Bài tập toán cao cấp Tập 1 part 6 pdf
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4.1. Hˆe
.
n phu
.
o
.
ng tr`ınh v´o
.
i n ˆa
’
nc´od
i
.
nh th´u
.
c kh´ac 0 139
Gia
’
i. 1) Lˆa
.
p ma trˆa
.
nmo
.
’
rˆo
.
ng v`a thu
.
.
chiˆe
.
n c´ac ph´ep biˆe
´
nd
ˆo
’
i:
A =
10−2
−3
−21 6
11
−15−4
−4
h
2
+2h
1
→ h
2
h
3
+ h
1
→ h
3
−→
10−2
−3
01 2
5
05−6
−7
−→
h
3
− 5h
2
→ h
3
10 −2
−3
01 2
5
00−16
−32
.
T`u
.
d´o suy ra
x
1
− 2x
3
= −3
x
2
+2x
3
=5
−16x
3
= −32
⇒ x
1
=1,x
2
=1,x
3
=2.
2) Lˆa
.
p ma trˆa
.
nmo
.
’
rˆo
.
ng v`a thu
.
.
chiˆe
.
n c´ac ph´ep biˆe
´
ndˆo
’
iso
.
cˆa
´
p:
2 −13−1
9
11−24
−1
32−13
0
5 −21−2
9
h
1
→ h
2
h
2
→ h
1
−→
11−24
−1
2 −13−1
9
32−13
0
5 −21−2
9
−→
h
2
− 2h
1
→ h
2
h
3
− 3h
1
→ h
3
h
4
− 5h
1
→ h
4
11−24
−1
0 −37 −9
11
0 −15 −9
3
0 −711−22
14
h
2
→ h
3
h
3
→ h
2
−→
140 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
−→
11−24
−1
0 −15 −9
3
0 −37 −9
11
0 −711−22
14
h
3
− 3h
2
→ h
3
h
4
− 7h
2
→ h
4
−→
11 −24
−1
0 −15−9
3
00 −818
2
00−24 41
−7
−→
h
4
− 3h
3
→ h
4
11−24
−1
0 −15 −9
3
00−818
2
00 0−13
−13
T`u
.
d´o suy ra r˘a
`
ng x
1
=1,x
2
= −2, x
3
=2,x
4
=1.
B
`
AI T
ˆ
A
.
P
Gia
’
i c´ac hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh sau
1.
x
1
− x
2
+2x
3
=11,
x
1
+2x
2
− x
3
=11,
4x
1
− 3x
2
− 3x
3
=24.
.(D
S. x
1
=9,x
2
=2,x
3
=2)
2.
x
1
− 3x
2
−4x
3
=4,
2x
1
+ x
2
− 3x
3
= −1,
3x
1
−2x
2
+ x
3
=11.
.(DS. x
1
=2,x
2
= −2, x
3
=1)
3.
2x
1
+3x
2
−x
3
=4,
x
1
+2x
2
+2x
3
=5,
3x
1
+4x
2
− 5x
3
=2.
.(D
S. x
1
= x
2
= x
3
=1)
4.1. Hˆe
.
n phu
.
o
.
ng tr`ınh v´o
.
i n ˆa
’
nc´od
i
.
nh th´u
.
c kh´ac 0 141
4.
x
1
+2x
2
+ x
3
=8,
−2x
1
+3x
2
− 3x
3
= −5,
3x
1
− 4x
2
+5x
3
=10.
.(D
S. x
1
=1,x
2
=2,x
3
=3)
5.
2x
1
+ x
2
− x
3
=0,
3x
2
+4x
3
= −6,
x
1
+ x
3
=1.
.(D
S. x
1
=1,x
2
= −2, x
3
=0)
6.
2x
1
− 3x
2
− x
3
+6 =0,
3x
1
+4x
2
+3x
3
+5 =0,
x
1
+ x
2
+ x
3
+2 =0.
.(D
S. x
1
= −2, x
2
=1,x
3
= −1)
7.
x
2
+3x
3
+6 =0,
x
1
− 2x
2
− x
3
=5,
3x
1
+4x
2
− 2x =13.
.(D
S. x
1
=3,x
2
=0,x
3
= −2)
8.
2x
1
− x
2
+ x
3
+2x
4
=5,
x
1
+3x
2
− x
3
+5x
4
=4,
5x
1
+4x
2
+3x
3
=2,
3x
1
− 3x
2
− x
3
−6x
4
= −6.
.
(D
S. x
1
=
1
3
, x
2
= −
2
3
, x
3
=1,x
4
=
4
3
)
9.
x
1
− 2x
2
+3x
3
− x
4
= −8,
2x
1
+3x
2
− x
3
+5x
4
=19,
4x
1
− x
2
+ x
3
+ x
4
= −1,
3x
1
+2x
2
− x
3
− 2x
4
= −2.
.
(D
S. x
1
= −
1
2
, x
2
=
3
2
, x
3
= −
1
2
, x
4
=3)
10.
x
1
− x
3
+ x
4
=3,
2x
1
+3x
2
− x
3
− x
4
=2,
5x
1
− 3x
4
= −6
x
1
+ x
2
+ x
3
+ x
4
=2.
.
(D
S. x
1
=0,x
2
=1,x
3
= −1, x
4
=2)
142 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
11.
2x
1
+3x
2
+8x
4
=0,
x
2
−x
3
+3x
4
=0,
x
3
+2x
4
=1,
x
1
+ x
4
= −24
.
(D
S. x
1
= −19, x
2
= 26, x
3
= 11, x
4
= −5)
12.
3x
1
+ x
2
− x
3
+ x
4
=0,
2x
1
+3x
2
− x
4
=0,
x
1
+5x
2
− 3x
3
=7,
3x
2
+2x
3
+ x
4
=2,
.
(D
S. x
1
= −1, x
2
=1,x
3
= −1, x
4
=1)
13.
x
1
− 2x
2
+ x
3
− 4x
4
− x
5
=13,
x
1
+2x
2
+3x
3
− 5x
4
=15,
x
2
− 2x
3
+ x
4
+3x
5
= −7,
x
1
− 7x
3
+8x
4
− x
5
= −30,
3x
1
−x
2
− 5x
5
=4.
.
(D
S. x
1
=1,x
2
= −1, x
3
=2,x
4
= −2, x
5
=0)
14.
x
1
+ x
2
+4x
3
+ x
4
− x
5
=2,
x
1
− 2x
2
− 2x
3
+3x
5
=0,
4x
2
+3x
3
− 2x
4
+2x
5
=2,
2x
1
−x
3
+3x
4
− 2x
5
= −2,
3x
1
+2x
2
− 5x
4
+3x
5
=3.
