bài giảng đại số tuyến tính và giải tích
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Chu
.
o
.
ng 1. MA TR
ˆ
A
.
N-D
-
I
.
NH TH
´
U
.
C (8+4)
I. Ma trˆa
.
n
* Cho m,n nguyˆen du
.
o
.
ng. Ta go
.
i ma trˆa
.
nc˜o
.
m × n l`a mˆo
.
tba
’
ng sˆo
´
gˆo
`
m m × n
sˆo
´
thu
.
.
cd¯u
.
o
.
.
cviˆe
´
t th`anh m h`ang, n cˆo
.
t c´o da
.
ng nhu
.
sau:
(a
i,j
)
m×n
=
a
1,1
a
1,2
a
1,n
a
2,1
a
2,2
a
2,n
a
m,1
a
m,2
a
m,n
trong d¯´o c´ac sˆo
´
thu
.
.
c
a
i,j
,i= 1,m,j = 1,n
d¯ u
.
o
.
.
cgo
.
il`ac´ac phˆa
`
ntu
.
’
cu
’
a ma trˆa
.
n,chı
’
sˆo
´
i chı
’
h`ang v`a chı
’
sˆo
´
j chı
’
cˆo
.
tcu
’
a
phˆa
`
ntu
.
’
ma trˆa
.
n.
* Ma trˆa
.
nc˜o
.
1 × n d¯ u
.
o
.
.
cgo
.
il`ama trˆa
.
n h`ang, ma trˆa
.
nc˜o
.
m × 1d¯u
.
o
.
.
cgo
.
il`ama
trˆa
.
ncˆo
.
t, ma trˆa
.
nc˜o
.
n ×n d¯ u
.
o
.
.
cgo
.
il`ama trˆa
.
n vuˆong cˆa
´
p n.
*Trˆen ma trˆa
.
n vuˆong cˆa
´
p n,d¯u
.
`o
.
ng ch´eo gˆo
`
m c´ac phˆa
`
ntu
.
’
a
i,i
,i= 1,n
d¯ u
.
o
.
.
cgo
.
il`ad¯ u
.
`o
.
ng ch´eo ch´ınh,d¯u
.
`o
.
ng ch´eo gˆo
`
m c´ac phˆa
`
ntu
.
’
a
i,n+1−i
,i= 1,n
d¯ u
.
o
.
.
cgo
.
il`ad¯ u
.
`o
.
ng ch´eo phu
.
cu
’
a ma trˆa
.
n.
* Ma trˆa
.
n vuˆong cˆa
´
p n c´o c´ac phˆa
`
ntu
.
’
n˘a
`
m ngo`ai d¯u
.
`o
.
ng ch´eo ch´ınh d¯ˆe
`
ub˘a
`
ng 0,
ngh˜ıa l`a:
a
i,j
=0, ∀i = j
d¯ u
.
o
.
.
cgo
.
il`ama trˆa
.
n ch´eo.
* Ma trˆa
.
n ch´eo c´o
a
i,i
=1,i = 1,n
d¯ u
.
o
.
.
cgo
.
il`ama trˆa
.
nd¯o
.
nvi
.
cˆa
´
p n,k´yhiˆe
.
u I
n
.
* Ma trˆa
.
nc˜o
.
m ×n c´o
a
i,j
=0, ∀i, j : i>j
d¯ u
.
o
.
.
cgo
.
il`ama trˆa
.
nbˆa
.
c thang.
* Ma trˆa
.
nc˜o
.
m × n c´o c´ac phˆa
`
ntu
.
’
d¯ ˆe
`
ub˘a
`
ng 0 d¯u
.
o
.
.
cgo
.
il`ama trˆa
.
n khˆong,k´y
hiˆe
.
u0
m,n
.
*Tago
.
i ma trˆa
.
n chuyˆe
’
nvi
.
A
T
=(a
j,i
)
n×m
=
a
1,1
a
2,1
a
m,1
a
1,2
a
2,2
a
m,2
a
1,n
a
2,n
a
m,n
Typeset by A
M
S-T
E
X
2
cu
’
a ma trˆa
.
n
A =(a
i,j
)
m×n
=
a
1,1
a
1,2
a
1,n
a
2,1
a
2,2
a
2,n
a
m,1
a
m,2
a
m,n
l`a ma trˆa
.
n c´o d¯u
.
o
.
.
ct`u
.
A b˘a
`
ng c´ach chuyˆe
’
n h`ang th`anh cˆo
.
t, cˆo
.
t th`anh h`ang.
* Hai ma trˆa
.
nc`ung c˜o
.
(a
i,j
)
m×n
v`a (b
i,j
)
m×n
d¯ u
.
o
.
.
cgo
.
il`ab˘a
`
ng nhau nˆe
´
u c´ac phˆa
`
n
tu
.
’
o
.
’
t`u
.
ng vi
.
tr´ıd¯ˆe
`
ub˘a
`
ng nhau:
a
i,j
= b
i,j
, ∀i = 1,m,∀j = 1,n.
+Tˆo
’
ng (hiˆe
.
u) cu
’
a hai ma trˆa
.
n c`ung c˜o
.
m × n l`a mˆo
.
t ma trˆa
.
nc˜o
.
m × n, trong d¯´o
phˆa
`
ntu
.
’
cu
’
a ma trˆa
.
ntˆo
’
ng (hiˆe
.
u) l`a tˆo
’
ng (hiˆe
.
u) c´ac phˆa
`
ntu
.
’
o
.
’
vi
.
tr´ıtu
.
o
.
ng ´u
.
ng:
(c
i,j
)
m×n
=(a
i,j
)
m×n
± (b
i,j
)
m×n
v´o
.
i
c
i,j
= a
i,j
± b
i,j
, ∀i = 1,m,∀j = 1,n.
+T´ıch vˆo hu
.
´o
.
ng cu
’
asˆo
´
thu
.
.
c α v´o
.
i ma trˆa
.
nc˜o
.
m ×n l`a ma trˆa
.
nc˜o
.
m ×n, trong d¯´o
mˆo
˜
i phˆa
`
ntu
.
’
l`a t´ıch cu
’
a α v´o
.
i phˆa
`
ntu
.
’
o
.
’
vi
.
tr´ıtu
.
o
.
ng ´u
.
ng cu
’
a ma trˆa
.
n ban d¯ˆa
`
u:
(c
i,j
)
m×n
= α.(a
i,j
)
m×n
v´o
.
i
c
i,j
= α.b
i,j
, ∀i = 1,m,∀j = 1,n.
+T´ıch vˆo hu
.
´o
.
ng c´o t´ınh phˆan bˆo
´
v´o
.
i ph´ep cˆo
.
ng c´ac ma trˆa
.
n: α.(A+B)=α.A+α.B,
v´o
.
i ph´ep cˆo
.
ng c´ac hˆe
.
sˆo
´
:(α + β).A = α.A + β.B, c´o t´ınh kˆe
´
tho
.
.
p:
α.(β · A)=(α.β) · A.
+T´ıch cu
’
a hai ma trˆa
.
n A =(a
i,j
)
m×n
v`a B =(b
j,k
)
n×q
l`a ma trˆa
.
n
C = A × B =(c
i,k
)
m×q
,
v´o
.
i
c
i,k
=
n
j=1
a
i,j
b
j,k
, ∀i = 1,m,∀k = 1,q.
V´ıdu
.
.
132
247
356
×
13
1 −1
32
=
1.1+3.1+2.31.3 −3.1+2.2
2.1+4.1+7.32.3 −4.1+7.2
3.1+5.1+6.33.3 −5.1+6.2
=
10 4
27 16
26 16
3
+Ph´ep nhˆan hai ma trˆa
.
n c´o t´ınh kˆe
´
tho
.
.
p: A × (B × C)=(A × B) × C, t´ınh phˆan
phˆo
´
id¯ˆo
´
iv´o
.
i ph´ep cˆo
.
ng:
A × (B + C)=A × B + A × C;(A + B) × C = A × C + B × C.
Ngo`ai ra, nˆe
´
u A c´o c˜o
.
m × n,th`ı
A × I
n
= I
m
×A = A.
