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Lecture6.InverseofMatrix

A~x= b:

Inonedimensioncase,i.e.,Ais1£1;then

Ax=b canbeeasilysolvedas x= b A 1 A b=A ¡1 bprovidedthatA6=0:

Denition7.1.AsquarematrixA

n£n issaidtobeinvertibleifthereexistsaunique matrixC n£n ofthesamesizesuchthat

AC=CA=I

n C=A ¡1

SupposenowA

n£n isinvertibleandC=A ¡1 isitsinversematrix.Thenthematrix equation A~x= b C=A ¡1 ;wehave

CA~x=C

b:

Bydenition,CA=A

¡1 A=I n :Itleadsto I n ~x=C b; whichisthesameas ~x=A ¡1 b:(1) discussionissummarizedas A~x= b hasauniquesolution ~x=A ¡1 b: 1

Example7.1(a)ShowAisinvertibleandA

¡1 =C;where A= 25

¡3¡7

;C=

¡7¡5

32
(b)Solve 2x 1 +5x 2 =1

¡3x

1

¡7x

2 =4: (c)Showthatthematrix B= 02 00 isNOTinvertible.

Solution:(a)Directcalculationsleadto

AC= 25

¡3¡7

¡7¡5

32
10 01 =I 2 CA=

¡7¡5

32
25

¡3¡7

10 01 =I 2

Bydenition,

C=A ¡1 A~x= 1 4 ~x=A ¡1 1 4

¡7¡5

32
1 4

¡27

11 (c)Bycalculation,wendthat B 2 02 00 02 00 00 00 BD=I 2 2

B(BC)=BI

2 B 2 C=B; whichimplies 0=B; sinceB 2

Theorem7.2A2£2matrix

A= ab cd det(A) def =ad¡bc6=0:

Whenad¡bc6=0;theinverseis

A ¡1 1 ad¡bc d¡b

¡ca

Example7.2.(a)FindA

¡1 if A= 25

¡3¡6

(b)Solve

2x+5y=1

¡3x¡6y=2

A ¡1 1 ad¡bc d¡b

¡ca

1 3

¡6¡5

32

¡2¡5=3

12=3

Wemayverifytheabovesolutionasfollows:

25

¡3¡6

¡2¡5=3

12=3 10 01 b;where A= 25

¡3¡6

b= 1 2

Thesolutionis

~x=A ¡1 b=

¡2¡5=3

12=3 1 2 16 3 7 3 3

²PropertiesifInvertibleMatrix:

1.A ¡1 isalsoinvertibleand(A ¡1 ¡1 =A: 2.A T isalsoinvertibleand A T ¡1 =(A ¡1 T (AB) ¡1 =B ¡1 A ¡1

Proof.(1)Bydenition,A

¡1 ifwecanndamatrixCsuchthat A ¡1 C=C A ¡1 =I:

TheaboveisindeedtrueifC=A:

(2)Taketransposesofallthreesidesof(A ¡1 )A=A(A ¡1 )=I; A ¡1 A T A A ¡1 T =I T =)A T A ¡1 T A ¡1 T A T =I =)A T C=CA T =I;whereC= A ¡1 T istheinverseofA T (3)LetC=B ¡1 A ¡1 :Since

C(AB)=(CA)B=

B ¡1 A ¡1 A B= B ¡1 A ¡1 A B= B ¡1 I B= B ¡1 B=I (AB)C=A(BC)=A B B ¡1 A ¡1 =A BB ¡1 A ¡1 =A IA ¡1 =AA ¡1 =I: (AB) ¡1 =C=B ¡1 A ¡1 (4)SinceA~x=

0hastheonlysolution

~x=A ¡1 0= 0; thushastobetheidentitymatrix. 4

Atype(1)elementarymatrixE

1 instance, 2 4 100
010 001 3 5 R 2 +¸R 1 !Rquotesdbs_dbs20.pdfusesText_26