.
(D
S. x
1
=
2
5
, x
2
= −
3
5
, x
3
=
4
5
, x
4
=0,x
5
=0)
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 143
4.2 Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
Tax´ethˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh gˆo
`
m m phu
.
o
.
ng tr`ınh v´o
.
i
n ˆa
’
n
a
11
x
1
+ a
12
x
2
+ ···+ a
1n
x
n
= b
1
,
a
21
x
1
+ a
22
x
2
+ ···+ a
2n
x
n
= b
2
,
a
m1
x
1
+ a
m2
x
2
+ ···+ a
mn
x
n
= b
m
,
(4.9)
v´o
.
i ma trˆa
.
nco
.
ba
’
n
A =
a
11
a
12
a
1n
a
m1
a
m2
a
mn
v`a ma trˆa
.
nmo
.
’
rˆo
.
ng
A =
a
11
a
12
a
1n
b
1
a
m1
a
m2
a
mn
b
m
Hiˆe
’
n nhiˆen r˘a
`
ng r(A) r(
A)v`ımˆo
˜
id
i
.
nh th´u
.
c con cu
’
a A d
ˆe
`
ul`adi
.
nh
th ´u
.
c con cu
’
a
A nhu
.
ng khˆong c´o diˆe
`
u ngu
.
o
.
.
cla
.
i. Ta luˆon luˆon gia
’
thiˆe
´
t
r˘a
`
ng c´ac phˆa
`
ntu
.
’
cu
’
a ma trˆa
.
n A khˆong dˆo
`
ng th`o
.
ib˘a
`
ng 0 tˆa
´
tca
’
.
Ngu
.
`o
.
i ta quy u
.
´o
.
cgo
.
idi
.
nh th´u
.
c con kh´ac 0 cu
’
amˆo
.
t ma trˆa
.
nm`a
cˆa
´
pcu
’
an´ob˘a
`
ng ha
.
ng cu
’
a ma trˆa
.
nd´ol`adi
.
nh th´u
.
c con co
.
so
.
’
cu
’
a n´o.
Gia
’
su
.
’
d
ˆo
´
iv´o
.
i ma trˆa
.
nd
˜a cho ta d˜acho
.
nmˆo
.
tdi
.
nh th´u
.
c con co
.
so
.
’
.
Khi d
´o c´ac h`ang v`a c´ac cˆo
.
t m`a giao cu
’
ach´ung lˆa
.
p th`anh di
.
nh th´u
.
c
con co
.
so
.
’
d´odu
.
o
.
.
cgo
.
il`ah`ang, cˆo
.
tco
.
so
.
’
.
D
-
i
.
nh ngh˜ıa. 1
+
Bˆo
.
c´o th´u
.
tu
.
.
n sˆo
´
(α
1
,α
2
, ,α
n
)du
.
o
.
.
cgo
.
i l`a nghiˆe
.
m
cu
’
ahˆe
.
(4.9) nˆe
´
u khi thay x = α
1
,x= α
2
, ,x= α
n
v`ao c´ac phu
.
o
.
ng
tr`ınh cu
’
a (4.9) th`ı hai vˆe
´
cu
’
amˆo
˜
iphu
.
o
.
ng tr`ınh cu
’
a (4.9) tro
.
’
th`anh
d
ˆo
`
ng nhˆa
´
t.
144 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
2+ Hˆe
.
(4.9) du
.
o
.
.
cgo
.
il`atu
.
o
.
ng th´ıch nˆe
´
u c´o ´ıt nhˆa
´
tmˆo
.
t nghiˆe
.
mv`a
go
.
il`akhˆong tu
.
o
.
ng th´ıch nˆe
´
u n´o vˆo nghiˆe
.
m.
3
+
Hˆe
.
tu
.
o
.
ng th´ıch d
u
.
o
.
.
cgo
.
il`ahˆe
.
x´ac d
i
.
nh nˆe
´
u n´o c´o nghiˆe
.
m duy
nhˆa
´
t v`a go
.
il`ahˆe
.
vˆo d
i
.
nh nˆe
´
u n´o c´o nhiˆe
`
uho
.
nmˆo
.
t nghiˆe
.
m.
D
-
i
.
nh l´y Kronecker-Capelli.
2
Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh (4.9)
tu
.
o
.
ng th´ıch khi v`a chı
’
khi ha
.
ng cu
’
a ma trˆa
.
nco
.
ba
’
nb˘a
`
ng ha
.
ng cu
’
a
ma trˆa
.
nmo
.
’
rˆo
.
ng cu
’
ahˆe
.
,t´u
.
cl`ar(A)=r(
A).
D
ˆo
´
iv´o
.
ihˆe
.
tu
.
o
.
ng th´ıch ngu
.
`o
.
itago
.
i c´ac ˆa
’
nm`ahˆe
.
sˆo
´
cu
’
ach´ung lˆa
.
p
nˆen di
.
nh th´u
.
c con co
.
so
.
’
cu
’
a ma trˆa
.
nco
.
ba
’
nl`aˆa
’
nco
.
so
.
’
, c´ac ˆa
’
n c`on
la
.
id
u
.
o
.
.
cgo
.
il`aˆa
’
ntu
.
.
do.
Phu
.
o
.
ng ph´ap chu
’
yˆe
´
udˆe
’
gia
’
ihˆe
.
tˆo
’
ng qu´at l`a:
1.
´
Ap du
.
ng quy t˘a
´
c Kronecker-Capelli.
2. Phu
.
o
.
ng ph´ap khu
.
’
dˆa
`
nc´acˆa
’
n (phu
.
o
.
ng ph´ap Gauss).
Quy t˘a
´
c Kronecker-Capelli gˆo
`
m c´ac bu
.
´o
.
c sau.
1
+
Kha
’
o s´at t´ınh tu
.
o
.
ng th´ıch cu
’
ahˆe
.
. T´ınh ha
.
ng r(
A)v`ar(A)
a) Nˆe
´
u r(
A) >r(A)th`ıhˆe
.
khˆong tu
.
o
.
ng th´ıch.
b) Nˆe
´
u r(
A)=r(A)=r th`ı hˆe
.
tu
.
o
.
ng th´ıch. T`ım di
.
nh th´u
.
c con
co
.
so
.
’
cˆa
´
p r n`ao d´o (v`a do vˆa
.
y r ˆa
’
nco
.
so
.
’
tu
.
o
.
ng ´u
.
ng xem nhu
.
du
.
o
.
.
c
cho
.
n) v`a thu du
.
o
.
