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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

;

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

; 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

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

THEOREM 8

An Inverse Formula

Let A beaninvertible n

n matrix. Then

1

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

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

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

ˇ

0 d ˇ

rectangle

A Geometric Proof

3–12

y

⎡0 ⎡

⎢d ⎢

⎣ ⎣

⎡ a⎡

⎢ ⎢

⎣ 0⎣

FIGURE 1

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

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

on AT .) Soitsufﬁcestoshowthattheseoperationsdonotaffectthevolumeofthe

parallelepipeddeterminedbythecolumnsof A.

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

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

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

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

.

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

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

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

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

parallelogram?

12. Usetheconceptofareaofaparallelogramtowriteastatementabouta 2 2 matrix A thatistrueifandonlyif A is

invertible.

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

1

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

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

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

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

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.]