II. D
-
i
.
nh th´u
.
c
* Cho E = {1, 2, 3, ,n}.Tago
.
i ho´an vi
.
cu
’
atˆa
.
p E l`a mˆo
.
t song ´anh f : E → E,
k´yhiˆe
.
u
f :
12 n
f(1) f(2) f(n)
hay
(f(1),f(2), ,f(n))
(c´o tˆa
´
tca
’
n! ho´an vi
.
kh´ac nhau).
V´ıdu
.
. Cho E = {1, 2, 3}.
´
Anh xa
.
f : E → E x´ac d¯i
.
nh bo
.
’
i: f(1) = 1,f(2) = 3,f(3) = 2
l`a mˆo
.
t ho´an vi
.
cu
’
a E,k´yhiˆe
.
ul`a
123
132
ho˘a
.
c
(1, 3, 2).
* Cho mˆo
.
t ho´an vi
.
f :
12 n
f(1) f(2) f(n)
ta th`anh lˆa
.
p c´ac c˘a
.
pth´u
.
tu
.
.
(f(i),f(j)) , ∀i = j,
s˜ec´oC
2
n
c˘a
.
pth´u
.
tu
.
.
nhu
.
thˆe
´
;mˆo
.
tc˘a
.
p(f(i),f(j)) d¯u
.
o
.
.
cgo
.
il`anghi
.
ch thˆe
´
nˆe
´
u
(i − j)(f(i) −f(j)) < 0.
Go
.
i N(f) l`a sˆo
´
c´ac nghi
.
ch thˆe
´
cu
’
a ho´an vi
.
f (c´o trong C
2
n
c˘a
.
pth´u
.
tu
.
.
trˆen).
V´ıdu
.
. T`ım sˆo
´
nghi
.
ch thˆe
´
cu
’
a ho´an vi
.
f :
12345
32154
.
4
T`u
.
ho´an vi
.
n`ay, ta c´o c´ac c˘a
.
pth´u
.
tu
.
.
(3, 2), (3, 1), (3, 5), (3, 4), (2, 1), (2, 5), (2, 4), (1, 5), (1, 4), (5, 4),
trong d¯´o ta c´o c´ac nghi
.
ch thˆe
´
:
(3, 2), (3, 1), (2, 1), (5, 4),
suy ra N(f)=4
* Cho ma trˆa
.
n(A)
n,n
. D
-
i
.
nh th´u
.
ccu
’
a A l`a mˆo
.
tsˆo
´
thu
.
.
c, k´yhiˆe
.
u v`a x´ac d¯i
.
nh nhu
.
sau:
det(A)=
f∈S
n
(−1)
N(f )
a
1,f(1)
a
2,f(2)
a
n,f(n)
trong d¯´o S
n
l`a tˆa
.
ptˆa
´
tca
’
n! ho`an vi
.
cu
’
a n phˆa
`
ntu
.
’
{1, 2, ,n}.Nhu
.
vˆa
.
y, d¯i
.
nh
th ´u
.
ccu
’
a ma trˆa
.
n A l`a mˆo
.
tsˆo
´
:
+b˘a
`
ng tˆo
’
ng d¯a
.
isˆo
´
cu
’
a n!ha
.
ng tu
.
’
da
.
ng
a
1,f(1)
a
2,f(2)
a
n,f(n)
+mˆo
˜
iha
.
ng tu
.
’
l`a t´ıch cu
’
a n phˆa
`
ntu
.
’
a
i,j
m`a mˆo
˜
i h`ang, mˆo
˜
icˆo
.
t pha
’
ic´omˆo
.
t
v`a chı
’
mˆo
.
t phˆa
`
ntu
.
’
tham gia v`ao t´ıch d¯´o.
+dˆa
´
ucu
’
amˆo
˜
iha
.
ng tu
.
’
phu
.
thuˆo
.
c v`ao sˆo
´
nghi
.
ch thˆe
´
cu
’
a ho´an vi
.
tu
.
o
.
ng ´u
.
ng.
*Tago
.
i d¯ i
.
nh th´u
.
ccˆa
´
p2l`a gi´a tri
.
t´ınh d¯u
.
o
.
.
ct`u
.
ba
’
ng 2 h`ang, 2 cˆo
.
tnhu
.
sau:
a
1,1
a
1,2
a
2,1
a
2,2
= a
1,1
a
2,2
− a
2,1
a
1,2
*Tago
.
i d¯ i
.
nh th´u
.
ccˆa
´
p3l`a gi´a tri
.
t´ınh d¯u
.
o
.
.
ct`u
.
ba
’
ng 3 h`ang, 3 cˆo
.
tnhu
.
sau:
a
1,1
a
1,2
a
1,3
a
2,1
a
2,2
a
2,3
a
3,1
a
3,2
a
3,3
= a
1,1
a
2,2
a
3,3
+ a
2,1
a
3,2
a
1,3
+ a
3,1
a
1,2
a
2,3
− a
3,1
a
2,2
a
1,3
− a
2,1
a
1,2
a
3,3
− a
1,1
a
3,2
a
2,3
+D
-
ˆe
’
t´ınh nhanh d¯i
.
nh th´u
.
ccˆa
´
p 3, ta viˆe
´
tcˆo
.
tth´u
.
nhˆa
´
t v`a th´u
.
hai tiˆe
´
p theo v`ao bˆen
pha
’
iba
’
ng n´oi trˆen:
a
1,1
a
1,2
a
1,3
a
1,1
a
1,2
a
2,1
a
2,2
a
2,3
a
2,1
a
2,2
a
3,1
a
3,2
a
3,3
a
3,1
a
3,2
th`ı 3 phˆa
`
ntu
.
’
lˆa
´
ydˆa
´
ucˆo
.
ng l`a t´ıch c´ac phˆa
`
ntu
.
’
n˘a
`
m trˆen c´ac d¯u
.
`o
.
ng ch´eo song song
v´o
.
id¯u
.
`o
.
ng ch´eo ch´ınh, ba phˆa
`
ntu
.
’
lˆa
´
ydˆa
´
utr`u
.
l`a t´ıch c´ac phˆa
`
ntu
.
’
n˘a
`
m trˆen c´ac
d¯ u
.
`o
.
ng ch´eo song song v´o
.
id¯u
.
`o
.
ng ch´eo phu
.
(quy t˘a
´
c Serrhus)
5
*Tago
.
i d¯ i
.
nh th´u
.
ccˆa
´
p n l`a gi´a tri
.
t´ınh d¯u
.
o
.
.
ct`u
.
ba
’
ng:
a
1,1
a
1,2
a
1,n
a
2,1
a
2,2
a
2,n
a
n,1
a
n,2
a
n,n
= a
1,1
D
1
− a
2,1
D
2
+ ···+(−1)
n+1
a
n,1
D
n
trong d¯´o D
k
l`a d¯i
.
nh th´u
.
ccˆa
´
p n − 1thud¯u
.
o
.
.
ct`u
.
ba
’
ng d¯˜a cho b˘a
`
ng c´ach bo
’
cˆo
.
t
th ´u
.
nhˆa
´
t v`a h`ang th´u
.
k, k = 1,n.
V´ıdu
.
.
1452
0331
2040
0021
=1.
331
040
021
− 0.
452
040
021
+2.
452
331
021
− 0.
452
331
040
=14
+D
-
i
.
nh th ´u
.
c khˆong thay d¯ˆo
’
inˆe
´
u ta d¯ˆo
’
i h`ang th`anh cˆo
.
t
+D
-
i
.
nh th ´u
.
cd¯ˆo
’
idˆa
´
unˆe
´
u ta d¯ˆo
’
ichˆo
˜
hai h`ang (ho˘a
.
c hai cˆo
.
t) v´o
.
i nhau
+D
-
i
.
nh th ´u
.
c c´o hai h`ang (ho˘a
.
c hai cˆo
.
t) ty
’
lˆe
.
v´o
.
i nhau nhau th`ı b˘a
`
ng 0
+Th`u
.
asˆo
´
chung cu
’
amˆo
.
t h`ang hay cˆo
.
tc´othˆe
’
d¯ u
.
a ra ngo`ai dˆa
´
ucu
’
ad¯i
.
nh th ´u
.
c
+D
-
i
.
nh th´u
.
c khˆong thay d¯ˆo
’
inˆe
´
u ta d¯ˆo
`
ng th`o
.
icˆo
.
ng v`ao c´ac phˆa
`
ntu
.
