Chuong 02

Chng 2 MA TRN Chng 2 MA TRN & H PT TUYN TNH& H PT TUYN TNH

ThS. L HONG TUN

Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn

Chng 2 MA TRN & H PTTT

F

MA TRN MATRIX - MATRICE

K hiu = trng sQ ( hoc

R , hoc C )

nh ngha mt ma trn kch thc (mxn) trn F l 1 bng s ch nht c m dng v n ct nh

sau

mnmmm

n

n

n

aaaa

aaaaaaaaaaaa

A

321

3333231

2232221

1131211 dng 1

dng 2

dng 3

dng m

ct 1

ct 2

ct 3

ct n



ijaMA TRN MATRIX -

MATRICE vi

trngF )1,1( njmi

vit gnnjmiijaA

11)(

khi m=n th

A gi l ma trn vung cp n trn F , v vit gn l

njiijaA ,1)(

k hiu

)(Fnm = tp hp tt c cc ma trn cp (mxn) trn F

)(Fn = tp hp tt c cc ma trn vung cp n trn F



)()(9/43/1056

15417/11085/23

4343 RQA

MA TRN MATRIX - MATRICE V d

5sin91103ln187/23

5

B

)(9/450327 51 CiiC )(

3ln5/9

2

14 C

ei

D



MA TRN

ZERO L ma trn c tt c cc h s u = 0, k hiu nm

n

V d

000000

32

000000000

3

CC PHP BIN I S CP TRN DNG CHO MA TRN C 3 php bin i s cp trn dng cho ma trn



0c

CC PHP BIN I S CP TRN DNG CHO MA TRN PHP BIN I S CP TRN DNG LOI 1

nhn dng th (i) vi s )( Fc

')()( AA ici

V d

987654321

A )2(7/3)2(

9877/187/157/12

321'A



987654321

A


hon v dng th (i) vi dng th (j)

')()( AA ji

V d

)3()1(

321654987

'A



)2(5)3()3(


thay dng (i) bng [dng (i) + c.dng (j)] 0c )( Fc

')]()[()( AA jcii

V d

987654321

A

211713654321

'A



mnmnmm

nn

nn

bxaxaxa

bxaxaxabxaxaxa

2211

22222121

11212111

H PHNG TRNH TUYN TNH Mt h pttt gm m phng trnh v n n s trn F l 1 h pt

c dng

(*)

, trong ija

ib)1,1( njmi l cc hng s cho

trc

nxxx ,,, 21 l n n s cn tm

( tuyn tnh cc n s dng bc 1 v c tnh cht ng thng )

0 cbyax



njmiijaA

11)(

H PHNG TRNH TUYN TNH

t

nx

xx

X2

1

mb

bb

B2

1

th h pt (*) c vit gn thnh 2 dng

BAX ( dng tch ma

trn )

)|( BA ( dng ma trn ha )



1)3/10(92)2(sin283)3(ln840)7/2(6)11/9(73

3514

74253

2415

54321

xxxexxxxxxxxxxxxxx

H PHNG TRNH TUYN TNH V d xt h

pttt

dng ma trn ha

1230

93/100112sin8204803ln7/2611/973

7

e

1x 2x 3x 4x 5x

hiu ngm



B

xxxxx

A

5

4

3

2

1

H PHNG TRNH TUYN TNH dng tch ma trn

, vi

93/100112sin8204803ln7/2611/973

e

A

1230

7B



BAX NGHIM CA 1 H PHNG TRNH TUYN

TNH Xt h pttt Ta ni b s

),,,( 21 nccc l 1 nghim ( 1 li gii ) ca h

nu tt c cc pt ca h u c tha khi ta thay th

nn cxcxcx ,,, 2211

Lu ch c ng 1 trong 3 trng hp sau xy ra cho 1 h pttt

TH1: h v nghim

TH2: h c nghim duy nht

TH3: h c v s nghim



25 xNGHIM CA 1 H PHNG TRNH TUYN

TNH V d a/ h )( RF c 1 nghim

5/2x

b/ h x0 )( RF v nghim

c/ h 00 x )( RF c nghim x thc ty ( v s nghim

)

d/ h

75432

yxyx

)( QF

c nghim duy nht

2,1 yx



)1(

NGHIM CA 1 H PHNG TRNH TUYN TNH V d

e/ h

2814547yx

yx)2(

)1(2)2()2(

1200547

yxyx

v nghim

)1(f/ h

3159153

yxyx

)2( )1(3)2()2(

000153

yxyx

hng ng

nghim

315ax

aya hu t ty



AXH PTTT THUN

NHT L h c dng ( ngha l 021 mbbb )

D thy h ny c 1 nghim l

021 nxxx

( nghim tm thng )

H thun nht

c nghim duy nht

v s nghim

Vd a/ h

0705302

yxyxyx

( gii)

0 yx( nghim duy nht )



