9



6. 2x1 C x2 C x3 D



x1 C



3

15. 4 1

2



17. Showthatif A is 2 2, thenTheorem 8givesthesame

formulafor A 1 asthatgivenbyTheorem 4inSection 2.2.



5x1 C 2x2 D 7



7



2



x2 D 1



3sx1 C 6sx2 D 2



InExercises11–16, computetheadjugateofthegivenmatrix, and

thenuseTheorem8togivetheinverseofthematrix.

2

3

2

3

0

2

1

1

1

3

0

05

2

15

11. 4 3

12. 4 2

1

1

1

0

1

0

2

3

2

3

3

5

4

3

6

7

0

15

2

15

13. 4 1

14. 4 0

2

1

1

2

3

4



InExercises19–22, findtheareaoftheparallelogramwhose

verticesarelisted.

19. .0; 0/, .5; 2/, .6; 4/, .11; 6/

20. .0; 0/, . 1; 3/, .4; 5/, .3; 2/

21. . 1; 0/, .0; 5/, .1; 4/, .2; 1/

22. .0; 2/, .6; 1/, . 3; 1/, .3; 2/

23. Findthevolumeoftheparallelepipedwithonevertexat

theoriginandadjacentverticesat .1; 0; 2/, .1; 2; 4/, and

.7; 1; 0/.

24. Findthevolumeoftheparallelepipedwithonevertexat

theoriginandadjacentverticesat .1; 4; 0/, . 2; 5; 2/, and

. 1; 2; 1/.

25. Usetheconceptofvolumetoexplainwhythedeterminantof

a 3 3 matrix A iszeroifandonlyif A isnotinvertible. Do

notappealtoTheorem 4inSection 3.2. [Hint: Thinkabout

thecolumnsof A.]

26. Let T W Rm ! Rn bealineartransformation, andlet p bea

vectorand S asetin Rm . Showthattheimageof p C S under

T isthetranslatedset T .p/ C T .S/ in Rn .



Chapter 3 Supplementary Exercises



27. Let S be the parallelogram determined by the vectors

Ä

Ä

Ä

2

2

6

2

and b2 D

, andlet A D

.

b1 D

3

5

3

2

Compute the area of the image of S under the mapping

x 7! Ax.

Ä

Ä

4

0

28. Repeat Exercise 27 with b1 D

, b2 D

, and

7

1

Ä

7

2

AD

.

1

1

29. Findaformulafortheareaofthetrianglewhoseverticesare

0, v1 , and v2 in R2 .

30. Let R bethetrianglewithverticesat .x1 ; y1 /, .x2 ; y2 /, and

.x3 ; y3 /. Showthat

2

3

x1

y1

1

1

y2

15

fareaoftriangleg D det 4 x2

2

x3

y3

1

[Hint: Translate R totheoriginbysubtractingoneofthe

vertices, anduseExercise29.]

31. Let T W R3 ! R3 bethelineartransformationdetermined

3

2

a

0

0

b

0 5, where a, b , and c are

bythematrix A D 4 0

0

0

c

positivenumbers. Let S betheunitball, whosebounding

surfacehastheequation x12 C x22 C x32 D 1.

a. Showthat T .S/ isboundedbytheellipsoidwiththe

x2

x2

x2

equation 12 C 22 C 32 D 1.

a

b

c

b. Usethefactthatthevolumeoftheunitballis 4 =3

todeterminethevolumeoftheregionboundedbythe

ellipsoidinpart(a).



185



32. Let S bethetetrahedronin R3 withverticesatthevectors 0,

e1 , e2 , and e3 , andlet S 0 bethetetrahedronwithverticesat

vectors 0, v1 , v2 , and v3 . Seethefigure.

x3

e3



x3

S



v3



x2



S'



v2



x2



e2

0



0

e1



x1



v1

x1



a. Describealineartransformationthatmaps S onto S 0 .

b. Findaformulaforthevolumeofthetetrahedron S 0 using

thefactthat



fvolumeof Sg D .1=3/fareaofbaseg fheightg



33. [M] TesttheinverseformulaofTheorem 8forarandom

4 4 matrix A. Useyourmatrixprogramtocomputethe

cofactorsofthe 3 3 submatrices, constructtheadjugate,

andset B D .adj A/=.det A/. Thencompute B inv.A/,

where inv.A/ istheinverseof A ascomputedbythematrix

program. Usefloatingpointarithmeticwiththemaximum

possiblenumberofdecimalplaces. Reportyourresults.

34. [M] TestCramer’sruleforarandom 4 4 matrix A anda

random 4 1 vector b. Computeeachentryinthesolutionof

Ax D b, andcomparetheseentrieswiththeentriesin A 1 b.