’
cu
’
amˆo
.
t h`ang
(hay mˆo
.
tcˆo
.
t) n`ao d¯´o c´ac phˆa
`
ntu
.
’
cu
’
amˆo
.
t h`ang (hay mˆo
.
tcˆo
.
t) kh´ac nhˆan v´o
.
ic`ung
mˆo
.
tsˆo
´
.
V´ıdu
.
. Gia
’
iphu
.
o
.
ng tr`ınh:
11 1 1
11−x 1 1
112−x 1
11 1 n− x
=0.
D
-
i
.
nh th´u
.
co
.
’
vˆe
´
tr´ai cu
’
aphu
.
o
.
ng tr`ınh l`a d¯a th´u
.
cbˆa
.
c n nˆen c´o khˆong qu´a n nghiˆe
.
m
kh´ac nhau. Thay x =0,x =1,x =2, ,x = n − 1 v`ao d¯i
.
nh th´u
.
c, ta luˆon c´o hai
h`ang v´o
.
i c´ac phˆa
`
ntu
.
’
b˘a
`
ng 1, nˆen d¯i
.
nh th´u
.
cb˘a
`
ng 0. Vˆa
.
yphu
.
o
.
ng tr`ınh c´o n nghiˆe
.
m
x =0,x=1,x =2, ,x = n − 1.
*D
-
i
.
nh th´u
.
ccu
’
a ma trˆa
.
n vuˆong A =(a
i,j
)
n×n
,k´yhiˆe
.
u det(A) l`a d¯i
.
nh th´u
.
ccˆa
´
p n
cu
’
aba
’
ng
a
1,1
a
1,2
a
1,n
a
2,1
a
2,2
a
2,n
a
n,1
a
n,2
a
n,n
v`a c´o t´ınh chˆa
´
t:
+ det(αA)=α
n
. det ( A)
+ det(A × B) = det(A). det(B)
III. Ma trˆa
.
n nghi
.
ch d¯a
’
o
6
* Ma trˆa
.
n A =(a
i,j
)
n×n
d¯ u
.
o
.
.
cgo
.
il`ama trˆa
.
n kha
’
nghi
.
ch nˆe
´
utˆo
`
nta
.
i ma trˆa
.
n A
−1
sao cho:
A × A
−1
= A
−1
× A = I
n
.
Khi d¯´o, ma trˆa
.
n A
−1
d¯ u
.
o
.
.
cgo
.
il`ama trˆa
.
n nghi
.
ch d¯a
’
o cu
’
a A.
+ Ma trˆa
.
n A kha
’
nghi
.
ch khi v`a chı
’
khi
det A =0.
* Cho A =(a
i,j
)
m×n
.Mˆo
.
t d¯ i
.
nh th´u
.
c con cˆa
´
p k (1 ≤ k ≤ n) cu
’
a A l`a mˆo
.
td¯i
.
nh
th ´u
.
cta
.
o th`anh t `u
.
ma trˆa
.
n A b˘a
`
ng c´ach bo
’
d¯ i m − k h`ang v`a n − k cˆo
.
t.
* Cho ma trˆa
.
n vuˆong cˆa
´
p n kha
’
nghi
.
ch
A =
a
1,1
a
1,2
a
1,n
a
2,1
a
2,2
a
2,n
a
n,1
a
n,2
a
n,n
Phˆa
`
nb`ud¯a
.
isˆo
´
cu
’
a phˆa
`
ntu
.
’
a
i,j
,l`asˆo
´
A
i,j
=(−1)
i+j
D
i,j
trong d¯´o D
i,j
l`a d¯i
.
nh
th ´u
.
ccˆa
´
p n −1cu
’
aba
’
ng thu d¯u
.
o
.
.
ct`u
.
ma trˆa
.
n A b˘a
`
ng c´ach ga
.
ch bo
’
h`ang th´u
.
i v`a
cˆo
.
tth´u
.
j.
+ Cho A l`a ma trˆa
.
n vuˆong kha
’
nghi
.
ch cˆa
´
p n v`a ∆ = det A = 0. Khi d¯´o ma trˆa
.
n
nghi
.
ch d¯a
’
ocu
’
a A d¯ u
.
o
.
.
c x´ac d¯i
.
nh mˆo
.
t c´ach duy nhˆa
´
tbo
.
’
i:
A
−1
=
1
∆
A
i,j
T
=
1
∆
A
1,1
A
2,1
D
n,1
A
1,2
A
2,2
D
n,2
A
1,n
A
2,n
D
n,n
V´ıdu
.
. Ma trˆa
.
n nghi
.
ch d¯a
’
ocu
’
a
A =
1 −11
211
112
l`a:
A
−1
=
1
5
13−2
−31 1
1 −23
v`ı:
∆ = det A = (1)(1)(2)+(2)(1)(1)+(1)(−1)(1)−(1)(1)(1)−(2)(−1)(2)−(1)(1)(1) = 5 =0
7
v`a:
A
1,1
=(−1)
1+1
11
12
=1;A
1,2
=(−1)
1+2
21
12
= −3; A
1,3
=(−1)
1+3
21
11
=1;
A
2,1
=(−1)
2+1
−11
12
=3;A
2,2
=(−1)
2+2
11
12
=1;A
2,3
=(−1)
2+3
1 −1
11
= −2
A
3,1
=(−1)
3+1
−11
11
= −2;
A
3,2
=(−1)
3+2
11
21
=1;
A
3,3
=(−1)
3+3
1 −1
21
=3
+T´ınh chˆa
´
t:
− Cho A kha
’
d¯ a
’
ov`ak = 0, th`ı: (kA)
−1
=
1
k
A
−1
− Cho A, B c`ung cˆa
´
p v`a kha
’
d¯ a
’
o, th`ı: (A ×B)
−1
= B
−1
× A
−1
− Cho A kha
’
d¯ a
’
oth`ıA
−1
c˜ung kha
’
d¯ a
’
ov`a
A
−1
−1
= A
Phˆa
`
n I.4: Ha
.
ng cu
’
a ma trˆa
.
n
*Tago
.
i ha
.
ng cu
’
a ma trˆa
.
n A =(a
i,j
)
m×n
,k´yhiˆe
.
u r(A) l`a cˆa
´
p cao nhˆa
´
tcu
’
a c´ac
d¯ i
.
nh th ´u
.
c con kh´ac 0 cu
’
a A.
+Ha
.
ng cu
’
a ma trˆa
.
n0
m×n
l`a 0, ha
.
ng cu
’
a ma trˆa
.
n A =(a)v´o
.
i a = 0 l`a 1.
+Ha
.
ng cu
’
a ma trˆa
.
n khˆong thay d¯ˆo
’
i qua c´ac ph´ep biˆe
´
nd¯ˆo
’
iso
.
cˆa
´
p sau d¯ˆay:
a. D
-
ˆo
’
ichˆo
˜
hai h`ang ho˘a
.
c hai cˆo
.
t cho nhau;
b. Nhˆan mˆo
.
t h`ang (hay mˆo
.
tcˆo
.
t) v´o
.
imˆo
.
tsˆo
´
kh´ac 0;
c. Cˆo
.
ng v`ao mˆo
.
t h`ang (hay mˆo
.
tcˆo
.
t) v´o
.
imˆo
.
t h`ang (hay mˆo
.
tcˆo
.
t) kh´ac nhˆan
v´o
.
imˆo
.
tsˆo
´
.
D
-
ˆe
’
t`ım ha
.
ng cu
’
a ma trˆa
.
n A
mtimesn
, c´o thˆe
’
d`ung c´ac phu
.
o
.
ng ph´ap sau:
+ Phu
.
o
.
ng ph´ap theo d¯i
.
nh ngh˜ıa: t´ınh c´ac d¯i
.
nh th´u
.
c con t`u
.
cˆa
´
p 2 tro
.