}0872{ zyxH PTTT THUN

NHT V d b/ h

giiCF nghi

m

287

,bax

bzay( a,b phc ty )

v s nghim

PHNG PHP GII H PTTT Khi gii 1 h pttt trn F (dng ma trn ha) ta c th s

dng cc php bin i s cp trn dng m khng lm thay i tp hp nghimca h



FPHNG PHP GAUSS-

JORDAN Xt 1 h pttt c m phng trnh v n n s, trn vit di dng ma trn

ha)|( BA

Ta thc hin cc bc sau y Dng cc php bin i s cp trn dng thch hp

xy dng trong ma trn A cc ct chun

0

001

1

E

0

010

2

E

0

100

3

E theo th t

( t tri qua phi )



kE

PHNG PHP GAUSS-JORDAN qu trnh chun ha cc ct phi tun th 2 nguyn

tc Khi ang xy dng ct chun

th khng c lm bin i

cc ct chun

121 ,,, kEEE trc

Nu ti ct ang xt

khng xy dng c

kE

th ta chuyn qua ct k cn bn phi

Qu trnh chun ha cc ct s kt thc khi ta xt xong ct cui cng ca ma

trn A C ng 1 trong 3 trng hp sau xy ra



)|000( a

PHNG PHP GAUSS-JORDAN

TH1: khi ang chun ha cc ct, nu c th bin i dng no

v dng , vi

0a

th h V NGHIM



0

0

00000

00000

TH2: khi chun ha xong cc ct, ta c h c dng

100000100000001000001000001

n

n

cc

ccc

1

3

2

1

cc dng zero c thc hoc khng

c

n ct

chun lin tip nhau

PHNG PHP GAUSS-JORDAN



nn cxcxcxcx ,,,, 332211

PHNG PHP GAUSS-JORDAN H c NGHIM DUY NHT

TH3: khi chun ha xong cc ct, ta thu c

r ct chun

rEEE ,,, 21 )( nr xen k vi )( rn ct khng chun ha

c

h c V S NGHIM nh sau

Cc n ng vi cc ct khng chun ha c l n t do, ly gi tr ty trn F, c (n-r) n

nh vy



m

k

k

k

u

uuu

uu

U

1

1

2

1

PHNG PHP GAUSS-JORDAN Cc n cn li (ng vi cc ct chun ha c) c tnh

theo cc n t do nh cc pt khng tm thng h cui cng

NHN BIT 1 CT C CHUN HA C HAY KHNG Ta mun chun ha ct

0

010

00

kE

c dng to 1 ch

okE

khng c dng to 1 ch

okE



01 mkk uuu NHN BIT 1 CT C CHUN HA C HAY

KHNG TH1: nu khng th chun hathnh

U kE

TH2: nu c (t nht) 1 s

0 trong cc s mkk uuu ,,, 1 th chun ha c

thnh

U kE

V d

00002703

U

00010000

5EKL: U khng th chun ha thn

h5E

a/



23701803

V

NHN BIT 1 CT C CHUN HA C HAY KHNG V

d

b/

00010000

5E

KL: V chun ha cthnh

5E



RGII H PTTT V d (nghim duy nht): gii h sau

trn

8232423282236232

1243

3412

2134

4321

xxxxxxxxxxxxxxxx

vit di dng ma trn ha, ta c

8486

1232212332122321

)1(2)4()4()1(3)3()3()1(2)2()2(

201446

54708104018502321



)2(7)4()4()2(4)3()3()2(2)1()1()3()2()2(

GII H PTTT

90541026

5418003618007210

16701

)3(7)1()1()3(2)2()2(

)3(181)3(

)3()4()4(

36345

18000210030102001



)4(2)3()3()4(3)2()2()4(2)1()1(

)4(181)4(

GII H PTTT

2121

1000010000100001

Vit li h cui cng

21

21

4

3

2

1

xxxx

( nghim duy nht )



3213

85723537121121375231

GII H PTTT V d ( v nghim ) Q: gii h sau

trn

)1(3)4()4()1(2)3()3()4()2()2(

6423

292013110191311709693075231

)2(11)4()4()2(7)3()3(

)2(31)2(

)2()1()1(



3/403/263/25

4220002110003231021701

GII H PTTT

)3(2)4()4(

400000

h v nghim



1712

74242436240121113011

GII H PTTT V d ( v s nghim ) Q: gii h sau

trn

)1(2)4()4()4(2)3()3(

)1()2()2(

5532

510220105101001222013011

)2()1()1()2(

21)2(

)2(5)3()3()2()4()4(



8102/32/1

4120005150002/111102/12101

GII H PTTT

)3(12)4()4()3()2()2()3(2)1()1(

)3(151)3(

03/26/56/5

000003/110006/501106/70101



3/23/6/5)6/5(6/5)6/7(

54

532

531

xxxxxxxx

GII H PTTT Lc ny h c th c vit li l

Nghim ( v s nghim )

32

3

65

65

65

67

,

4

2

1

53

bx

bax

bax

bxax

(a,b hu t ty )

( do ct 3 v ct 5 khng chun ha c )



0

000

'1

a

E

CC CT BN CHUN Dng tng qut ca m ct bn chun ( mi ct c m

dng ) l

0

00

'2

cb

E

0

0'3

fed

E

m

m

m

uu

uuu

E

1

3

2

1

'

ty , 0, trn F

121 ,,,,,,, muuuecb l cc s ty

trn F



00002

'1E

CC CT BN CHUN Lu cc ct chun cng l cc ct bn chun c

bit V d vit 5 ct bn chun cp 5

00071

'2E

003

0

'3

E

02ln503/2

' 44E

1sin0168

'5E



)(FA nmHNG CA MT MA

TRN Xt ma trn

, th

A 'Abn chun ha ti a cc ct( chun ha ti a cc ct )

Ta ni 'A l dng bc thang ca ma trn A

t )(Ar = hng ca A = s dng khc zero ca

'A



54A

HNG CA MT MA TRN V

d a/

bn chun ha ti a

00000730000620010425

'A

1'E 2'E 3'E

'A c 3 dng khc zero 3)( Ar

53B

b/

bn chun ha ti a

30000419000265/20

'B3)( Br

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Chuong 02

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