Sec 3.5 Inverses of Matrices

IAA 1* Where A is nxn

Finding the inverse of A:

IA 1AISeq or row operations

Finding the inverse of A:

100***

010***

001***

******

******

*****1

*****0

*****0

*****1

*****0

****10

*****1

****00

****10

*****1

***100

****10

*****1

***100

***010

***0*1

***100

***010

***001

A

1A

ExampleFind inverse

1223

1341

511

A

Properties

ABBA )1CBACBA )()( )2

CABBCA )()( )3 ACABCBA )( )4

BCACCBA )( )5

Fact1: AB in terms of columns of B

],,,[ 21 nbbbB

],,,[ 21 nAbAbAbAB

Fact1: Ax in terms of columns of AT

nxxxx ],,,[ 21

naxaxaxAx 22211

Basic unit vector:

,

0

1

0

,

0

0

1

21

ee

0

1

0

je J-th location

????jAe

????AI

Def: A is invertable if

There exists a matrix B such that IBAAB

TH1: the invers is unique

TH2: the invers of 2x2 matrix

dc

baA

ac

bdA

det

1bcad det Example

Find inverse

95

64A

TH3: Algebra of inverse

If A and B are invertible, then

AAA -111 )( and invertable is 1

nnn AAAn )()( and invertible is ,0 11 2

111)( ABAB3

11111)( ABCDABCD4

TH4: solution of Ax = b

vector-n a is andmatrix invertiblenxn is bA

bAx bAx 1 sol uique a has

Example Solve

1895

664

yx

yx

Def: E is elementary matrix if

1) Square matrix nxn

2) Obtained from I by a single row operation

01

101E

102

010

001

2E

10

01I

100

010

001

I

21 RR

312 RR

Which of these matrices are elementary matrices

01

151E

100

010

021

2E

0001

0100

0010

1000

4E

010

100

021

3E

1100

0100

0011

0001

5E

1000

1100

0010

0001

5E

REMARK: Let E corresponds to a certain elem row operation.

It turns out that if we perform this same operation on matrix A , we get the product matrix EA

102

010

001

1E

100

010

001

I312 RR

102

413

211

A312 RR

520

413

211

1A AEA 11

520

413

211

1A

413

520

211

2A32 RR

010

100

001

2E ????

NOTE: Every elementary matrix is invertible

102

010

001

1A

100

010

001

I312 RR

100

010

001

I312 RR

102

010

001ˆ

3E

33ˆ EEI

102

010

001

3E

001

010

100

1E

100

010

001

I31 RR

??11 E

100

050

001

2E

100

010

001

I25R

??12 ETypeI TypeII

TypeIII??13 E

001

010

1001

1E

100

00

001

511

2E

102

010

0011

3En??observatioAny

Sec 3.5 Inverses of Matrices

IA 1AI

TH6: A is row equivalent to

identity matrix I

*1A *2A Row operation 1 Row operation 2 Row operation 3 Row operation k

AEA 11 122 AEA 1 kkAEI

AEEEEI kk 121

1211 EEEEA kk

1121

EEEEA kk

111

12

11

kk EEEEA

211 kkk AEA

1-A into I transforms

also I intoA sform that tranoperationsrow elementary of sequence the:Note

A

is invertible

A is invertable

A is a product of elementary

matrices

Example Solve

4253

1365

2234

zyx

zyx

zyx

4

1

2

253

365

234

z

y

x

9117

221

3431A

39

8

14

4

1

21A

z

y

x

Solving linear system

Example Solve

025303650234

zyxzyxzyx

0

0

0

253

365

234

z

y

xWhat is the

solution

Quiz2: SAT in Class (3.3+3.4)

1) Given a matrix A find the reduced row echelon form

0113

0011

0212

A

2) Use the method of Gauss-Jordan elimination to solve the following system (find the solution in vector form (i.e) as a linear combination of vectors)

01000

00100

00091

Quiz3: Sund online (3.3+3.4)

Example Solve

9117

221

3431A

Matrix Equation

253

365

234

A

1425

5147

6213

B

BAX

BAX 1

In certain applications, one need to solve a system Ax = b of n equations in n unknowns several times but with different vectors b1, b2,..

11 bAx 22 bAx kk bAx

kk bbbAxAxAx 2121

kk bbbxxxA 2121

BAX Matrix Equation

Definition: A is nonsingular matrix if the system has only

the trivial solution

0Ax

0x

101

010

011

A

ExampleShow that A is nonsingular

RECALL: Definitions

invertible Row equivalent

nonsingular

AInvertible

Theorem7:(p193) Arow equivalent

I

Anonsingular

Ax = b

Every n-vector b

has unique sol

Ax = b

Every n-vector b

is consistent

Ax = 0

The system

has only the trivial sol

Ais a product of

elementary matrices

TH7: A is an nxn matrix. The following is equivalent

(a) A is invertible

(b) A is row equivalent to the nxn identity matrix I

(c) Ax = 0 has the trivial solution

(d) For every n-vector b, the system A x = b has a unique solution

(e) For every n-vector b, the system A x = b is consistent

Problems (page194)

34) Show that a diagonal matrix is inverible if and only if each diagonal element is nonzero. In this case , state concisely how the invers matrix is obtained.

35) Let A be an nxn matrix with either a row or a column consisting only of zeros. Show that A is not invertible.

41) Show that the i-th row of the product AB is Ai B, where Ai is the i-th row of the matrix A.

AInvertiblenot

True & False Arow equivalent

I

Anonsingular

Ax = b

Every n-vector b

has unique sol

Ax = b

Every n-vector b

is consistent

Ax = 0

The system

has only the trivial sol

Ais a product of

elementary matrices

? ??? ? ?