’
lˆen. Gia
’
su
.
’
ma trˆa
.
nc´o1d¯i
.
nh th´u
.
c con cˆa
´
p r kh´ac 0, t´ınh tiˆe
´
p c´ac d¯i
.
nh th´u
.
ccˆa
´
p r +1,nˆe
´
u
tˆa
´
tca
’
d¯ ˆe
`
ub˘a
`
ng 0 th`ı kˆe
´
t luˆa
.
nha
.
ng ma trˆa
.
nl`ar,nˆe
´
uc´od¯i
.
nh th´u
.
ccˆa
´
p r +1 kh´ac
0 th`ı t´ınh tiˆe
´
p c´ac d¯i
.
nh th ´u
.
ccˆa
´
p r +2,c´u
.
nhu
.
thˆe
´
d¯ ˆe
´
nd¯i
.
nh th ´u
.
ccˆa
´
pl´o
.
n nhˆa
´
t
V´ıdu
.
. T`ım ha
.
ng cu
’
a ma trˆa
.
n
A =
1235
3249
1014
Ta c´o d¯i
.
nh th ´u
.
c con cˆa
´
p2:
12
32
= −4 = 0, v`a c´ac d¯i
.
nh th ´u
.
ccˆa
´
p3:
123
324
101
=0;
125
329
104
=0;
135
349
114
=0;
235
249
014
=0
suy ra r(A)=2
8
+ Phu
.
o
.
ng ph´ap d`ung ph´ep biˆe
´
nd¯ˆo
’
iso
.
cˆa
´
p: biˆe
´
nd¯ˆo
’
i ma trˆa
.
nvˆe
`
da
.
ng bˆa
.
c
thang
B =
b
1,1
b
1,2
b
1,r
b
1,n
0 b
2,2
b
2,r
b
2,n
00 b
r,r
b
r,n
00 0 0
00 0 0
v´o
.
i b
i,j
=0, ∀i>jhay i>rv`a b
ii
=0,i = 1,r th`ı r(A)=r(B)=r.
V´ıdu
.
. T`ım ha
.
ng ma trˆa
.
n
A =
13205
269712
−2 −524 5
148420
A
h2−2h1;h3+2h1;h4−h1
−→
1320 5
0057 2
016415
016415
h4−h3;h2↔h3
−→
1320 5
016415
0057 2
0000 0
suy ra r(A)=3
+ Ngo`ai ra, c´o thˆe
’
t`ım ma trˆa
.
n nghi
.
ch d¯a
’
o qua c´ac ph´ep biˆe
´
nd¯ˆo
’
iso
.
cˆa
´
p:
lˆa
.
p ma trˆa
.
n khˆo
´
i A|E (E c`ung c˜o
.
v´o
.
i A, thu
.
.
chiˆe
.
n c´ac ph´ep biˆe
´
nd¯ˆo
’
iso
.
cˆa
´
p CHI
’
TR
ˆ
EN H
`
ANG, nˆe
´
ud¯u
.
ad¯u
.
o
.
.
cvˆe
`
da
.
ng E|B th`ı B l`a nghi
.
ch d¯a
’
ocu
’
a A.
V´ıdu
.
. A|E =
1 −11| 100
211| 010
112| 001
h2−2h1,h3−h1
−→
1 −11| 100
03−1 |−210
02 1|−101
h1−h3,h2+h3
−→
1 −30| 20−1
050|−31 1
021|−10 1
h2(
1
5
)
−→
1 −30| 20−1
010|−3/51/51/5
021|−101
h1+3h2,h3−2h2
−→
100| 1/53/5 −2/5
010|−3/51/51/5
011| 1/5 −2/53/5
thu d¯u
.
o
.
.
ckˆe
´
t qua
’
nhu
.
c˜u.
B
`
AI T
ˆ
A
.
P
1.1. Khˆong t´ınh, ch´u
.
ng minh c´ac d¯i
.
nh th ´u
.
c sau chia hˆe
´
t cho 17:
204
527
255
;
323
20 9 1
55 2 5
1.2. Ch´u
.
ng minh c´ac d¯˘a
’
ng th ´u
.
c sau d¯ˆay (khˆong t´ınh d¯i
.
nh th ´u
.
cb˘a
`
ng d¯i
.
nh ngh˜ıa):
9
a.
0 xyz
x 0 zy
yz0 x
xyz0
=
01 1 1
10z
2
y
2
1 z
2
0 x
2
1 y
2
x
2
0
v´o
.
i xyz =0
b.
1 xyz
1 yzx
1 zxy
=(x − y)(y −z)(z − x)
c.
111
xyz
x
3
y
3
z
3
=(x + y + z)(x − y)(y − z)(z −x)
1.3. T`ım x sao cho:
a.
33− x −x
273
x +1 3x − 7 x
=0 b.
xx+1 x +2
x +3 x +4 x +5
x +6 x +7 x +8
=0
c.
1 xx
2
31 x
45 1
< 0 d.
x 12
1 x 3
x +1 2 −4
> 0
1.4. T´ınh c´ac d¯i
.
nh th´u
.
c sau:
0110
0011
1001
1100
;
1 xxx
1 a 00
10b 0
100c
;
x 1111
1 x 111
11x 11
111x 1
1111x
;
a + xx x
ab+ xx
xxc+ x
;
x
2
+1 xy xz
xy y
2
+1 yz
xz yz z
2
+1
;
1 xx
2
x
3
x
3
x
2
x 1
12x 3x
2
4x
3
4x
3
3x
2
2x 1
;
0 xyz
x 0 zy
yz0 x
xyz0
;
2 x 1 x
1 x 2 x
21xx
xx21
;
x 0 y 0
0 z 0 t
y 0 z 0
0 t 0 x
;
axx−x −x
x 2aa 00
xa2a 00
−x 002aa
−x 00 a 2a
;
12 3 n
21 2 n− 1
32 1 n− 2
nn− 1 n −2 1
;
xaa a
axa a
aax a
aaaax
;
cos(x
1
− y
1
) cos(x
1
− y
2
) cos(x
1
− y
n
)
cos(x
2
− y
1
) cos(x
2
− y
2
) cos(x
2
− y
n
)
cos(x
n
−y
1
) cos(x
n
− y
2
) cos(x
n
− y
n
)
;
011 11
10x x x
1 x 0 x x
1 xx 0 x
1 xx x0
;
10
1+x
1
y
1
1+x
1
y
2
1+x
1
y
n
1+x
2
y
1
1+x
2
y
2
1+x
2
y
n
1+x
n
y
1
1+x
n
y
2
1+x
n
y
n
;
a
1
−a
2
0 00
0 a
2
−a
3
00
00a
3
00
00 0 a
n−1
−a
n
11 1 11+a
n
1.5. Cho A =
212
301
012
v`a B =
1 −2
46
5 −3
.T`ım A
2
,AB,A
−1
.
1.6. T`ım c´ac ma trˆa
.
n
2 −1
3 −2
n
;
a 1
0 a
n
;
cos x −sin x
sin x cos x
n
1.7. Cho A =
12
21
.T`ım f(A)v´o
.
i f(x)=x
2
− 4x +3,f(x)=x
2
− 2x +1.
1.8.
a. Cho A =
211
312
1 −10
v`a B =
12−2
23 1
12 2
.
1. T`ım A
−1
,B
−1
.
2. T`ım f(A),f(B)v´o
.
i f(x)=x
2
−x − 1
b. T`ım ma trˆa
.
n nghi
.
ch d¯a
’
ocu
’
a A =
2100
3200
1134
2 −123
; B =
13−57
01 2 −3
00 1 2
00 0 1
.
1.9.
a. T`ım ma trˆa
.
n vuˆong cˆa
´
p hai c´o b`ınh phu
.
o
.
ng b˘a
`
ng ma trˆa
.
n khˆong.
b. T`ım ma trˆa
.
n vuˆong cˆa
´
p hai c´o b`ınh phu
.
o
.
ng b˘a
`
ng ma trˆa
.
nd¯o
.
nvi
.
