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Page1sur9

MATRIXDETERMINANTS

Summary

Uses

1Ǧ ReminderǦDefinitionandcomponentsofamatrix

Amatrixisarectangulartableofform

Page2of9

Theelementsarematrixentriesܽ

,distinctfromܽ

2Ǧ Thematrixdeterminant

Avaluecalledthedeterminantofܣ

correspondstoeverysquarematrixܣ insteadonitscalculation.

3Ǧ Calculationofthedeterminantfora૛ൈ૛matrix

Letusconsiderthematrixܣ

Thedeterminantofthematrixܣ

Page3of9

Example

Giventhematrix

ThedeterminantofAis

4Ǧ Exercise

Solutions:a)Ͳ17b)0c)5d)11

agreaterdimension),youwillfirstneed tolearnafewconcepts...

5Ǧ Definitionofaminor

Theminorܯ

andthesecondcolumnofܣ

Theminorܯ

andthesecondcolumnofܣ

Page4of9

6Ǧ Definitionofacofactor

Thecofactor,ܥ

,ofamatrixܣ withthepossibleexceptionoftheirsign.

Letusagainconsiderthematrix

Wehavealreadyshownthattheminorܯ

,is

Theminorܯ

andthecofactorܥ areofdifferentsigns.

Theminorܯ

is

Thistime,theminor

andthecofactor areidentical.

Thisprocessiscalledancofactorexpansion.

7Ǧ Cofactorexpansion-amethodtocalculatethedeterminant

.Thedeterminantisobtainedbycofactor expansionasfollows:

Choosearoworacolumnofܣ

containingthemostzeros)...

Multiplyeachoftheelementsܽ

oftherow(orcolumn)chosenbyits correspondingcofactor,ܥ

Addtheseresults.

Page5of9

8Ǧ Calculatethedeterminantfora૜ൈ૜matrix

Fora͵ ൈ ͵matrix,thiswouldmeanthatbychoosingtomakeanexpansionalongthe firstrow,thedeterminantwouldbe calculate

Example

Whatisthedeterminantofmatrixܣ

Solution

Letusfollowthe

Choosearoworacolumnofܣ

elementsofthefirstrowareܽ ൌ͵thatwemultiplywiththe correspondingcofactors,i.e.

Finally,weneedtocalculate

Page6of9

andܥ .The correspondingcofactorforܽ is

Thedeterminantofܣ

9Ǧ Alternativemethodtocalculatedeterminants

avoidtheuseofcofactors. signtothepositionܽ vertically.

Choosearoworcolumnofܣ

Multiplyeachelementofܽ

whichܽ is. thefirststep.

Page7of9

Example

Giventhematrixܣ

rowhasthemostzeros,but...)

Finally,therespective

10Ǧ Exercise

Solutions:a)24b)Ͳ12c)Ͳ66d)0

Page8of9

11Ǧ Determinantsofsquarematricesofdimensions4x4and

greater ,thedeterminantisobtainedbyfollowinga cofactorexpansionasfollows:

Chosearoworcolumnofܣ

containsthemostzeros)...

Multiplyeachoftheelementsܽ

oftherow(orcolumn)chosen,bythe correspondingcofactorܥ

Addtheresults.

ofa i th rowandthej th columnofܣ

Example

CalculatethedeterminantofmatrixA

expansionalongthefourthcolumn,which meansthat

Asܽ

andܽ arezero,itisuselesstofindܥ andܥ .Thecofactorsܥ andܥ will benecessary...

Page9of9

Weletthereaderverifythatܥ

ൌͳͺetܥ

Exercise

along a) Thefirstrow b) Thethirdcolumnquotesdbs_dbs20.pdfusesText_26