Chuong 02
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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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
)()(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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
MA TRN
ZERO L ma trn c tt c cc h s u = 0, k hiu nm
n
V d
000000
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CC PHP BIN I S CP TRN DNG CHO MA TRN C 3 php bin i s cp trn dng cho ma trn
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
987654321
A
CC PHP BIN I S CP TRN DNG CHO MA TRN PHP BIN I S CP TRN DNG LOI 2
hon v dng th (i) vi dng th (j)
')()( AA ji
V d
)3()1(
321654987
'A
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
)2(5)3()3(
CC PHP BIN I S CP TRN DNG CHO MA TRN PHP BIN I S CP TRN DNG LOI 3
thay dng (i) bng [dng (i) + c.dng (j)] 0c )( Fc
')]()[()( AA jcii
V d
987654321
A
211713654321
'A
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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 )
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
)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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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 )
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
}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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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 )
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
)|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
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
0
0
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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
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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/
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
23701803
V
NHN BIT 1 CT C CHUN HA C HAY KHNG V
d
b/
00010000
5E
KL: V chun ha cthnh
5E
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
)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
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
)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 )
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
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)2(11)4()4()2(7)3()3(
)2(31)2(
)2()1()1(
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
3/403/263/25
4220002110003231021701
GII H PTTT
)3(2)4()4(
400000
h v nghim
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
1712
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GII H PTTT V d ( v s nghim ) Q: gii h sau
trn
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)1()2()2(
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21)2(
)2(5)3()3()2()4()4(
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
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
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
3/23/6/5)6/5(6/5)6/7(
54
532
531
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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 )
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
0
000
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a
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CC CT BN CHUN Dng tng qut ca m ct bn chun ( mi ct c m
dng ) l
0
00
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cb
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m
m
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ty , 0, trn F
121 ,,,,,,, muuuecb l cc s ty
trn F
-
Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
00002
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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
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003
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1sin0168
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Trng H Cng Ngh Thng Tin HQG Tp.HCM http://www.uit.edu.vn
Chng 2 MA TRN & H PTTT
)(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
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Chng 2 MA TRN & H PTTT
54A
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d a/
bn chun ha ti a
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bn chun ha ti a
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