9 - solving linear systems

7/23/2019 9 - Solving Linear Systems

1/25

2014 Baylor UniversitySlide 1

Fundamentals of Engineering AnalysisEGR 1302 The Inverse Matrix


2/25


Solving Systems of Linear Equations

The Inverse atri!

"n!nand Bn!nare Square atri#es of the same $rder%

" & B ' In

B & " ' In

B is #alled (The Inverse of ")

B ' "*1

" & "*1

' I

in "lge+ra, the equivalent is 11

=

x

x


3/25

2014 Baylor UniversitySlide -

Solving Systems of Linear Equations

Sum of .rodu#ts

The same equation #an re/resent any$rder%

=

2

1

2

1

2221

1211

d

d

x

x

aa

aa

1212111 dxaxa =+

2222121 dxaxa =+

dxA =


4/25


Solving this System for !1, !2

Be#omes

Su+tra#t

22a

12a

1212111 dxaxa =+

2222121 dxaxa =+

1222221212211 daxaaxaa =+

2122122211221 daxaaxaa =+

212122112212211 !" dadaxaaaa =

211121212212211 !" dadaxaaaa +=

!" 12212211

2121221

aaaa

dadax

=

!" 12212211

2111212

aaaa

dadax

+=


5/25

2014 Baylor UniversitySlide

Solving this System for !1, !2 #ont%

Sum of .rodu#ts

" ne3 atri! ()

fa#tor out

the denominator

+

=

= 211121

212122

122122112

1

!"

1

dada

dada

aaaax

x

x

=

2

1

1121

1222

122122112

1

!"

1

d

d

aa

aa

aaaax

x


6/25


6e no3 have a solution for the Inverse of a 2!2 atri!

The Solution to

dxA =

=

2

1

1121

1222

122122112

1 !"

1dd

aaaa

aaaaxx

dAxAA 11 =

xIxAAwhere 1

=

dAx 1=

=

1121

1222

12212211

1

!"

1

aa

aa

aaaaA


7/25 2014 Baylor UniversitySlide 7

The 8eterminant

The 8eterminant of " '

=

1121

1222

12212211

1

!"

1

aa

aa

aaaa

A

=

2221

1211

aa

aaA12212211 aaaa

12212211det aaaaA =



:ules for ;inding the Inverse of a 2!2 atri!

Rule 1: Swap the Main Diagonal

Rule 2: Change Signs on

the Back Diagonal

Rule 3: Divide by the Deterinant

=

1121

1222

12212211

1

!"

1

aa

aa

aaaaA

=

1121

1222

12212211

1

!"

1

aa

aa

aaaa

A

=

2221

1211

aa

aaA


9/25 2014 Baylor UniversitySlide umeri#al E!am/le

Be#omes

=

213$

!!13"$2"11A

$32 21 =+ xx

2$21

=+ xx

=

2

$

$1

32

2

1

x

x

=

2

$

11%211%1

11%311%$1A

=

11%&

11%10x



This is the

definition of the

inverse for any

matri!=

The ?eneral larger than 2!2 Solution to

The 2!2 #ase@

"lready sho3n that this is det" all this the (adAoint) or adA"

The general #ase@

8efinition of adAoint@

dxA =

dAx 1=

=

1121

1222

12212211

1

!"

1

aa

aa

aaaaA

A

CfA

A

AA

T

==!det"

!"ad'1

=

1112

2122

aa

aaCfA

==

1121

1222!"ad'

aa

aaCfAA T



Gaussian Elimination ( )olutions of *inear

)ystems

Fundamentals of Engineering Analysis

EGR 1302


13/25 2014 Baylor UniversitySlide 1-

The >eed for a ?eneral Solution to Linear Systems

Unique Solution, all

#ross at the same

/oint, the (solution)

Det ! "

#e need a ethod o$

$inding a general solution

when the coe$$icient

atri% & is Singular'

the system

of equations

1x

2x

3x

+2 321 =+ xxx

232 321 =+ xxx ,2 321 =+ xxx



?aussian Elimination * " general solution ethodology

Elementary :o3 $/erations@

1% ulti/ly +y a #onstant *

2% S3a/ t3o ro3s *

-% :e/la#e a ro3 +y adding to it another &ro3 *

6e 3ill use three eleentary row operationsto solve this set of

linear equations +y ?aussian Elimination%

Same as multi/lying +oth

sides of an equation

$rder doesnCt matter

Same as elimination

ii rkr

-=

ijji rrrr .

jii rkrr - +=

+2$

+3212

321

321

321

=+

=+=++

xxx

xxxxxx



Using Elementary :o3 $/erations

to Solve +y ?aussian Elimination

Ste/ 1@ Use :ule - to eliminate from ro3s 2 D -@

1% ee/ :o3 1 the same

-%

2%

Ste/ 2@ Use :ule - to eliminate from ro3 -@


-%


Ste/ -@ Use :ule 1 to redu#e all #oeffi#ients to 1@


-%

2%

+2$

+32

12

321

321

321

=+

=+

=++

xxx

xxx

xxx

&320

,+0

12

32

32

321

=+

=++

=++

xx

xx

xxx

1x

13

-

3 rrr =12

-

2 2 rrr +=

+

$/

+

2300

,+0

12

3

32

321

=+

=++

=++

x

xx

xxx2x

23

-

3+

2rrr =&320

,+0

12

32

32

321

=+

=++

=++

xx

xx

xxx

200

+

,

+

10

12

3

32

321

=++

=++

=++

x

xx

xxx

3-

323+ rr =+

$/+

2300

,+0

12

3

32

321

=+

=++

=++

x

xx

xxx

2

-

2+

1rr =

12 =x

11=x



The "ugmented atri!