.
chˆe
.
phu
.
o
.
ng tr`ınh tu
.
o
.
ng du
.
o
.
ng gˆo
`
m r phu
.
o
.
ng tr`ınh
v´o
.
i n ˆa
’
nm`a(r ×n)-ma trˆa
.
nhˆe
.
sˆo
´
cu
’
an´och´u
.
a c´ac phˆa
`
ntu
.
’
cu
’
ad
i
.
nh
th ´u
.
c con co
.
so
.
’
d˜a c h o
.
n. C´ac phu
.
o
.
ng tr`ınh c`on la
.
i c´o thˆe
’
bo
’
qua.
2
+
T`ım nghiˆe
.
mcu
’
ahˆe
.
tu
.
o
.
ng d
u
.
o
.
ng thu d
u
.
o
.
.
c
a) Nˆe
´
u r = n, ngh˜ıa l`a sˆo
´
ˆa
’
nco
.
so
.
’
b˘a
`
ng sˆo
´
ˆa
’
ncu
’
ahˆe
.
th`ı hˆe
.
c´o
nghiˆe
.
m duy nhˆa
´
t v`a c´o thˆe
’
t`ım theo cˆong th´u
.
c Cramer.
b) Nˆe
´
u r<n, ngh˜ıa l`a sˆo
´
ˆa
’
nco
.
so
.
’
b´e ho
.
nsˆo
´
ˆa
’
ncu
’
ahˆe
.
th`ı ta
chuyˆe
’
n n − r sˆo
´
ha
.
ng c´o ch´u
.
aˆa
’
ntu
.
.
do cu
’
a c´ac phu
.
o
.
ng tr`ınh sang
vˆe
´
pha
’
idˆe
’
thu du
.
o
.
.
chˆe
.
Cramer dˆo
´
iv´o
.
i c´ac ˆa
’
nco
.
so
.
’
. Gia
’
ihˆe
.
n`ay ta
thu du
.
o
.
.
c c´ac biˆe
’
uth´u
.
ccu
’
a c´ac ˆa
’
nco
.
so
.
’
biˆe
’
udiˆe
˜
n qua c´ac ˆa
’
ntu
.
.
do.
2
L. Kronecker (1823-1891) l`a nh`a to´an ho
.
cD´u
.
c,
A. Capelli (1855-1910) l`a nh`a to´an ho
.
c Italia.
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 145
D´o l`a nghiˆe
.
mtˆo
’
ng qu´at cu
’
ahˆe
.
. Cho n −r ˆa
’
ntu
.
.
do nh˜u
.
ng gi´a tri
.
cu
.
thˆe
’
t`uy ´y ta t`ım d
u
.
o
.
.
c c´ac gi´a tri
.
tu
.
o
.
ng ´u
.
ng cu
’
aˆa
’
nco
.
so
.
’
.T`u
.
d
´o t h u
du
.
o
.
.
c nghiˆe
.
m riˆeng cu
’
ahˆe
.
.
Tiˆe
´
p theo ta tr`ınh b`ay nˆo
.
i dung cu
’
aphu
.
o
.
ng ph´ap Gauss.
Khˆong gia
’
mtˆo
’
ng qu´at, c´o thˆe
’
cho r˘a
`
ng a
11
= 0. Nˆo
.
i dung cu
’
a
phu
.
o
.
ng ph´ap Gauss l`a nhu
.
sau.
1
+
Thu
.
.
chiˆe
.
n c´ac ph´ep biˆe
´
ndˆo
’
iso
.
cˆa
´
p trˆen c´ac phu
.
o
.
ng tr`ınh cu
’
a
hˆe
.
dˆe
’
thu du
.
o
.
.
chˆe
.
tu
.
o
.
ng du
.
o
.
ng m`a b˘a
´
tdˆa
`
ut`u
.
phu
.
o
.
ng tr`ınh th´u
.
hai
mo
.
iphu
.
o
.
ng tr`ınh d
ˆe
`
u khˆong ch´u
.
aˆa
’
n x
1
.K´yhiˆe
.
uhˆe
.
n`ay l`a S
(1)
.
2
+
C˜ung khˆong mˆa
´
ttˆo
’
ng qu´at, c´o thˆe
’
cho r˘a
`
ng a
22
= 0. La
.
i thu
.
.
c
hiˆe
.
n c´ac ph´ep biˆe
´
ndˆo
’
iso
.
cˆa
´
p trˆen c´ac phu
.
o
.
ng tr`ınh cu
’
ahˆe
.
S
(1)
(tr `u
.
ra phu
.
o
.
ng tr`ınh th´u
.
nhˆa
´
tdu
.
o
.
.
cgi˜u
.
nguyˆen!) nhu
.
d˜a l`am trong bu
.
´o
.
c
1
+
ta thu du
.
o
.
.
chˆe
.
tu
.
o
.
ng du
.
o
.
ng m`a b˘a
´
tdˆa
`
ut`u
.
phu
.
o
.
ng tr`ınh th´u
.
ba
mo
.
iphu
.
o
.
ng tr`ınh d
ˆe
`
u khˆong ch´u
.
aˆa
’
n x
2
,
3
+
Sau mˆo
.
tsˆo
´
bu
.
´o
.
ctac´othˆe
’
g˘a
.
pmˆo
.
t trong c´ac tru
.
`o
.
ng ho
.
.
p sau
dˆa y .
a) Thˆa
´
yngaydu
.
o
.
.
chˆe
.
khˆong tu
.
o
.
ng th´ıch.
b) Thu du
.
o
.
.
cmˆo
.
thˆe
.
“tam gi´ac”. Hˆe
.
n`ay c´o nghiˆe
.
m duy nhˆa
´
t.
c) Thu d
u
.
o
.
.
cmˆo
.
t“hˆe
.
h`ınh thang” da
.
ng
a
11
x
1
+ a
12
x
2
+ + a
1n
x
n
= h
1
,
b
22
x
2
+ + b
2n
x
n
= h
2
,
b
rr
x
r
+ ···+ b
rn
x
n
= h
r
,
0=h
r+1
,
0=
h
m
.
Nˆe
´
u c´ac sˆo
´
h
r+1
, ,h
m
kh´ac 0 th`ı hˆe
.
vˆo nghiˆe
.
m. Nˆe
´
u h
r+1
=
··· =
h
m
=0th`ıhˆe
.
c´o nghiˆe
.
m. Cho x
r+1
= α, ,x
m
= β th`ı
thu du
.
o
.
.
chˆe
.
Cramer v´o
.
iˆa
’
nl`ax
1
, ,x
r
. Gia
’
ihˆe
.
d´o ta thu du
.
o
.