.
1.10. T`ım ma trˆa
.
n X sao cho:
12
34
× X =
35
59
; X ×
3 −2
5 −4
=
−12
56
;
12−3
32−4
2 −10
×X =
1 −30
10 2 7
10 7 8
; X×
11−1
21 0
1 −11
=
1 −13
432
1 −25
;
21
32
× X ×
−32
5 −3
=
−24
3 −1
;
41
3 −1
× X ×
21
53
=
50
61
;
X ×
111
011
001
− 2
21−1
30 6
=
105
−1 −21
;
122
254
245
× X +
35
76
21
=3
15
−12
−20
;
11
111 1
011 1
001 1
000 1
× X =
123 n
012 n− 1
001 n− 2
000 1
.
1.11. T`ım ha
.
ng cu
’
a ma trˆa
.
n sau:
2 1112
104−1
11 4 56 −5
2 −15−6
;
2111
1311
1141
1115
1111
;
13205
269712
−2 −524 5
148420
;
12314
32111
11 1 6
23−15
11 0 3
;
135791
1 −23−452
2 11 12 25 22 4
;
31−311
2 −17−32
13−253
3 −27−53
1.12. Biˆe
.
n luˆa
.
n theo a sˆo
´
ha
.
ng cu
’
a c´ac ma trˆa
.
n sau:
−12 1
2 a −2
3 −6(a + 3)( a +7)
;
1 a −12
2 −1 a 5
110−61
;
a 114
1 a 13
12a 14
;
31 1 4
a 4101
17173
22 4 3
;
1436
−1011
21−10
02a 4
;
12−132
2 −1 a
2
04
31 227
12 a 11
1.13. T`ım c´ac gi´a tri
.
cu
’
a m d¯ ˆe
’
:
a. r(A)=2v´o
.
i A =
34 5 7 1
26−34 2
4 2 13 10 0
5 0 21 13 m
b. r(A)=3v´o
.
i A =
123−11
321−11
231 1 1
55202m +1
c. r(A)=3v´o
.
i A =
1436
−10 1 1
21−10
02m 4
d. r(A)=2v´o
.
i A =
3114
m 4101
17173
2243
e. r(A)=2v´o
.
i A =
m 111
11m 1
111m
1 m 11
12
-ooOoo-
13
Chu
.
o
.
ng 2. H
ˆ
E
.
PHU
.
O
.
NG TR
`
INH TUY
ˆ
E
´
NT
´
INH (2+2)
I. C´ac d¯i
.
nh ngh˜ıa
*Tago
.
i hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
nt´ınh m phu
.
o
.
ng tr`ınh n ˆa
’
n l`a hˆe
.
c´o da
.
ng
a
1,1
x
1
+ a
1,2
x
2
+ ···+ a
1,n
x
n
= b
1
a
2,1
x
1
+ a
2,2
x
2
+ ···+ a
2,n
x
n
= b
2
a
m,1
x
1
+ a
m,2
x
2
+ ···+ a
m,n
x
n
= b
m
(1)
trong d¯´o a
i,j
,b
i
(i = 1,m,j = 1,n) l`a c´ac hˆe
.
sˆo
´
(thu
.
.
c ho˘a
.
cph´u
.
c), x
1
,x
2
, ,x
n
l`a c´ac
ˆa
’
nsˆo
´
.Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh d¯u
.
o
.
.
cgo
.
il`ac´o nghiˆe
.
m (hay tu
.
o
.
ng th´ıch)nˆe
´
u
tˆa
.
p nghiˆe
.
mcu
’
a n´o kh´ac rˆo
˜
ng.
+Hˆe
.
(1) c´o thˆe
’
d¯ u
.
o
.
.
cviˆe
´
tdu
.
´o
.
ida
.
ng ma trˆa
.
n AX = B trong d¯´o:
A =
a
1,1
a
1,2
a
1,n
a
2,1
a
2,2
a+2,n
a
m,1
a
m,2
a
m,n
; X =
x
1
x
2
.
.
. x
n
; B =
b
1
b
2
.
.
. b
m
hay
du
.
´o
.
ida
.
ng ma trˆa
.
nmo
.
’
rˆo
.
ng:
A =
a
1,1
a
1,2
a
1,n
b
1
a
2,1
a
2,2
a+2,n b
2
a
m,1
a
m,2
a
m,n
b
m
, khi d¯´o ha
.
ng
r(A)cu
’
a A d¯ u
.
o
.
.
cgo
.
il`aha
.
ng cu
’
ahˆe
.
phu
.
o
.
ng tr`ınh (1)
II. Hˆe
.
Cramer
*Hˆe
.
(1) c´o sˆo
´
phu
.
o
.
ng tr`ınh b˘a
`
ng sˆo
´
nghiˆe
.
m(m = n) v`a d¯i
.
nh th´u
.
c det(A)=0d¯u
.
o
.
.
c
go
.
il`ahˆe
.
Cramer.
+Hˆe
.
Cramer c´o nghiˆe
.
m duy nhˆa
´
td¯u
.
o
.
.
c x´ac d¯i
.
nh nhu
.
sau: ∀i = 1,n,x
i
=
D
i
D
, trong
d¯ ´o D = det(A), c`on D
i
l`a d¯i
.
nh th´u
.
c thu d¯u
.
o
.
.
ct`u
.
D b˘a
`
ng c´ach thay cˆo
.
tth´u
.
i b˘a
`
ng
cˆo
.
thˆe
.
sˆo
´
tu
.
.
do.
V´ıdu
.
. Gia
’
ihˆe
.
:
x
1
+2x
2
+3x
3
=6
2x
1
− x
2
+ x
3
=2
3x
1
+ x
2
− 2x
3
=2
Do D =
12 3
2 −11
31−2
=30=0,hˆe
.
c´o nghiˆe
.
m duy nhˆa
´
t(1, 1, 1):
x =
1
30
62 3
2 −11
21−2
=1;y =
1
30
16 3
22 1
32−2
=1;z =
1
30
126
2 −12
312
=1
III. C´ac d¯i
.
nh l´y vˆe
`
nghiˆe
.
mcu
’
ahˆe
.
(Kronecker-Kapeli)
+ (1) c´o nghiˆe
.
m (tu
.
o
.
ng th´ıch) khi v`a chı
’
khi r(A)=r(A).
+ (1) c´o nghiˆe
.
m duy nhˆa
´
t (x´ac d¯i
.
nh) khi v`a chı
’
khi r(A)=r(
A)=n.
+nˆe
´
u r(A)=r(
A)=r<nth`ı (1) c´o vˆo sˆo
´
nghiˆe
.
m v`a c´ac th`anh phˆa
`
n nhiˆe
.
m phu
.
thuˆo
.
c n − r tham sˆo
´
tu `y ´y.
14
V´ıdu
.
. Biˆe
.
n luˆa
.
n theo a sˆo
´
nghiˆe
.
mcu
’
ahˆe
.
:
ax
1
+ x
2
+ x
3
=1
x
1
+ ax
2
+ x
3
=1
x
1
+ x
2
+ ax
3
=1
D`ung c´ac ph´ep biˆe
´
nd¯ˆo
’
iso
.
cˆa
´
pd¯ˆe
’
x´ac d¯i
.
nh ha
.
ng cu
’
a A v`a A
A =
a 11| 1
1 a 1 | 1
11a | 1
h1↔h3
−→
11a | 1
1 a 1 | 1
a 11| 1
h2−h1
−→
h3−ah1
11 a | 1
0 a − 11−a | 0
01− a 1 − a
2
| 1 − a
h3+h2
−→
11 a | 1
0 a −11− a | 0
002−a −a
2
| 1 −a
+Nˆe
´
u2− a − a
2
= 0, c´o 2 tru
.
`o
.
ng ho
.
