Obviously, if the ranks of two matrices are equal, they have the same normal forms, and so they are equivalent; conversely, if two matrices are equivalent, their ranks are equal too. That is ：Theory: matrices A and B are equivalent if and only if r(A)=r(B).

!!! Please remember:we need to figure out if the ranks of matrices are equal only and we will know if they are equivalent.

Non-degenerate MatrixDefinition: if the rank of square matrix A is equal to its order, we call A a non-degenerate matrix. Otherwise, degenerate matrix.( non-degeneratenon-singular;

degeneratesingular ）E----non-degenerate matrix O----degenerate matrix

Theory: A is a non-degenerate matrix, then the normal form of A is an identity matrix E with the same size

EA

The rank of matrix is an important numerical character of matrix.

Corollary 1: the following propositions are equivalent:

;degenerate-non is)( Ai ;)( EAii singular;-non is )( Aiii

)matrix. elemantaryan is (;)( 21 im PPPPAiv

)()()(Theory

iiiiii )singular.-non is isThat

,0,)((

A

AnAr

:)()( ivii ,EAsuch that ,,,,,, 121 mll PPPPP ，

mll PEPPPPA 121 mll PPPPP 121 :)()( iiiv mPPPA 21

EPPP m21 EA

matrix elementary

)()()()( iviiiiii

Prove:

Corollary 2:Matrices A and B are equivalent if and onlyif there are m-order and n-order non-Degenerate matrices P,Q, such that nnmmnm QBPA And we also have ：If P,Q are non-degenerate, then

r(A) = r(PA) = r(PAQ) = r(AQ)

e.g.).( then ,

301

020

201

,2)(Let 34 ABrBAr

,3)( Br ,degenerate-non is B 2)()( ArABr

The Inverse of a Matrix

.

.1such that ,,0 111 aaaaaa

EBAABBA such that,matrix?,matrix

Definition ： if A is an n-order square matrix, and there is anothern-order square matrix B such that AB=BA=E, we say that B is an inverse of A, and A is invertible.

（ 1 ） The inverse of matrix is unique.Let B,C are all inverses of A, then B=EB=(CA)B=C(AB)=CE=C

1ADenote the inverse of A as

（ 2 ） Not all square matrices are invertible.

For example

00

01A is not invertible. ,1

dc

baA

0000

01 ba

dc

ba

10

0110

It’s impossible. So A is not invertible.

The questions to answer:

1. When the matrix is invertible?

2. How to find the inverse?

Review ： adjoint matrix nnijaA

nnnn

n

n

AAA

AAA

AAA

A

21

22212

12111ij

ij

a

A

ofcominor

algebraic theis

Adjoint matrix

to?attention paid be should

what , use When we A The order of algebraic cominor!

The adjoint matrix of 2-order matrix A .

dc

baA

ac

bdA

AA

nnnn

n

n

nnnn

n

n

AAA

AAA

AAA

aaa

aaa

aaa

21

22212

12111

21

22221

11211

A

A

EA AA

EAAAAA

It’s an important formula.

Formula :

Theory: An n-order square matrix A is invertible if and only if A

AA

11.0A

Prove:,invertible isA For 1 EAA ”“

sideeach oft determinan thefindcan We

111 EAAAA 0 A

,0For A”“ EAAAAA

EAAA

AA

A )1

()1

(

AA

A11 Keep in mind!

.degenerate-non is

singular-non is invertible is

A

AA

e.g.1..of inverse theFind

dc

baA

AA

A11

ac

bd

bcad

1Solution ：

)0( bcad

e.g.2.);,(),(1 jiEjiE ));

1(())((1

kiEkiE

))(,())(,(1 kjiEkjiE

Prove: EjiEjiE ),(),(

),(),(1 jiEjiE By the same method,we can prove others

That is, the inverses of elementary matrices are elementary matrices of the same size.

——This is the 3rd property of elementary matrices 。Exercises: Find the inverse.

12

11.1 A

21

11.2 B

10

22.3C

12

11

3

1.1 1A

11

12.2 1B

20

21

2

1.3 1C

102

123

111

A

？？ ？ How to find the inverse of

Properties of the Inverse

;1

invertible is )( 1

AAAi

;)(,invertible isinvertible is )( 111 AAAAii

;)()( 1 ABEBAorEABiii

;)())(( 11 TT AAiv

;))(( 111 ABABv

).invertible is ,0(,1

))(( 11 AkAk

kAvi

))(()( 11 ABABv E ;)( 111 ABAB

Methods to Find the InverseMethod 1 ： inverse. thefind tomatrix adjoint theUse A

Method 2 ：Use elementary operations to find the inverse.

,invertible is invertible is 1 AA sPPPA 211

EAPPP s 211

21AEPPP s

)()( 1operations row

AEEA

.

102

123

111

of inverse theFind 1e.g.

A：

124100

013210

001111

124100

235010

112001

100102

010123

001111

)(

EA

102320

013210

001111

1A

124

235

112

?,

153

132

5431

AA

131

7185

112981A

1

001

0001

00001

321

2

aaaa

aa

a

A

nnn

?1 A

Ex

1000

0010

0001

00001

1

a

a

a

A

Method 3: use the definition.

. Find .0,2 e.g. 11

1

Aaa

a

a

A n

n

：

Guest ：

na

aB

1

1

1

. that prove to

needonly Weright that Is

EAB ？

1AB

：Solution

na

a

1

na

a

1

1

1

E

na

aA

1

1

11

1

1

. find and

,invertible is that prove2 satisfies let 3..1

2

A

AOEAAAge ，：

EAA 22 EEAA 2)(

EEA

A

2 2

1 EAA

Method 4: prove B is the inverse of A by definition.

: thatprove),integer. positive a is (,Let .4e.g. kOAk 121)( kAAAEAE

))(( 12 kAAAEAE )( 1212 kkk AAAAAAAE

kAE E

Applications of the inverse—— to solve matrix equations.

.invertible is ,.1 ABAX BAXI 1:Solution

XBPPP s 21EAPPP s 21

)()(

operations RowXEBA

.invertible is ,.2 ABXA

）（

operations elementary of method the:Solution II

sPPPA 211

1:Solution BAXI

）（

.operations elementary

of method the:Solution II

sPPPA 211 XPPBP s 21 EPPAP s 21

X

E

B

A

operationsColumn

.invertible are ,,.3 CABAXC 11:Solution BCAXI

:Solution II

BAXC 1

1BAAX

When we solve matrix equations, remember that before figuring out the solutions, reduce the matrices at first.

. determine , and , have weIf .1 BBAABA

ABAB ABEA )( AEAB 1)(

, find tooperations elementary use alsocan We

B )()(operations row

BEAEA

. determine , and , have weIf.2 2 XXAEAXA

))((2 EAEAEAXAX ))(()( EAEAXEA

. then ,invertible is ifOnly EAXEA

).9()3( determine , have weIf .3 21 EAEAA

)9()3( 21 EAEA )3)(3()3( 1 EAEAEA

EA 3