.
c
146 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
nghiˆe
.
m x
1
= x
1
; x
2
= x
2
, ,x
r
= x
r
v`a nghiˆe
.
mcu
’
ahˆe
.
d˜achol`a
(x
1
, x
2
, ,x
r
,α, ,β).
Lu
.
u´yr˘a
`
ng viˆe
.
c gia
’
ihˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh b˘a
`
ng phu
.
o
.
ng
ph´ap Gauss thu
.
.
cchˆa
´
t l`a thu
.
.
chiˆe
.
n c´ac ph´ep biˆe
´
nd
ˆo
’
iso
.
cˆa
´
p trˆen c´ac
h`ang cu
’
a ma trˆa
.
nmo
.
’
rˆo
.
ng cu
’
ahˆe
.
du
.
an´ovˆe
`
da
.
ng tam gi´ac hay da
.
ng
h`ınh thang.
C
´
AC V
´
IDU
.
V´ı d u
.
1. Gia
’
ihˆe
.
phu
.
o
.
ng tr`ınh
3x
1
− x
2
+ x
3
=6,
x
1
−5x
2
+ x
3
=12,
2x
1
+4x
2
= −6,
2x
1
+ x
2
+3x
3
=3,
5x
1
+4x
3
=9.
Gia
’
i. 1. T`ım ha
.
ng cu
’
a c´ac ma trˆa
.
n
A =
3 −11
1 −51
240
213
504
,
A =
3 −11
6
1 −51
12
240
−6
213
3
504
9
Ta thu d
u
.
o
.
.
c r(
A)=r(A) = 3. Do d´ohˆe
.
tu
.
o
.
ng th´ıch.
Ta cho
.
nd
i
.
nh th´u
.
c con co
.
so
.
’
l`a
∆=
1 −51
240
213
v`ı∆=36=0v`ar(A) = 3 v`a c´ac ˆa
’
nco
.
so
.
’
l`a x
1
,x
2
,x
3
.
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 147
2. Hˆe
.
phu
.
o
.
ng tr`ınh d
˜a cho tu
.
o
.
ng d
u
.
o
.
ng v´o
.
ihˆe
.
x
1
− 5x
2
+ x
3
=12,
2x
1
+4x
2
= −6,
2x
1
+ x
2
+3x
3
=3.
Sˆo
´
ˆa
’
nco
.
so
.
’
b˘a
`
ng sˆo
´
ˆa
’
ncu
’
ahˆe
.
nˆen hˆe
.
c´o nghiˆe
.
m duy nhˆa
´
tl`ax
1
=1,
x
2
= −2, x
4
=1.
V´ı du
.
2. Gia
’
ihˆe
.
phu
.
o
.
ng tr`ınh
x
1
+2x
2
− 3x
3
+4x
4
=7,
2x
1
+4x
2
+5x
3
− x
4
=2,
5x
1
+10x
2
+7x
3
+2x
4
=11.
Gia
’
i. T`ım ha
.
ng cu
’
a c´ac ma trˆa
.
n
A =
12−34
24 5 −1
510 7 2
,
A =
12−34
7
24 5 −1
2
510 7 2
11
Tathud
u
.
o
.
.
c r(
A)=r(A) = 2. Do d´ohˆe
.
tu
.
o
.
ng th´ıch.
Ta c´o thˆe
’
lˆa
´
ydi
.
nh th´u
.
c con co
.
so
.
’
l`a
∆=
2 −3
45
v`ı∆=22= 0 v`a cˆa
´
pcu
’
ad
i
.
nh th´u
.
c=r(A) = 2. Khi cho
.
n ∆ l`am
di
.
nh th´u
.
c con, ta c´o x
2
v`a x
3
l `a ˆa
’
nco
.
so
.
’
.
Hˆe
.
d˜a cho tu
.
o
.
ng du
.
o
.
ng v´o
.
ihˆe
.
x
1
+2x
2
− 3x
3
+4x
4
=7,
2x
1
+4x
2
+5x
3
− x
4
=2
hay
2x
2
− 3x
3
=7−x
1
− 4x
4
,
4x
2
+5x
3
=2−2x
1
+ x
4
.
148 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
2. Ta c´o thˆe
’
gia
’
ihˆe
.
theo quy t˘a
´
c Cramer. D˘a
.
t x
1
= α, x
4
= β ta
c´o
2x
2
−3x
3
=7−α − 4β,
4x
2
+5x
3
=2−2α + β.
Theo cˆong th´u
.
c Cramer ta t`ım du
.
o
.
.
c
x
2
=
7 −α − 4β −3
2 −2α + β 5
22
=
41 −11α −17β
22
,
x
3
=
27− α −4β
42−2α + β
22
=
−24 + 18β
22
·
Do d´otˆa
.
pho
.
.
p c´ac nghiˆe
.
mcu
’
ahˆe
.
c´o da
.
ng
α;
41 −11α −17β
22
;
9β − 12
11
; β
∀α,β ∈ R
V´ı d u
.
3. B˘a
`
ng phu
.
o
.
ng ph´ap Gauss h˜ay gia
’
ihˆe
.
phu
.
o
.
ng tr`ınh
4x
1
+2x
2
+ x
3
=7,
x
1
− x
2
+ x
3
= −2,
2x
1
+3x
2
−3x
3
=11,
4x
1
+ x
2
−x
3
=7.
Gia
’
i. Trong hˆe
.
d
˜a cho ta c´o a
11
=4=0nˆen dˆe
’
cho tiˆe
.
ntadˆo
’
ichˆo
˜
hai phu
.
o
.
ng tr`ınh d
ˆa
`
u v`a thu du
.
o
.
.
chˆe
.
tu
.
o
.
ng d
u
.
o
.
ng
x
1
− x
2
+ x
3
= −2,
4x
1
+2x
2
+ x
3
=7,
2x
1
+3x
2
−3x
3
=11,
4x
1
+ x
2
−x
3
=7.
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 149
Tiˆe
´
p theo ta biˆe
´
ndˆo
’
i ma trˆa
.
nmo
.
’
rˆo
.
ng
A =
1 −11
−2
42 1
7
23−3
11
41−1
7
h
2
− 4h
1
→ h
2
h
3
− 2h
1
→ h
3
h
4
− 4h
1
→ h
4
−→
1 −11
−2
06−3
15
05−5
15
05−5
15
h
4
− h
3
→ h
4
→
−→
1 −11
−2
06−3
15
05−5
15
00 0
0
h
2
× 5 → h
2
h
3
× 6 → h
3
−→
−→
1 −11
−2
030−15
75
030−30
90
00 0
0
−→
h
3
− h
2
→ h
3
1 −11
−2
030−15
75
00−15
15
00 0
0
.