#an +e re/resented as

(augmented) matri!

Ste/ 1 * 1% ee/ :o3 1 the same

-%

2%

Ste/ 2 * 1% ee/ :o3 1 the same

-%


Ste/ - * 1% ee/ :o3 1 the same

-%

2%(Row )chelon *or+

+2$1+132

1121

+2$+32

12

321

321

321

=+=+

=++

xxxxxx

xxx

13-

3 rrr =

12

-

2 2 rrr +=

&320

,1+0

1121

23

-

3

+

2rrr =

+

$/

+

2300

,1+0

1121

3

-

3

23

+rr =

2

-

2

+

1rr =

2100+

,

+

110

1121



:edu#ed :o3 E#helon ;orm of the "ugmented atri!

Using (Ba#3ards Su+stitution)on the :o3 E#helon ;orm

,dentity Matri%

(Reduced Row )chelon *or+

$+serve dire#tly that

31

-

1 rrr =

32-

2 +1 rrr =

2100+

,

+

1

10

1121

2100

1010

1021

21

-

1 2rrr =

2100

1010

1001

2

1

1

3

2

1

=

=

=

x

x

x



Using the TI*9< to do ?aussian Elimination

>ote that #al#ulator #om/utes a different :E; result,+y using a different algorithm, +ut the ans3er is still #orre#t%

Save the augmented matri! as varia+le (!-4)

Reduced Row )chelon *or -

use the $unction (rre$./

Row )chelon *or -

use the $unction (re$./

TI*9< :esult anual :esult

+2$1+132

1121

2100+

,

+

110

1121


19/25 2014 Baylor UniversitySlide 1ote@ det"'0

0his syste has no solution

(inconsistent+

22

,33

3

321

321

321

=+

=+

=+

xxx

xxx

xxx

2211,313

3111

1320

$/$0

3111

13-

3

12

-

2 3

rrr

rrr

=

=

23

-

3

2

1rrr =

1000

$/$0

3111

10 3=x



The Infinite Solution

hange the System

Same :o3

o/erations

0his syste has in$inite

solutions depending

on the value o$

iplies

det" still equals Fero

2211

,313

1111

=3x

1320

2/$0

1111

13

-

3

12

-

2 3

rrr

rrr

=

=

0000

2/$0

1111

23

-

32

1rrr =

+

=

2

3

2

12

1

2

3

3

2

1

x

x

x

00 3=x

2

3

2

1.2/$ 22 +==+ xx

2

1

2

3.1!

2

3

2

1" 11 ==++ xx



The Three ?eneral Solutions

1' niue ! X3e%ists as a single value

2' 4one ! 4o X3e%ists

3' ,n$inite ! X3e%ists as any value

k100

10

1

0det A

k000

10

1

0det =A

000010

1

0det =A



?ra/hi# E!am/les of the Three ?eneral Solutions

rre$./

rre$./

rre$./

in$inite

uniue

no solution

planes

parallel

never

intersect

all planes

intersect

on the sae line

single point

o$ intersection

12+3$

,/312102102

=+

=+=++

zyx

zyxzyx

22

,33

1

=+

=+

=+

zyx

zyx

zyx

22

2233

1

=+

=+

=+

zyx

zyx

zyx

,&,&06

,&,103

,&,

,+1

100

010001

0000

10

01

21

23

23

21

1000

010

001

23

21


23/25 2014 Baylor UniversitySlide 2-

E!am/le of the Three ?eneral Solutions

?iven 5

1% Unique Solution

-% Infinite Solutions

2% >o Solutions

Unno3ns@

1a

1

1

+

=

a

bz

ba11

3212

2321

13

-

3

12

-

22

rrr

rrr

=

=23

-3 rrr =

+

1100

1$30

2321

ba

!1"3det += aA

1=b1=a0

0=z

1b1=a0

kz=

z

y

x


24/25


The Gomogeneous Set of Linear Equations

S/e#ial ase

$nly 2 /ossi+le solutions@

, im/lying infinite solutions

6hen k', infinite solutions e!ist,

other3ise, there is no solution%

rre$./

, a trivial solution, orEither

03

02

0

321

21

321

=+

=+

=++

xkxx

xx

xxx

031

0021

0111

k

0,00

0110

0111

k

,det =kA

0=xA

=3x

0det =A

0=x

== 22 .0 xx

2.0 11 ==++ xx


25/25

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9 - solving linear systems

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