.
p:
a = 1 th`ı: A −→
111| 1
000| 0
000| 0
⇒ r(A)=r(A)=1< 3, hˆe
.
c´o vˆo sˆo
´
nghiˆe
.
m phu
.
thuˆo
.
c 2 tham sˆo
´
tu `y ´y.
a = −2 th`ı:
A −→
11−2 | 1
0 −33| 0
00 0| 3
⇒ r(A)=2<r(
A =3,hˆe
.
vˆo
nghiˆe
.
m.
+Nˆe
´
u2−a −a
2
=0⇔ a =1,a = −2, th`ı r(A)=r(A)=3,hˆe
.
c´o nghiˆe
.
m duy nhˆa
´
t.
IV. Phu
.
o
.
ng ph´ap gia
’
ihˆe
.
+ C´ac ph´ep biˆe
´
nd¯ˆo
’
iso
.
cˆa
´
p cho hˆe
.
tu
.
o
.
ng d¯u
.
o
.
ng (tu
.
o
.
ng ´u
.
ng v´o
.
i c´ac ph´ep biˆe
´
nd¯ˆo
’
i
theo h`ang cu
’
a ma trˆa
.
nmo
.
’
rˆo
.
ng):
− D
-
ˆo
’
ichˆo
˜
hai phu
.
o
.
ng tr`ınh cho nhau (d¯ˆo
’
ichˆo
˜
hai h`ang cu
’
a ma trˆa
.
n)
− Nhˆan hai vˆe
´
cu
’
aphu
.
o
.
ng tr`ınh n`ao d¯´o v´o
.
imˆo
.
tsˆo
´
kh´ac 0 (nhˆan c´ac phˆa
`
ntu
.
’
trˆen mˆo
.
t h`ang cu
’
a ma trˆa
.
nv´o
.
imˆo
.
tsˆo
´
kh´ac 0)
− Cˆo
.
ng t`u
.
ng vˆe
´
cu
’
amˆo
.
tphu
.
o
.
ng tr`ınh v´o
.
imˆo
.
tphu
.
o
.
ng tr`ınh kh´ac nhˆan v´o
.
i
mˆo
.
tsˆo
´
(cˆo
.
ng mˆo
.
t h`ang v´o
.
ibˆo
.
isˆo
´
mˆo
.
t h`ang kh´ac)
1.
´
Ap du
.
ng d¯i
.
nh l´y Carmer
Nˆe
´
uhˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh l`a hˆe
.
Cramer, c´o thˆe
’
´ap du
.
ng d¯i
.
nh l´y Carmer
ho˘a
.
c t`ım ma trˆa
.
n A
−1
, suy ra X = A
−1
B.
V´ıdu
.
. Gia
’
ib˘a
`
ng phu
.
o
.
ng ph´ap ma trˆa
.
n nghi
.
ch d¯a
’
o:
2x +3y +2z =9
x +2y − 3z =14
3x +4y − z =16
Do det(A)=
23 2
12−3
34 1
= −6 =0nˆe
`
hˆe
.
l`a Cramer.
V´o
.
i A
−1
=
1
det(A)
A
1,1
A
2,1
A
3,1
A
1,2
A
2,2
A
3,2
A
1,3
A
2,3
A
3,3
=
1
−6
14 5 −13
−10 −48
−21 1
nˆen X = A
−1
B = −
1
6
14 5 −13
−10 −48
−21 1
9
14
16
=
2
3
−2
, suy ra
x =2
y =3
z = −2.
15
2. Phu
.
o
.
ng ph´ap Gauss (khu
.
’
dˆa
`
nˆa
’
nsˆo
´
)
D`ung c´ac ph´ep biˆe
´
nd¯ˆo
’
iso
.
cˆa
´
p theo c´ac h`ang, biˆe
´
nd¯ˆo
’
i ma trˆa
.
nmo
.
’
rˆo
.
ng
A th`anh
ma trˆa
.
n A
1
c´o nhiˆe
`
u phˆa
`
ntu
.
’
0 (nhu
.
ma trˆa
.
nbˆa
.
c thang), khi d¯´o r(A)=r(A
1
)v`a
r(A)=r(A
1
).
+nˆe
´
u r(A
1
) <r(A
1
), th`ı hˆe
.
vˆo nghiˆe
.
m
+nˆe
´
u r(A
1
)=r(A
1
)=r th`ı lˆa
.
phˆe
.
phu
.
o
.
ng tr`ınh m´o
.
i (tu
.
o
.
ng d¯u
.
o
.
ng hˆe
.
d¯˜a cho) sau
kho bo
’
c´ac h`ang m`a mo
.
i phˆa
`
ntu
.
’
d¯ ˆe
`
ub˘a
`
ng 0. Gia
’
ihˆe
.
n`ay (r phu
.
o
.
ng tr`ınh, n ˆa
’
n
sˆo
´
)b˘a
`
ng c´ach cho
.
n r ˆa
’
nco
.
ba
’
nv`an −r ˆa
’
n khˆong co
.
ba
’
n (thay b˘a
`
ng tham sˆo
´
tu `y
´y), nˆe
´
u r = n th`ı hˆe
.
c´o nghiˆe
.
m duy nhˆa
´
t.
V´ıdu
.
. Gia
’
i c´ac hˆe
.
phu
.
o
.
ng tr`ınh sau:
x
1
− 3x
2
+2x
3
= −1
x
1
+9x
2
+6x
3
=3
x
1
+3x
2
+5x
3
=1
A =
1 −32−1
1963
1351
h
2
−h
1
−→
h
3
−h
1
1 −321
01244
0632
h
2
×1/2
−→
h
3
−h
2
1 −32−1
0311
0010
,
suy ra
x
1
− 3x
2
+2x
3
= −1
3x
2
+ x
3
=1
x
3
=0
⇒
x
1
= −1+3x
2
− 2x
3
3x
2
=1−x
3
x
3
=0
⇒
x
1
=0
x
2
=
1
3
x
3
=0
x
1
− 3x
2
+2x
3
− x
4
=2
2x
1
+7x
2
− x
3
= −1
4x
1
+ x
2
+3x
3
− 2x
4
=1
B =
1 −32−12
27−10−1
41 3−21
h
2
−2h
1
−→
h
3
−4h
1
1 −32−12
013−52−5
013−52−7
h
3
−h
2
−→
1 −32−12
013−52−5
00 0 0−2
=
B
1
.Dor(B)=r(B
1
)=2< 3=r(B
1
)=r(B), hˆe
.
vˆo nghiˆe
.
m.
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
4x
1
+9x
2
+10x
3
+5x
4
=2
C =
15 4 31
2 −12−10
53 8 11
491052
h
3
−h
1
−2h
2
−→
h
4
−2h
1
−h
2
15431
2 −12−10
00000
00000
h
2
−2h
1
−→
bo
’
h
3
,h
4
15 4 3 1
0 −11 −6 −7 −2
,t´u
.
c l`a:
16
x
1
+5x
2
+4x
3
+3x
4
=1
−11x
2
− 6x
3
− 7x
4
= −2.
Cho
.
n x
3
= α, x
4
= β, ta suy ra:
x
1
= −
14
11
α +
2
11
β +
1
11
x
2
= −
6
11
α −
7
11
β +
2
11
x
3
= α
x
4
= β
V´ıdu
.
2. Gia
’
i v`a biˆe
.
n luˆa
.
n theo a:
ax + y + z =1
x + ay + z = a
x + y + az = a
2
A =
a 11 1
1 a 1 a
11aa
2
h
3
↔h
1
−→
11aa
2
1 a 1 a
a 11 1
h
2
−h
1
−→
h
3
−ah
1
11 aa
2
0 a − 11−aa− a
2
01− a 1 − a
2
1 − a
3
h
3
+h
2
−→
11 aa
2
0 a − 11− aa−a
2
002− a − a
2
1+a − a
2
− a
3
, suy ra:
*Nˆe
´
u2− a − a
2
=0⇔ (a =1)∨ (a = −2)
+Nˆe
´
u a = 1, th`ı A → (1 1 1 1), tu
.
o
.
ng d¯u
.
o
.
ng v´o
.
i x + y + z = 1 nˆen c´o vˆo sˆo
´
nghiˆe
.
m
da
.
ng (1 − α − β; 1; 1) v´o
.
i α, β tu`y ´y.