Writethecommand(orkeystrokes)foryourmatrixprogram

thatusesCramer’sruletoproducethesecondentryof x.

35. [M] IfyourversionofMATLAB hasthe flops command,

useittocountthenumberoffloatingpointoperationstocompute A 1 forarandom 30 30 matrix. Comparethisnumber

withthenumberofflopsneededtoform .adj A/=.det A/.



SOLUTION TO PRACTICE PROBLEM

ˇ

ˇ

Ä

ˇ

1 5 ˇˇ

Theareaof S is ˇˇ det

D 14; and det A D 2. ByTheorem 10, theareaofthe

3 1 ˇ

imageof S underthemapping x 7! Ax is

jdet Aj fareaof Sg D 2 14 D 28



CHAPTER 3 SUPPLEMENTARY EXERCISES

1. MarkeachstatementTrueorFalse. Justifyeachanswer.

Assumethatallmatricesherearesquare.

a. If A isa 2 2 matrixwithazerodeterminant, thenone

columnof A isamultipleoftheother.

b. If two rows of a 3

det A D 0.

c. If A isa 3



3 matrix A are the same, then



3 matrix, then det 5A D 5 det A.



d. If A and B are n n matrices, with det A D 2 and

det B D 3, then det.A C B/ D 5.

e. If A is n



n and det A D 2, then det A3 D 6.



f. If B isproducedbyinterchangingtworowsof A, then

det B D det A.

g. If B isproducedbymultiplyingrow3of A by5, then

det B D 5 det A.



186



Determinants



CHAPTER 3



h. If B is formed by adding to one row of A a linear

combinationoftheotherrows, then det B D det A.

i. det AT D



det A.



j. det. A/ D

k. det ATA



0.



l. Anysystemof n linearequationsin n variablescanbe

solvedbyCramer’srule.

m. If u and v arein R2 and det Œ u v  D 10, thenthearea

ofthetriangleintheplanewithverticesat 0, u, and v is

10.

o. If A isinvertible, then det A



1



D det A.



p. If A isinvertible, then .det A/.det A



1



/ D 1.



UserowoperationstoshowthatthedeterminantsinExercises2–4

areallzero.

ˇ

ˇ

ˇ

ˇ

ˇ 12

ˇ1

13

14 ˇˇ

a

b C c ˇˇ

ˇ

ˇ

b

a C c ˇˇ

16

17 ˇˇ

3. ˇˇ 1

2. ˇˇ 15

ˇ 18

ˇ1

c

aCbˇ

19

20 ˇ

ˇ

ˇ a

ˇ

4. ˇˇ a C x

ˇaCy



b

bCx

bCy



ˇ

c ˇˇ

c C x ˇˇ

cCyˇ



ComputethedeterminantsinExercises5and6.

ˇ

ˇ

ˇ9

1

9

9

9 ˇˇ

ˇ

ˇ9

0

9

9

2 ˇˇ

ˇ

0

0

5

0 ˇˇ

5. ˇˇ 4

ˇ9

0

3

9

0 ˇˇ

ˇ

ˇ6

0

0

7



ˇ

ˇ4

ˇ

ˇ0

ˇ

6. ˇˇ 6

ˇ0

ˇ

ˇ0



8

1

8

8

8



8

0

8

8

2



8

0

8

3

0



det T D .b



a/.c



a/.c



b/



10. Let f .t/ D det V , with x1 , x2 , x3 alldistinct. Explainwhy

f .t/ isacubicpolynomial, showthatthecoefficientof t 3 is

nonzero, andfindthreepointsonthegraphof f .



det A.



n. If A3 D 0, then det A D 0.



9. Userowoperationstoshowthat



ˇ

5 ˇˇ

0 ˇˇ

7 ˇˇ

0 ˇˇ





7. Showthattheequationofthelinein R2 throughdistinct

points .x1 ; y1 / and .x2 ; y2 / canbewrittenas

2

3

1

x

y

x1

y1 5 D 0

det 4 1

1

x2

y2

8. Finda 3 3 determinantequationsimilartothatinExercise 7

thatdescribestheequationofthelinethrough .x1 ; y1 / with

slope m.

Exercises9and10concerndeterminantsofthefollowing Vandermondematrices.

2

3

2

3

1

t

t2

t3

2

1

a

a

6

7

61

x1

x12

x13 7

6

7

7

T D 41

b

b 2 5; V .t/ D 6

61

2

37

x

x

x

2

4

5

2

2

1

c

c2

1

x3

x32

x33



11. Determinetheareaoftheparallelogramdeterminedbythe

points .1; 4/, . 1; 5/, .3; 9/, and .5; 8/. Howcanyoutell

thatthequadrilateraldeterminedbythepointsisactuallya

parallelogram?

