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Recall...DenitionA square matrixAisinvertible(ornonsingular) if9matrix

Bsuch thatAB=IandBA=I. (We sayBis aninverseofA.)

RemarkNot all square matrices are invertible.

Theorem.IfAis invertible, then its inverse is unique.

RemarkWhenAis invertible, we denote its inverse asA1.Theorem.If A is annninvertible matrix, then the system of

linear equations given byA~x=~bhas the unique solution~x=A1~b. Proof.AssumeAis an invertible matrix. Then we haveMatrixinverses

Recall...

DefinitionAsq uarematrixAisinvertible(ornonsingular)if?matrixB suchthatAB=IandBA=I.(We sayBisan inverseofA.)

RemarkNotallsq uarematr icesareinvertible.

Theorem.IfAisinvertible, thenitsinverseis unique.

RemarkWhenAisinv ertible,wedenoteitsinversea sA

-1 Theorem.IfAisann×ninvertiblematrix,then thesystemoflinear equationsgivenbyA?x= bhastheunique solution?x=A -1 b.

Proof.AssumeAisan inve rtiblematrix.Thenwehave

A(A -1 b)=(AA -1 b=I b= b.

Theorem(Propertiesofmatrixin verse).

(a)IfAisinvertible, thenA -1 isitselfinver tibleand (A -1 -1 =A. (b)IfAisinvertible andc?=0isasc alar,then cAisinvertible and (cA) -1 1 c A -1 (c)IfAandBarebothn×ninvertiblematrices, thenABisinvertible and(AB) -1 =B -1 A -1 "socksandshoesr ule"-similar totransposeofAB generalizationtopro ductofnmatrices (d)IfAisinvertible, thenA T isinvertible and(A T -1 =(A -1 T Topr ove(d),weneedto showthatthe matrix Bthatsatisfies BA T =IandA T

B=IisB=(A

-1 T

Lecture8Math40,Spring"1 2,P rof.KindredPag e1

by associativity of matrix mult. by def'n of inverse by def'n of identityThus,~x=A1~bis a solution toA~x=~b. Suppose~yis another solution to the linear system. It follows thatA~y=~b, but multiplying both sides byA1gives~y=A1~b=~x.

Theorem(Properties of matrix inverse).

(a) If Ais invertible, thenA1is itself invertible and(A1)1=A. (b) If Ais invertible andc6= 0is a scalar, thencAis invertible and (cA)1=1c A1. (c) If AandBare bothnninvertible matrices, thenABis invertible and(AB)1=B1A1.Lecture 8 Math 40, Spring '12, Prof. Kindred Page 1 \socks and shoes rule" { similar to transpose ofAB generalization to product ofnmatrices (d) If Ais invertible, thenATis invertible and(AT)1= (A1)T. To prove (d), we need to show that the matrixBthat satises BA

T=IandATB=IisB= (A1)T.

Proof of (d).AssumeAis invertible. ThenA1exists and we have (A1)TAT= (AA1)T=IT=I and A

T(A1)T= (A1A)T=IT=I:

SoATis invertible and (AT)1= (A1)T.

Recall...How do we compute the inverse of a matrix, if it exists? Inverse of a22matrix:Consider the special case whereAis a

22 matrix withA= [a bc d]. Ifadbc6= 0, thenAis invertible and its

inverse is A

1=1adbc

db c a How do we nd inverses of matrices that are larger than 22 matrices? Theorem.If some EROs reduce a square matrixAto the identity matrix

I, then the same EROs transformItoA1.?

AI I A -1 EROsIf we can transformAintoI, then we will obtainA1. If we cannot do so, thenAis not invertible.Lecture 8 Math 40, Spring '12, Prof. Kindred Page 2 Can we capture the eect of an ERO through matrix multiplication? DenitionAnelementary matrixis any matrix obtained by doing an

ERO on the identity matrix.

Examples

1$R2on 44 identity2

6640 1 0 0

1 0 0 0

0 0 1 0

0 0 0 13

7 775R

14R3on 33 identity2

41 04
0 1 0

0 0 13

Notice that

2 41 04
0 1 0

0 0 13

52
4a

11a12a13

21a22a23

31a32a333

5 =2 4a

114a31a124a32a134a33

21a22a23

31a32a333

5 Left mult. ofAby row vector is a linear comb. of rows ofA. RemarkAn elementary matrixEis invertible andE1is elementary matrix corresponding to the \reverse" ERO of one associated withE. ExampleIfEis 2nd elementary matrix above, then \reverse" ERO is R

1+ 4R3andE1=2

41 0 4

0 1 0

0 0 13

5 RemarkWhen ndingA1using Gauss-Jordan elimination of [AjI], if we keep track of EROs, and ifE1;E2;:::;Ekare corresponding elem. matrices, then we have E kEk1E1A=I=)A=E11E1 k1E1 k:Lecture 8 Math 40, Spring '12, Prof. Kindred Page 3

Theorem(Fundamental Thm of Invertible Matrices).