T`u
.
d
´othudu
.
o
.
.
chˆe
.
tu
.
o
.
ng d
u
.
o
.
ng
x
1
− x
2
+ x
3
= −2
30x
2
−15x
3
=75
−15x
3
=15
v`a do d
´othudu
.
o
.
.
c nghiˆe
.
m x
1
=1,x
2
=2,x
3
= −1.
V´ı du
.
4. Gia
’
ihˆe
.
phu
.
o
.
ng tr`ınh
x
1
+ x
2
+ x
3
+ x
4
+ x
5
= −1,
2x
1
+2x
2
+3x
4
+ x
5
=1,
2x
3
+2x
4
− x
5
=1,
−2x
3
+4x
4
− 3x
5
=7,
6x
3
+3x
4
− x
5
= −1.
150 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
Gia
’
i. 1) B˘a
`
ng c´ac ph´ep biˆe
´
ndˆo
’
iso
.
cˆa
´
p (chı
’
thu
.
.
chiˆe
.
n trˆen c´ac
h`ang !) ma trˆa
.
nmo
.
’
rˆo
.
ng
A d
u
.
o
.
.
cd
u
.
avˆe
`
ma trˆa
.
nbˆa
.
c thang
A −→
11111
−1
00−21−1
3
00 0 3−2
4
00000
0
00000
0
.
2) Ma trˆa
.
n n`ay tu
.
o
.
ng ´u
.
ng v´o
.
ihˆe
.
phu
.
o
.
ng tr`ınh
x
1
+ x
2
+ x
3
+ x
4
+ x
5
= −1,
−2x
3
+ x
4
− x
5
=3,
3x
4
− 2x
5
=4.
hˆe
.
n`ay tu
.
o
.
ng du
.
o
.
ng v´o
.
ihˆe
.
d˜a cho v`a c´o x
1
,x
3
,x
4
l `a ˆa
’
nco
.
so
.
’
, c`on
x
2
,x
5
l `a ˆa
’
ntu
.
.
do.
3) Chuyˆe
’
n c´ac sˆo
´
ha
.
ng ch´u
.
aˆa
’
ntu
.
.
do sang vˆe
´
pha
’
i ta c´o
x
1
+ x
3
+ x
4
= −1 −x
2
−x
5
,
−2x
3
+ x
4
=3+x
5
,
3x
4
=4+2x
5
.
4) Gia
’
ihˆe
.
n`ay (t`u
.
du
.
´o
.
i lˆen) ta thu d
u
.
o
.
.
c nghiˆe
.
mtˆo
’
ng qu´at
x
1
=
−3 − 3x
2
−x
5
2
,
x
3
=
−5 − x
5
6
,x
4
=
4+2x
5
3
·
V´ı d u
.
5. Gia
’
ihˆe
.
phu
.
o
.
ng tr`ınh
x
1
+3x
2
+5x
3
+7x
4
+9x
5
=1,
x
1
−2x
2
+3x
3
− 4x
4
+5x
5
=2,
2x
1
+11x
2
+12x
3
+25x
4
+22x
5
=4.
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 151
Gia
’
i. Ta thu
.
.
chiˆe
.
n c´ac ph´ep biˆe
´
nd
ˆo
’
iso
.
cˆa
´
p trˆen c´ac h`ang cu
’
ama
trˆa
.
nmo
.
’
rˆo
.
ng:
A =
13 5 7 9
1
1 −23−45
2
2 11 12 25 22
4
h
2
− h
1
→ h
2
h
3
−2h
1
→ h
3
−→
−→
13579
1
0 −5 −2 −11 −4
1
052114
2
h
3
+ h
2
→ h
3
−→
−→
13579
1
0 −5 −2 −11 −4
1
00000
3
T`u
.
d
´o suy r˘a
`
ng r(
A)=3;r(A) = 2 v`a do vˆa
.
y r(
A) >r(A)v`ahˆe
.
d˜a cho khˆong tu
.
o
.
ng th´ıch. .
V´ı du
.
6. Gia
’
i v`a biˆe
.
n luˆa
.
nhˆe
.
phu
.
o
.
ng tr`ınh theo tham sˆo
´
λ:
λx
1
+ x
2
+ x
3
=1,
x
1
+ λx
2
+ x
3
=1,
z
1
+ x
2
+ λx
3
=1.
Gia
’
i. Ta c´o
A =
λ 11
1 λ 1
11λ
⇒ detA =(λ + 2)(λ −1)
2
= D,
tiˆe
´
p theo dˆe
˜
d`ang thu d
u
.
o
.
.
c
D
x
1
= D
x
2
= D
x
3
=(λ − 1)
2
.
1
+
Nˆe
´
u D =0,t´u
.
cl`anˆe
´
u(λ + 2)(λ −1)
2
=0⇔ λ = −2v`aλ =1
th`ı hˆe
.
d˜a cho c´o nghiˆe
.
m duy nhˆa
´
t v`a theo c´ac cˆong th´u
.
c Cramer ta c´o
x
1
= x
2
= x
3
=
1
λ +2
·
152 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
2
+
Nˆe
´
u λ = −2th`ıD = 0 v`a ta c´o
A =
−21 1
1 −21
11−2
⇒ r(A)=2
−21
1 −2
=0
,
A =
−21 1
1
1 −21
1
11−2
1
.
B˘a
`
ng c´ach thu
.
.
chiˆe
.
n c´ac ph´ep biˆe
´
ndˆo
’
iso
.
cˆa
´
p trˆen c´ac ma trˆa
.
n
A ta
thu d
u
.
o
.
.
c r(
A)=3.
Do d
´ov´o
.
i λ = −2th`ır(
A) >r(A)v`ahˆe
.
vˆo nghiˆe
.
m.
3
+
Nˆe
´
u λ = 1 th`ı detA =0v`adˆe
˜
thˆa
´
yr˘a
`
ng r(
A)=r(A)=1< 3
(sˆo
´
ˆa
’
ncu
’
ahˆe
.
l`a 3). T`u
.
d´o suy ra hˆe
.
c´o vˆo sˆo
´
nghiˆe
.
m phu
.
thuˆo
.
c hai
tham sˆo
´
: x
1
+ x
2
+ x
3
=1.
V´ı d u
.
7. Gia
’
iv`abiˆe
.
n luˆa
.
nhˆe
.
phu
.
o
.
ng tr`ınh theo tham sˆo
´
λx
1
+ x
2
+ x
3
=1,
x
1
+ λx
2
+ x
3
= λ,
x
1
+ x
2
+ λx
3
= λ
2
.
