2/28/2015 InverseofaMatrix

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InverseofaMatrix

Pleasereadour IntroductiontoMatrices first.

What is the Inverse of a Matrix?TheInverseofaMatrixisthesameideaasthe reciprocal ofanumber:

ReciprocalofaNumber

Butwedon'twrite (becausewedon'tdividebyaMatrix!),insteadwewriteA1fortheinverse:

(Infact canalsobewrittenas81)

Andthereareothersimilarities:

Whenyoumultiplyanumberbyitsreciprocalyouget1

8( )=1

WhenyoumultiplyaMatrixbyitsInverseyougettheIdentityMatrix(whichislike"1"forMatrices):

AA1=I

/1 A

/1 8

/1 8

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Italsoworkswhentheinversecomesfirst:( )8=1andA1A=I

IdentityMatrix

Note:the"IdentityMatrix"isthematrixequivalentofthenumber"1":

A3x3IdentityMatrixItis"square"(hassamenumberofrowsascolumns),

Ithas1sonthediagonaland0severywhereelse.

It'ssymbolisthecapitalletterI.

TheIdentityMatrixcanbe22insize,or33,44,etc...

DefinitionSowehaveadefinitionofaMatrixInverse...

TheInverseofAisA1onlywhen:

AA1=A1A=I

SometimesthereisnoInverseatall.

2x2 MatrixOK,howdowecalculatetheInverse?

Well,fora2x2MatrixtheInverseis:

/1 8

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Inotherwords:swapthepositionsofaandd,putnegativesinfrontofbandc,anddivideeverythingbythe determinant (adbc).

Letustryanexample:

Howdoweknowthisistherightanswer?

Rememberitmustbetruethat:AA1=I

So,letuschecktoseewhathappenswhenwe multiplythematrix byitsinverse:

And,hey!,weendupwiththeIdentityMatrix!Soitmustberight.

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Itshouldalsobetruethat:A1A=I

Whydon'tyouhaveagoatmultiplyingthese?SeeifyoualsogettheIdentityMatrix:

Why Would We Want an Inverse?BecausewithMatriceswedon'tdivide!Seriously,thereisnoconceptofdividingbyaMatrix.

ButwecanmultiplybyanInverse,whichachievesthesamething.

Imagineyoucouldn'tdividebynumbers,andsomeoneasked"HowdoIshare10appleswith2people?"

Butyoucouldtakethereciprocalof2(whichis0.5),soyoucouldanswer:

100.5=5

Theyget5appleseach

ThesamethingcanbedonewithMatrices:

SaythatyouknowMatrixAandB,andwanttofindMatrixX:

XA=B

ItwouldbenicetodividebothsidesbyA(togetX=B/A),butrememberwecan'tdivide.

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ButwhatifwemultiplybothsidesbyA1?

XAA1=BA1

AndweknowthatAA1=I,so:

XI=BA1

WecanremoveI(forthesamereasonwecouldremove"1"from1x=abfornumbers):

X=BA1

Andwehaveouranswer(assumingwecancalculateA1)

Inthatexamplewewereverycarefultogetthemultiplicationscorrect,becausewithMatricestheorderofmultiplicationmatters.ABisalmostneverequaltoBA.

A Real Life ExampleAgrouptookatriponabus,at$3perchildand$3.20peradultforatotalof$118.40.

Theytookthetrainbackat$3.50perchildand$3.60peradultforatotalof$135.20.

Howmanychildren,andhowmanyadults?

First,letussetupthematrices(becarefultogettherowsandcolumnscorrect!):

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Thisisjustliketheexampleabove:

XA=B

Sotosolveitweneedtheinverseof"A":

Nowwehavetheinversewecansolveusing:

X=BA1

Therewere16childrenand22adults!

Theansweralmostappearslikemagic.Butitisbasedongoodmathematics.

Calculationslikethat(butusingmuchlargermatrices)helpEngineersdesignbuildings,areusedinvideogamesandcomputeranimationstomakethingslook3dimensional,andmanyotherplaces.

Itisalsoawaytosolve SystemsofLinearEquations .

Thecalculationsaredonebycomputer,butthepeoplemustunderstandtheformulas.

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Order is Important

Saythatyouaretryingtofind"X"inthiscase:

AX=B

Thisisdifferenttotheexampleabove!XisnowafterA.

WithMatricestheorderofmultiplicationusuallychangestheanswer.DonotassumethatAB=BA,itisalmostnevertrue.

Sohowdowesolvethisone?Usingthesamemethod,butputA1infront:

A1AX=A1B

AndweknowthatA1A=I,so:

IX=A1B

WecanremoveI:

X=A1B

Andwehaveouranswer(assumingwecancalculateA1)

Whydon'twetryourexamplefromabove,butwiththedatasetupthiswayaround.(Yes,youcandothis,justbecarefulhowyousetitup.)

ThisiswhatitlookslikeasAX=B:

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Itlookssoneat!IthinkIpreferitlikethis.

Alsonotehowtherowsandcolumnsareswappedover("Transposed")comparedtothepreviousexample.

Tosolveitweneedtheinverseof"A":

ItisliketheInversewegotbefore,butTransposed(rowsandcolumnsswappedover).

Nowwecansolveusing:

X=A1B

Sameanswer:16childrenand22adults.

So,Matricesarepowerfulthings,buttheydoneedtobesetupcorrectly!

The Inverse May Not ExistFirstofall,tohaveanInversetheMatrixmustbe"Square"(samenumberofrowsandcolumns).

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Butalsothedeterminantcannotbezero(oryouwouldendupdividingbyzero).Howaboutthis:

2424?Thatequals0,and1/0isundefined.Wecannotgoanyfurther!ThisMatrixhasnoInverse.

SuchaMatrixiscalled"Singular",whichonlyhappenswhenthedeterminantiszero.

Anditmakessense...lookatthenumbers:thesecondrowisjustdoublethefirstrow,anddoesnotaddanynewinformation.

Imagineinourexampleabovethatthepricesonthetrainwereexactly,say,50%higher...wewouldn'tbeanyclosertofiguringouthowmanyadultsandchildren...weneedsomethingdifferent.

Andthedeterminantneatlyworksthisout.

Bigger MatricesTheinverseofa2x2iseasy...comparedtolargermatrices(suchasa3x3,4x4,etc).

Forthoselargermatricestherearethreemainmethodstoworkouttheinverse:

InverseofaMatrixusingElementaryRowOperations(GaussJordan)

InverseofaMatrixusingMinors,CofactorsandAdjugate

Useacomputer(suchastheMatrixCalculator)

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ConclusionTheInverseofAisA1onlywhenAA1=A1A=I

TofindtheInverseofa2x2Matrix:swapthepositionsofaandd,putnegativesinfrontofbandc,anddivideeverythingbythedeterminant(adbc).

SometimesthereisnoInverseatall

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