+Nˆe
´
u a = −2, th`ı
A →
11−24
0 −33−6
00 0 3,
suy ra r(A)=2< 3=r(A)nˆen hˆe
.
vˆo
nghiˆe
.
m.
*Nˆe
´
u2− a − a
2
=0⇔ (a =1)∧ (a = −2)
A
h
2
:a−1
−→
h
3
:2−a−a
2
11 aa
2
01−1 −a
00 1
(a +1)
2
a +2
, nˆen hˆe
.
d¯˜a cho tu
.
o
.
ng ´u
.
ng v´o
.
i:
x + y + az = a
2
y − z = −a
z =
(a +1)
2
a − 2
⇔
x
1
= −
a +1
a − 2
x
2
=
1
a +2
x
3
=
(a +1)
2
a +2
V´ıdu
.
3. Gia
’
i v`a biˆe
.
n luˆa
.
n theo a, b:
ax + y + z =1
x + by + z =3
x +2by + z =4
D = det(A)=
a 11
1 b 1
12b 1
=(1− a)b; D
x
=
411
3 b 1
42b 1
= −2b +1;
D
y
=
a 41
131
141
=1− a; D
z
=
a 14
1 b 3
12b 4
=4b − 2ab − 1
17
+Nˆe
´
u D =(1−a)b =0⇔
a =1
b =0
,hˆe
.
l`a Cramer, c´o nghiˆe
.
m duy nhˆa
´
t:
x
1
=
−2b +1
(1 − a)b
x
2
=
1
b
x
3
=
4b − 2ab − 1
(1 − a)b
+Nˆe
´
u a =1,hˆe
.
tro
.
’
th`anh:
x + y + z =1
x + by + z =3
x +2by + z =4
⇔
x + y + z =4
(b − 1)y = −1
(2b − 1)y =0
, th`ı::
− Nˆe
´
u2b − 1=0⇔ b =
1
2
:
x + y + z =0
y =2
⇔
x =2−α
y =2
z = α
, α tu`y ´y.
− Nˆe
´
u2b−1 =0⇔ b =
1
2
:
x + y + z =4
(b − 1)y = −1
y =0
vˆo nghiˆe
.
mv`ı(b−1)0 = −1
+Nˆe
´
u b =0,hˆe
.
tro
.
’
th`anh:
ax − y + z =4
x + z =3
x + z =4
vˆo nghiˆe
.
m
V. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh thuˆa
`
n nhˆa
´
t
* Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
nt´ınh thuˆa
`
n nhˆa
´
t l`a hˆe
.
c´o da
.
ng
AX = 0 (I I)
(B l`a ma trˆa
.
n to`an sˆo
´
0), khi d¯´o r(A)=r(
A), hˆe
.
luˆon luˆon c´o nghiˆe
.
m:
+nˆe
´
u r(A)=n,hˆe
.
c´o nghiˆe
.
m duy nhˆa
´
t nghiˆe
.
mtˆa
`
m thu
.
`o
.
ng x
1
= x
2
= ··· =
x
n
=0;
+nˆe
´
u r(A) <n,hˆe
.
c´o vˆo sˆo
´
nghiˆe
.
m, c´ac th`anh phˆa
`
ncu
’
a nghiˆe
.
m phu
.
thuˆo
.
c n −r(A)
tham sˆo
´
, nˆen c´o nghiˆe
.
m kh´ac nghiˆe
.
m khˆong (nghiˆe
.
m khˆong tˆa
`
m thu
.
`o
.
ng).
+V´o
.
ihˆe
.
c´o n phu
.
o
.
ng tr`ınh, n ˆa
’
nsˆo
´
,hˆe
.
c´o nghiˆe
.
m khˆong tˆa
`
mthu
.
o
.
ng khi v`a chı
’
khi
det(A) = 0 v`a c´o nghiˆe
.
m duy nhˆa
´
ttˆa
`
mthu
.
o
.
ng khi v`a chı
’
khi det(A)=0.
V´ıdu
.
. T`ım a d¯ ˆe
’
hˆe
.
ax
1
+ x
2
+ ···+ x
n−1
+ x
n
=0
x
1
+ ax
2
+ ···+ x
n−1
+ x
n
=0
=0
x
1
+ x
2
+ ···+ ax
n−1
+ x
n
=0
x
1
+ x
2
+ ···+ x
n−1
+ ax
n
=0.
c´o nghiˆe
.
m khˆong tˆa
`
m
thu
.
o
.
ng
18
det(A)=
a 1 11
1 a 11
11 a 1
11 1 a
h
1
+
h
i
=
i=1
a + n − 1 a + n −1 a+ n −1 a + n −1
1 a 11
11 a 1
11 1 a
=(a + n − 1)
11 11
1 a 11
11 a 1
11 1 a
h
i
−h
1
=
i=1
(a + n − 1)
11 11
0 a − 1 00
00 a− 10
00 0 a − 1
=(a + n − 1)(a − 1)
n−1
Hˆe
.
c´o nghiˆe
.
m khˆong tˆa
`
mthu
.
`o
.
ng khi det(A)=0⇔
a =1−n
a =1.
+nˆe
´
u
(α
1
; α
2
; ; α
n−1
; α
n
)v`a(β
1
; β
2
; ; β
n−1
; β
n
)
l`a nghiˆe
.
mcu
’
ahˆe
.
(I I) th`ı
∀h, k ∈ R :(hα
1
+ kβ
1
; hα
2
+ kβ
2
; ; hα
n−1
+ kβ
n−1
; hα
n
+ kβ
n
)
c˜ung l`a nghiˆe
.
mhˆe
.
(I I).
+Tru
.
`o
.
ng ho
.
.
p r(A) <n(sˆo
´
ˆa
’
ncu
’
ahˆe
.
)th`ır(A)ˆa
’
nco
.
ba
’
nd¯u
.
o
.
.
cbiˆe
’
udiˆe
˜
n qua
n −r(A)ˆa
’
n khˆong co
.
ba
’
n (lˆa
´
y gi´a tri
.
tu `y ´y). Nˆe
´
ucho
.
n n −r(A)ˆa
’
n khˆong co
.
ba
’
n
tu
.
o
.
ng ´u
.
ng theo n − r(A) th`anh phˆa
`
ncu
’
a n − r(A)bˆo
.
sˆo
´
:
(1; 0; 0; ; 0); (0; 1; 0; ; 0); (0; 0; 1; ; 0); ; (0; 0; 0; ;1)
th`ı n −r(A) nghiˆe
.
mcu
.
thˆe
’
cu
’
ahˆe
.
(I I) d¯u
.
o
.
.
cgo
.
il`amˆo
.
thˆe
.
nghiˆe
.
mco
.
ba
’
ncu
’
a
hˆe
.
.
V´ıdu
.
. T`ım hˆe
.
nghiˆe
.
mco
.
ba
’
ncu
’
a
x
1
+2x
2
− 2x
3
+ x
4
=0
2x
1
+4x
2
+2x
3
− x
4
=0
x
1
+2x
2
+4x
3
− 2x
4
=0
4x
1
+8x
2
− 2x
3
+ x
4
=0.
19
A =
12−21
24 2 −1
12 4 −2
48−21
h
2
−2h
1
−→
h
3
−h
1
h
4
−4h
1
12−21
00 6 −3
00 6 −3
00 6 −3
h
3
−h
2
−→
h2:2
h
4
−h
2
12−21
00 2 −1
´u
.
ng
v´o
.
ihˆe
.
:
x
1
+2x
2
− 2x
3
+ x
4
=0
2x
3
− x
4
=0
⇔
x
1
= −2x
2
x
4
=2x
3
.
+ Cho
.
n(x
2
,x
3
)=(1, 0), ta c´o: nghiˆe
.
m(−2; 1;0; 0)
+ Cho
.
n(x
2
,x
3
)=(0, 1), ta c´o: nghiˆe
.
m (0; 0; 1; 2)
* Gia
’
ith´ıch c´ach t`ım ma trˆa
.
nghi
.
ch d¯a
’
oo
.