Gia
’
i. Di
.
nh th´u
.
ccu
’
ahˆe
.
b˘a
`
ng
D =
λ 11
1 λ 1
11λ
=(λ − 1)
2
(λ +2).
Nˆe
´
u D =0⇔ λ
1
=1,λ
2
= −2th`ıhˆe
.
c´o nghiˆe
.
m duy nhˆa
´
t. Ta t´ınh
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 153
D
x
1
, D
x
2
, D
x
3
:
D
x
1
=
111
λλ1
λ
2
1 λ
= −(λ −1)
2
(λ +1),
D
x
2
=
λ 11
1 λ 1
1 λ
2
λ
=(λ − 1)
2
,
D
x
3
=
λ 11
1 λλ
11λ
2
=(λ − 1)
2
(λ +1)
2
.
T`u
.
d´o theo cˆong th´u
.
c Cramer ta thu du
.
o
.
.
c
x
1
= −
λ +1
λ +2
,x
2
=
1
λ +2
,x
3
=
(λ +1)
2
λ +2
·
Ta c`on x´et gi´a tri
.
λ =1v`aλ = −2.
Khi λ =1hˆe
.
d
˜a cho tro
.
’
th`anh
x
1
+ x
2
+ x
3
=1,
x
1
+ x
2
+ x
3
=1,
x
1
+ x
2
+ x
3
=1.
Hˆe
.
n`ay c´o vˆo sˆo
´
nghiˆe
.
m phu
.
thuˆo
.
c hai tham sˆo
´
.Nˆe
´
ud
˘a
.
t x
2
= α,
x
3
= β th`ı
x
1
=1 − α − β,
α, β ∈ R,
v`a nhu
.
vˆa
.
ytˆa
.
pho
.
.
p nghiˆe
.
mc´othˆe
’
viˆe
´
tdu
.
´o
.
ida
.
ng (1 − α −
β; α; β; ∀α, β ∈ R).
Khi λ = −2th`ıhˆe
.
d
˜a cho tro
.
’
th`anh
−2x
1
+ x
2
+ x
2
=2,
x
1
− 2x
2
+ x
3
= −2,
x
1
+ x
2
− 2x
3
=4.
154 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
B˘a
`
ng c´ach cˆo
.
ng ba phu
.
o
.
ng tr`ınh la
.
iv´o
.
i nhau ta thˆa
´
yngayhˆe
.
d
˜acho
vˆo nghiˆe
.
m.
V´ı d u
.
8. X´et hˆe
.
phu
.
o
.
ng tr`ınh
x
1
+2x
2
+ λx
3
=3,
3x
1
− x
2
−λx
3
=2,
2x
1
+ x
2
+3x
3
= µ.
V´o
.
i gi´a tri
.
n`ao cu
’
a c´ac tham sˆo
´
λ v`a µ th`ı
1) hˆe
.
c´o nghiˆe
.
m duy nhˆa
´
t?
2) hˆe
.
vˆo nghiˆe
.
m?
3) hˆe
.
c´o vˆo sˆo
´
nghiˆe
.
m?
Gia
’
i. Ta viˆe
´
t c´ac ma trˆa
.
n
A =
12 λ
3 −1 −λ
21 3
;
A =
12 λ
3
3 −1 −λ
2
21 3
µ
Ta c´o
D = detA =
12 λ
3 −1 −λ
21 3
=2λ − 21.
T`u
.
d
´o
1
+
Hˆe
.
d˜a cho c´o nghiˆe
.
m duy nhˆa
´
t khi v`a chı
’
khi
detA =0⇔ λ =
21
2
,µt`uy ´y.
2
+
Dˆe
’
hˆe
.
vˆo nghiˆe
.
mdˆa
`
u tiˆen n´o pha
’
i tho
’
a m˜an
detA =0⇔ λ =
21
2
·
Khi λ =
21
2
th`ı detA = 0 v`a do vˆa
.
y
r(A) < 3.
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 155
V`ıdi
.
nh th´u
.
c
12
3 −1
= −7 =0nˆen:
r(A)=2 khi λ =
21
2
·
Theo d
i
.
nh l´y Kronecker-Capelli hˆe
.
d˜a cho vˆo nghiˆe
.
m khi v`a chı
’
khi
r(
A) >r(A)=2.
Ta t`ım d
iˆe
`
ukiˆe
.
ndˆe
’
hˆe
.
th ´u
.
c n`ay tho
’
a m˜an. Cu
.
thˆe
’
l`a t`ım r(
A) khi
λ =
21
2
.Tac´o
A =
12
21
2
3
3 −1 −
21
2
2
21 3
µ
h
1
× 2 → h
1
h
2
× 2 → h
2
−→
−→
24 21
6
6 −2 −21
4
21 3
µ
h
2
− 3h
1
→ h
2
h
3
− h
1
→ h
3
−→
−→
24 21
6
0 −14 −84
−14
0 −3 −18
µ − 6
h
2
×
−1
14
→ h
2
−→
−→
24 21
6
01 6
1
0 −3 −18
µ − 6
h
3
+3h
1
→ h
3
−→
2421
6
01 6
1
00 0
µ −3
T`u
.
kˆe
´
t qua
’
biˆe
´
ndˆo
’
i ta thu du
.
o
.
.
c
r(
A)=
2nˆe
´
u µ =3,
3nˆe
´
u µ =3,
156 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
V`ı r( A) = 2 nˆen hˆe
.
d˜a cho vˆo nghiˆe
.
mnˆe
´
u
λ =
21
2
v`a µ =3.
3
+
Hˆe
.
d˜a cho c´o vˆo sˆo
´
nghiˆe
.
m khi v`a chı
’
khi
r(
A)=r(A)=r<3
t´u
.
c l`a khi ha
.
ng cu
’
a A v`a
A b˘a
`
ng nhau nhu
.
ng b´e ho
.
nsˆo
´
ˆa
’
ncu
’
ahˆe
.
l`a
3. T`u
.
lˆa
.
p luˆa
.
n trˆen suy r˘a
`
ng hˆe
.
c´o vˆo sˆo
´
nghiˆe
.
mnˆe
´
u
r(
A)=r(A)=2⇔
λ =
21
2
,
µ =3.
Khi d
´ohˆe
.
d˜a cho tu
.
o
.
ng du
.
o
.
ng v´o
.
ihˆe
.
2x
1
+4x
2
=6−21α,
6x
1
− 2x
2
=4+21α.
α = x
3
,
v`a nghiˆe
.
mcu
’
an´ol`a
1+
3
2
α, 1 − 6α, α
∀α ∈ R
.