12. Usetheconceptofareaofaparallelogramtowriteastatementabouta 2 2 matrix A thatistrueifandonlyif A is

invertible.

13. Showthatif A isinvertible, then adj A isinvertible, and

1

.adj A/ 1 D

A

det A

[Hint: Givenmatrices B and C , whatcalculation(s)would

showthat C istheinverseof B‹

14. Let A, B , C , D , and I be n n matrices. Usethedefinitionorpropertiesofadeterminanttojustifythefollowing

formulas. Part (c)isusefulinapplicationsofeigenvalues

(Chapter 5).

Ä

Ä

A

0

I

0

a. det

D det A

b. det

D det D

0

I

C

D

Ä

Ä

A

0

A

B

c. det

D .det A/.det D/ D det

C

D

0

D

15. Let A, B , C , and D be n n matriceswith A invertible.

a. Findmatrices X and Y toproducetheblockLU factorization

Ä

Ä

Ä

A

B

I

0

A

B

D

C

D

X

I

0

Y

andthenshowthat

Ä

A

B

det

D .det A/ det.D

C

D



CA



1



B/



b. Showthatif AC D CA, then

Ä

A

B

det

D det.AD CB/

C

D

16. Let J be the n n matrix of all 1’s, and consider

A D .a b/I C bJ ; thatis,

2

3

a

b

b

b

6b

a

b

b7

6

7

6b

b

a

b7

AD6:

7

:

:

:

::

6:

::

::

:: 7

:

4:

5



b



b



b



a



Confirmthat det A D .a b/ Œa C .n 1/b asfollows:

a. Subtractrow2fromrow1, row3fromrow2, andsoon,

andexplainwhythisdoesnotchangethedeterminantof

thematrix.

n 1



Chapter 3 Supplementary Exercises



b. Withtheresultingmatrixfrompart(a), addcolumn1to

column2, thenaddthisnewcolumn2tocolumn3, andso

on, andexplainwhythisdoesnotchangethedeterminant.

c. Findthedeterminantoftheresultingmatrixfrom(b).

17. Let A betheoriginalmatrixgiveninExercise16, andlet

2

3

a b

b

b

b

6 0

a

b

b7

7

6

6 0

b

a

b7

,

BD6 :

::

::

:: 7

::

7

6 :

:

4 :

:

:

:5



0



2



b

6b

6

6

C D 6 b:

6:

4:

b



b



b



b

a

b

::

:



b

b

a

::

:



b



b



3



::



:



a



b

b7

7

b7

:: 7

7

:5



a



Noticethat A, B , and C arenearlythesameexceptthatthe

firstcolumnof A equalsthesumofthefirstcolumnsof B

and C . A linearityproperty ofthedeterminantfunction,

discussedinSection3.2, saysthat det A D det B C det C .

UsethisfacttoprovetheformulainExercise16byinduction

onthesizeofmatrix A.

18. [M] ApplytheresultofExercise16tofindthedeterminants

ofthefollowingmatrices, andconfirmyouranswersusinga

matrixprogram.

3

2

3

2

8

3

3

3

3

3

8

8

8

63

8

3

3

37

7

6

68

3

8

87

7

63

6

3

8

3

37

7

6

5

48

8

3

8

43

3

3

8

35

8

8

8

3

3

3

3

3

8



187



19. [M] Useamatrixprogramtocomputethedeterminantsof

thefollowingmatrices.

2

3

2

3

1

1

1

1

1

1

1

61

2

2

27

6

7

41

2

25

41

2

3

35

1

2

3

1

2

3

4

2

3

1

1

1

1

1

61

2

2

2

27

6

7

61

2

3

3

37

6

7

41

2

3

4

45

1

2

3

4

5

Usetheresultstoguessthedeterminantofthematrixbelow,

andconfirmyourguessbyusingrowoperationstoevaluate

thatdeterminant.

2

3

1

1

1

1

61

2

2

27

6

7

61

2

3

37

6:

7

:

:

:

::

6:

::

::

:: 7

:

4:

5



1



2



3



n



20. [M] UsethemethodofExercise19toguessthedeterminant

of

2

3

1

1

1

1

61

7

3

3

3

6

7

61

7

3

6

6

6:

7

::

::

::

::

6:

7

:

4:

5

:

:

:

1

3

6

3.n 1/

Justifyyourconjecture. [Hint: UseExercise14(c)andthe

resultofExercise19.]



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