For annnmatrix, the following are equivalent:

(1)Ais invertible. (2)A~x=~bhas a unique solution for any~b2Rn. (3)A~x=~0has only the trivial solution~x= 0. (4)

The RREF of AisI.

(5)Ais product of elementary matrices.12345 Proof strategyProof. (1))(2):

Proven in rst theorem

of today's lecture(2))(3): IfA~x=~bhas unique sol'n for any~b2Rn, then in particular,A~x=~0 has a unique sol'n. Since~x=~0 is a solution toA~x=~0, it must be the unique one. (3))(4):

IfA~x=~0 has unique sol'n~x= 0,

then augmented matrix has no free variables and a leading one in every column:2 6 66410
10 10 3 7 775
so RREF ofAisI.(4))(5): E kE1A= RREF ofA=I and elem. matrices are invertible =)A=E11E1 k1E1 k: (5))(1):

SinceA=EkE1andEiinvertible

8i,Ais product of invertible matri-

ces so it is itself invertible. Lecture 8 Math 40, Spring '12, Prof. Kindred Page 4 Theorem.LetAbe a square matrix. IfBis a square matrix such that eitherAB=IorBA=I, thenAis invertible andB=A1. Proof.SupposeA;Barennmatrices and thatBA=I. Then consider the homogeneous systemA~x=~0. We have

B(A~x) =B~0 =)(BA)|{z}

I~x=~0 =)~x=~0:

SinceA~x=~0 has only the trivial solution~x=~0, by the Fundamental Thm of Inverses, we have thatAis invertible, i.e.,A1exists. Thus, (BA)A1=IA1=)B(AA1)|{z}

I=A1=)B=A1:

We leave the case ofAB=Ias an exercise.DenitionThe vectors~e1;~e2;:::;~en2Rn, where~eihas a one in its

ith component and zeros elsewhere, are calledstandard unit vectors.

ExampleThe 44 identity matrix can be expressed as

I 4=2 6

6641 0 0 0

0 1 0 0

0 0 1 0

0 0 0 13

7 775=2

4j j j j

1~e2~e3~e4

j j j j3 5 Theorem.If some EROs reduce a square matrixAto the identity matrixI, then the same EROs transformItoA1.

Why does this work?

Want to solveAX=I, withXunknownnnmatrix.

If~x1;:::;~xnare columns ofA, then want to solvenlinear systems A~x

1=~e1;:::;A~xn=~en. Can do so simultaneously using one

\super-augmented matrix."Lecture 8 Math 40, Spring '12, Prof. Kindred Page 5quotesdbs_dbs15.pdfusesText_21

[PDF] Matrix inverses - Harvey Mudd College

Matrix inverses

Bsuch thatAB=IandBA=I. (We sayBis aninverseofA.)

RemarkNot all square matrices are invertible.

Recall...

RemarkNotallsq uarematr icesareinvertible.

Theorem.IfAisinvertible, thenitsinverseis unique.

RemarkWhenAisinv ertible,wedenoteitsinversea sA

Proof.AssumeAisan inve rtiblematrix.Thenwehave

Theorem(Propertiesofmatrixin verse).

B=IisB=(A

Lecture8Math40,Spring"1 2,P rof.KindredPag e1

Theorem(Properties of matrix inverse).

T=IandATB=IisB= (A1)T.

T(A1)T= (A1A)T=IT=I:

SoATis invertible and (AT)1= (A1)T.

22 matrix withA= [a bc d]. Ifadbc6= 0, thenAis invertible and its

1=1adbc

I, then the same EROs transformItoA1.?

ERO on the identity matrix.

Examples

1$R2on 44 identity2

6640 1 0 0

1 0 0 0

0 0 1 0

0 0 0 13

14R3on 33 identity2

0 0 13

Notice that

0 0 13

11a12a13

21a22a23

31a32a333

114a31a124a32a134a33

21a22a23

31a32a333

1+ 4R3andE1=2

41 0 4

0 0 13

Theorem(Fundamental Thm of Invertible Matrices).

For annnmatrix, the following are equivalent:

The RREF of AisI.

Proven in rst theorem

IfA~x=~0 has unique sol'n~x= 0,

SinceA=EkE1andEiinvertible

8i,Ais product of invertible matri-

B(A~x) =B~0 =)(BA)|{z}

I~x=~0 =)~x=~0:

I=A1=)B=A1:

ExampleThe 44 identity matrix can be expressed as

6641 0 0 0

0 1 0 0

0 0 1 0

0 0 0 13

4j j j j

1~e2~e3~e4

Why does this work?

Want to solveAX=I, withXunknownnnmatrix.

1=~e1;:::;A~xn=~en. Can do so simultaneously using one