B
`
AI T
ˆ
A
.
P
Gia
’
i c´ac hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
1.
6x
1
+3x
2
+4x
3
=3;
3x
1
− x
2
+2x
3
=5.
(D
S. x
2
= −
7
5
, x
3
=
18 −15x
1
10
, x
1
t`uy ´y)
2.
x
1
− x
2
+ x
3
= −1,
2x
1
+ x
2
− x
3
=5.
(D
S. x
1
=
4+2x
3
3
, x
2
=
7 −x
3
3
, x
3
t`uy ´y)
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 157
3.
x
1
+ x
2
+2x
3
+ x
4
=1,
x
1
−2x
2
− x
4
= −2.
(DS. x
3
=
1
2
(−2x
1
+ x
2
− 1), x
4
= x
1
− 2x
2
+2,
x
1
,x
2
t`uy ´y)
4.
x
1
+5x
2
+4x
3
+3x
4
=1,
2x
1
− x
2
+2x
3
− x
4
=0,
5x
1
+3x
2
+8x
3
+ x
4
=1.
(D
S. x
1
= −
14
11
x
3
+
2
11
x
4
+
1
11
, x
2
= −
6
11
x
3
−
7
11
x
4
+
2
11
,
x
3
,x
4
t`uy ´y)
5.
3x
1
+5x
2
+2x
3
+4x
4
=3,
2x
1
+3x
2
+4x
3
+5x
4
=1,
5x
1
+9x
2
− 2x
3
+2x
4
=9.
(D
S. Hˆe
.
vˆo nghiˆe
.
m)
6.
x
1
+2x
2
+3x
3
=14,
3x
1
+2x
2
+ x
3
=10,
x
1
+ x
2
+ x
3
=6,
2x
1
+3x
2
− x
3
=5,
x
1
+ x
2
=3.
(D
S. x
1
=1,x
2
=2,x
3
=3)
7.
x
1
+3x
2
− 2x
3
+ x
4
+ x
5
=1,
x
1
+3x
2
− x
3
+3x
4
+2x
5
=3,
x
1
+3x
2
− 3x
3
− x
4
=2.
(D
S. Hˆe
.
vˆo nghiˆe
.
m)
8.
5x
1
+ x
2
− 3x
3
= −6,
2x
1
− 5x
2
+7x
3
=9,
4x
1
+2x
2
− 4x
3
= −7,
5x
1
− 2x
2
+2x
3
=1.
(D
S. x
1
= −
1
3
, x
2
=
1
6
, x
3
=
3
2
)
158 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
9.
x
1
+ x
2
+ x
3
+ x
4
=1,
x
1
+ x
2
− 2x
3
− x
4
=0,
x
1
+ x
2
− 4x
3
+3x
4
=2,
x
1
+ x
2
+7x
3
+5x
4
=3.
(D
S. x
1
=
2 − 3x
2
− 2x
4
3
, x
3
=
1 −2x
4
3
, x
2
,x
4
t`uy ´y)
10.
x
1
+2x
2
+3x
3
+4x
4
=5,
x
2
+2x
3
+3x
4
=1,
x
1
+3x
3
+4x
4
=2,
x
1
+ x
2
+5x
3
+6x
4
=1.
(D
S. x
1
=
15
4
, x
2
=
3
2
, x
3
= −
13
4
, x
4
=2)
11.
x
1
+2x
2
+3x
3
+4x
4
=30,
−x
1
+2x
2
−3x
3
+4x
4
=10,
x
2
− x
3
+ x
4
=3,
x
1
+ x
2
+ x
3
+ x
4
=10.
(D
S. x
1
=1,x
2
=2,x
3
=3,x
4
=4)
12.
5x
1
+ x
2
− 3x
3
= −6,
2x
1
− 5x
2
+7x
3
=9,
4x
1
+2x
2
− 4x
3
= −7,
5x
1
− 2x
2
+2x
3
=1.
(D
S. x
1
= −
1
3
, x
2
=
1
6
, x
3
=
3
2
)
13.
x
1
− x
2
+ x
3
− x
4
=4,
x
1
+ x
2
+2x
3
+3x
4
=8,
2x
1
+4x
2
+5x
3
+10x
4
=20,
2x
1
− 4x
2
+ x
3
− 6x
4
=4.
(D
S. x
1
=6−
3
2
x
3
− x
4
, x
2
=2−
1
2
x
3
− 2x
4
, x
3
v`a x
4
t`uy ´y)
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 159
14.
x
1
− 2x
2
+3x
3
−4x
4
=2,
3x
1
+3x
2
− 5x
3
+ x
4
= −3,
−2x
1
+ x
2
+2x
3
−3x
4
=5,
3x
1
+3x
3
− 10x
4
=8.
(D
S. Hˆe
.
vˆo nghiˆe
.
m)
15.
x
1
+2x
2
+3x
3
− 2x
4
=1,
2x
1
−x
2
− 2x
3
−3x
4
=2,
3x
1
+2x
2
− x
3
+2x
4
= −5,
2x
1
− 3x
2
+2x
3
+ x
4
=11.
(D
S. x
1
=
2
3
, x
2
= −
43
18
, x
3
=
13
9
, x
4
= −
7
18
)
16.
x
1
+2x
2
− 3x
3
+5x
4
=1,
x
1
+3x
2
− 13x
3
+22x
4
= −1,
3x
1
+5x
2
+ x
3
− 2x
4
=5,
2x
1
+3x
2
+4x
3
− 7x
4
=4.
(D
S. x
1
= −17x
3
+29x
4
+5,x
2
=10x
3
−17x
4
− 2, x
3
, x
4
t`uy ´y)
17.
x
1
− 5x
2
− 8x
3
+ x
4
=3,
3x
1
+ x
2
− 3x
3
− 5x
4
=1,
x
1
− 7x
3
+2x
4
= −5,
11x
2
+20x
3
− 9x
4
=2.
(D
S. Hˆe
.
vˆo nghiˆe
.
m)
18.
x
2
−3x
3
+4x
4
= −5,
x
1
− 2x
3
+3x
4
= −4,
3x
1
+2x
2
− 5x
4
=12,
4x
1
+3x
2
− 5x
3
=5.
(D
S. x
1
=1,x
2
=2,x
3
=1,x
4
= −1)
Kha
’
o s´at t´ınh tu
.
o
.
ng th´ıch cu
’
a c´ac hˆe
.
phu
.
o
.
ng tr`ınh sau d
ˆay
19.
x
1
+ x
2
+ x
3
− x
4
=0,
x
1
− x
2
−x
3
+ x
4
=1,
x
1
+3x
2
+3x
3
− 3x
4
=0.
160 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
(DS. Hˆe
.
khˆong tu
.
o
.
ng th´ıch)
20.
x
1
+ x
2
+ x
3
+ x
4
=1,
x
1
+ x
2
+2x
3
+ x
4
=0,
x
1
+ x
2
− x
3
+ x
4
=3.
(DS. Hˆe
.
tu
.
o
.
ng th´ıch)
21.
x
1
− 2x
2
+ x
3
+ x
4
=1,
x
1
− 2x
2
+ x
3
− x
4
= −1,
x
1
− 2x
2
+ x
3
+5x
4
=5.
(DS. Hˆe
.
tu
.
o
.
ng th´ıch)
22.
x
1
+ x
2
+ x
3
+ x
4
+ x
5
=7,
3x
1
+2x
2
+ x
3
+ x
4
− 3x
5
= −2,
x
2
+2x
3
+2x
4
+6x
5
=23,
5x
1
+4x
2
+3x
3
+3x
4
− x
5
=12.
(D
S. Hˆe
.
tu
.
o
.
ng th´ıch)
23.
2x
1
+ x
2
− x
3
+ x
4
=1,
3x
1
− 2x
2
+2x
3
− 3x
4
=2,
5x
1
+ x
2
−x
3
+2x
4
= −1,
2x
1
− x
2
+ x
3
−3x
4
=4.
(D
S. Hˆe
.
khˆong tu
.
o
.
ng th´ıch)
24.
3x
1
+ x
2
− 2x
3
+ x
4
−x
5
=1,
2x
1
− x
2
+7x
3
− 3x
4
+5x
5
=2,
x
1
+3x
2
− 2x
3
+5x
4
−7x
5
=3,
3x
1
− 2x
2
+7x
3
− 5x
4
+8x
5
=3.
(D
S. Hˆe
.
khˆong tu
.
o
.
ng th´ıch)
25.
5x
1
+7x
2
+4x
3
+5x
4
− 8x
5
+3x
6
=1,
2x
1
+3x
2
+3x
3
− 6x
4
+7x
5
− 9x
6
=2,
7x
1
+9x
2
+3x
3
+7x
4
− 5x
5
− 8x
6
=5.
(DS. Hˆe
.
tu
.
o
.
ng th´ıch)
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 161
Kha
’
o s´at t´ınh tu
.
o
.
ng th´ıch v`a gia
’
i c´ac hˆe
.
phu
.
o
.
ng tr`ınh (nˆe
´
uhˆe
.
tu
.
o
.
ng th´ıch)
26.
2x
1
− x
2
+3x
3
=3,
3x
1
+ x
2
− 5x
3
=0,
4x
1
− x
2
+ x
4
=3,
x
1
+3x
2
− 13x
3
= −6.
(D
S. x
1
=1,x
2
=2,x
3
=1)
27.
2x
1
−x
2
+ x
3
−x
4
=1,
2x
1
− x
2
− 3x
4
=2,
3x
1
− x
3
+ x
4
= −3,
2x
1
+2x
2
− 2x
3
+5x
4
= −6.
(D
S. x
1
=0,x
2
=2,x
3
=
5
3
, x
4
= −
4
3
)
28.
2x
1
+ x
2
+ x
3
=2,
x
1
+3x
2
+ x
3
=5,
x
1
+ x
2
+5x
3
= −7,
2x
1
+3x
2
− 5x
3
=14.
(D
S. x
1
=1,x
2
=2,x
3
= −2)
29.
2x
1
+3x
2
+4x
3
+3x
4
=0,
4x
1
+6x
2
+9x
3
+8x
4
= −3,
6x
1
+9x
2
+9x
3
+4x
4
=8.
(D
S. x
1
=
7
2
−
3x
2
2
, x
3
= −1, x
4
= −1, x
2
t`uy ´y)
30.
3x
1
+3x
2
− 6x
3
− 2x
4
= −1,
6x
1
+ x
2
− 2x
4
= −2,
6x
1
− 7x
2
+21x
3
+4x
4
=3,
9x
1
+4x
2
+2x
4
=3,
12x
1
− 6x
2
+21x
3
+2x
4
=1.
(D
S. x
1
=
7
5
, x
2
= −4, x
3
= −
11
5
, x
4
=
16
5
)
162 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
31.
x
1
+ x
2
+2x
3
+3x
4
=1,
3x
1
− x
2
−x
3
− 2x
4
= −4,
2x
1
+3x
2
−x
3
− x
4
= −6,
x
1
+2x
2
+3x
3
−x
4
= −4.
(D
S. x
1
= x
2
= −1, x
3
=0,x
4
=1)
32.
x
1
+2x
2
+3x
3
−2x
4
=6,
2x
1
− x
2
−2x
3
− 3x
4
=8,
3x
1
+2x
2
− x
3
+2x
4
=4,
2x
1
− 3x
2
+2x
3
+ x
4
= −8.
(D
S. x
1
=1,x
2
=2,x
3
= −1, x
4
= −2)
33.
x
2
− 3x
3
+4x
4
= −5,
x
1
− 2x
3
+3x
4
= −4,
3x
1
+2x
2
− 5x
4
=12,
4x
1
+3x
2
− 5x
3
=5.
(D
S. x
1
=1,x
2
=2,x
3
=1,x
4
= −1)
34.
x
1
+ x
2
− x
3
+ x
4
=4,
2x
1
− x
2
+3x
3
−2x
4
=1,
x
1
− x
3
+2x
4
=6,
3x
1
− x
2
+ x
3
−x
4
=0.
(D
S. x
1
=1,x
2
=2,x
3
=3,x
4
=4)
35.
x
1
+ x
2
+ x
3
+ x
4
=0,
x
2
+ x
3
+ x
4
+ x
5
=0,
x
1
+2x
2
+3x
4
=2,
x
2
+2x
3
+3x
4
= −2,
x
3
+2x
4
+3x
5
=2.
(D
S. x
1
=1,x
2
= −1, x
3
=1,x
4
= −1, x
5
=1)
4.2. Hˆe
.
t`uy ´y c´ac phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh 163
36.
3x
1
−x
2
+ x
3
+2x
5
=18,
2x
1
−5x
2
+ x
4
+ x
5
= −7,
x
1
− x
4
+2x
5
=8,
2x
2
+ x
3
+ x
4
−x
5
=10,
x
1
+ x
2
− 3x
3
+ x
4
=1.
(D
S. x
1
=5,x
2
=4,x
3
=3,x
4
=1,x
5
=2)