’
phˆa
`
n IV, chu
.
o
.
ng 1
Cho ma trˆa
.
n vuˆong A =
a
1,1
a
1,2
a
1,3
a
1,n
a
2,1
a
2,2
a
2,3
a
2,n
a
n,1
a
n,2
a
n,3
a
n,n
c´o det( A) =0. X´et hˆe
.
n phu
.
o
.
ng tr`ınh 2n ˆa
’
n:
a
1,1
x
1
+ a
1,2
x
2
+ a
1,3
x
3
+ ···+ a
1,n
x
n
+ x
n+1
=0
a
2,1
x
1
+ a
2,2
x
2
+ a
2,3
x
3
+ ···+ a
2,n
x
n
+ x
n+2
=0
a
3,1
x
1
+ a
3,2
x
2
+ a
3,3
x
3
+ ···+ a
3,n
x
n
+ x
n+3
=0
a
n,1
x
1
+ a
n,2
x
2
+ a
n,3
x
3
+ ···+ a
n,n
x
n
+ x
n+1
=0
c´o da
.
ng ma trˆa
.
n
A × X + X
=0⇔ A × X = −X
(1)
v´o
.
i X =
x
1
x
2
x
3
.
.
.
x
n
v`a X
=
x
n+1
x
n+2
x
n+3
.
.
.
x
2n
v`ı det(A) =0, ∃A
−1
nˆen: (1)⇔ X = −A
−1
× X
⇔ X + A
−1
×X
= 0 (*)
Hˆe
.
c´o ma trˆa
.
nhˆe
.
sˆo
´
:
a
1,1
a
1,2
a
1,3
a
1,n
| 100 0
a
2,1
a
2,2
a
2,3
a
2,n
| 010 0
a
3,1
a
3,2
a
3,3
a
3,n
| 001 0
a
n,1
a
n,2
a
n,3
a
n,n
| 000 1
=(A|E)
Gia
’
su
.
’
qua c´ac ph´ep biˆe
´
nd¯ˆo
’
iso
.
cˆa
´
p trˆen c´ac h`ang, ta d¯u
.
ad¯u
.
o
.
.
c ma trˆa
.
nvˆe
`
da
.
ng
100 0 | b
1,1
b
1,2
b
1,3
b
1,n
010 0 | b
2,1
b
2,2
b
2,3
b
2,n
001 0 | b
3,1
b
3,2
b
3,3
b
3,n
000 1 | b
n,1
b
n,2
b
n,3
b
n,n
=(E|B)
20
´u
.
ng v´o
.
ihˆe
.
:
x
1
+ b
1,1
x
n+1
+ b
1,2
x
n+2
+ b
1,3
x
n+3
+ ···+ b
1,n
x2n =0
x
2
+ b
2,1
x
n+1
+ b
2,2
x
n+2
+ b
2,3
x
n+3
+ ···+ b
2,n
x2n =0
x
3
+ b
3,1
x
n+1
+ b
3,2
x
n+2
+ b
3,3
x
n+3
+ ···+ b
3,n
x2n =0
x
n
+ b
n,1
x
n+1
+ b
n,2
x
n+2
+ b
n,3
x
n+3
+ ···+ b
n,n
x2n =0
c´o da
.
ng X + B × X
= 0, suy ra B = A
−1
B
`
AI T
ˆ
A
.
P
2.1. Gia
’
ic´achˆe
.
phu
.
o
.
ng tr`ınh sau:
3x − 5y +2z +4t =2
7x − 4y + z +3t =5
5x +7y − 4z − 6t =3
2x + y −z =1
x − y + z =2
4x +3y + z =3
x + y − 3z = −1
2x + y − 2z =1
x +2y − 3z =1
x + y + z =3
2x +3y − z +5t =0
3x − y +2z −7t =0
4x + y − 3z +6t =0
x − 2y +4z − 7t =0
x − 2y +3z − 4t =4
y − z + t = −3
x +3y − 3t =1
−7y +3z +3t = −3
2x + y − 3z =4
x +2y + z =1
3x − 3y +2z =11
x +3y +4z =8
2x + y − z =2
2x +6y −5z =4
x − y +2z − 3t =1
x +4y − z − 2t = −2
x − 4y +3z − 2t = −2
x − 8y +5z − 2t = −2
2x +3y − z + t =2
2x +3y + z =4
2x +3y +2z =3
2x +3y =5
3x +4y +5z +7t =1
2x +6y − 3z +4t =2
4x +2y +13z +10t =0
2x +21z +13t =3
x + y +5z = −7
x +3y + z =5
2x + y + z =2
2x +3y − 3z =14
2x − 5y +4z +3t =0
3x − 4y +7z +5t =0
4x − 9y +8z +5t =0
3x − 2y +5z − 3t =0
3x + y − 3z + t =1
2x − y +7z − 3t =2
x +3y − 2z +5t =3
3x − 2y +7z − 5t =3
x +2y +3z − t =1
3x +2y + z − t =1
2x +3y + z + t =1
5x +5y +5z =2
8x +6y +5z +2t =21
3x +3y +2z + t =10
4x +2y +3z+=8
3x +5y + z + t =15
7x +4y +5z +2t =18
x
1
+ x
2
=1
x
1
+ x
2
+ x
3
=4
x
2
+ x
3
+ x
4
= −3
x
3
+ x
4
+ x
5
=2
x
4
+ x
5
= −1
x
1
+2x
2
+3x
3
+4x
4
=0
7x
1
+14x
2
+20x
3
+27x
4
=0
5x
1
+10x
2
+16x
3
+19x
4
= −2
3x
1
+5x
2
+6x
3
+13x
4
=5
2.2. Gia
’
iv`abiˆe
.
n luˆa
.
n theo a c´ac hˆe
.
sau:
21
(a +1)x + y + z =1
x +(a +1)y + z = a
x + y +(a +1)z = a
2
ax + y + z + t =1
x + ay + z + t = a
x + y + az + t = a
2
x − y + az + t = a
x + ay − z + t = −1
ax + ay − z − t = −1
x + y + z + t = −a
2 −11−1
2 −10−3
30−11
22−2 a
×
x
y
z
t
=
1
2
−3
−6
2.3. Cho hˆe
.
phu
.
o
.
ng tr`ınh
ax
1
− 3x
2
+ x
3
= −2
ax
1
+ x
2
+2x
3
=3
3x
1
+2x
2
+ x
3
= b.
a. T`ım a d¯ ˆe
’
hˆe
.
trˆen l`a hˆe
.
Cramer; ´u
.
ng v´o
.
i gi´a tri
.
cu
’
a a v`u
.
a t`ım, t`ım nghiˆe
.
mcu
’
ahˆe
.
theo b.
b. T`ım a, b d¯ ˆe
’
hˆe
.
trˆen vˆo nghiˆe
.
m.
c. T`ım a, b d¯ ˆe
’
hˆe
.
trˆen c´o vˆo sˆo
´
nghiˆe
.
m, t`ım nghiˆe
.
mtˆo
’
ng qu´at cu
’
ahˆe
.
.
2.4. T`ım m d¯ ˆe
’
c´ac hˆe
.
phu
.
o
.
ng tr`ınh sau d¯ˆay:
a. c´o nghiˆe
.
m
23 1
37−6
58 1
×
x
y
x
=
7
−2
m
;
36
48
27
×
x
y
=
−9
12
m
;
32 5
24 6
57m
×
x
y
z
=
1
3
5
;
375
231
693
×
x
y
z
=
−m
2
5
;
3x +4y +5z +7t =1
2x +6y − 3z +4t =2
4x +2y +13z +10t = m
5x +21z +13t =3
;
mx +2y +3z +2t =3
2x + my +3z +2t =3
2x +3y + mz +2t =3
2x +3y +2z + mt =3
2x +3y +2z +3t = m
b. vˆo nghiˆe
.
m:
2x − y + z − t =1
2x − y − 3t =2
3x − z + t = −3
2x +2y − 2z + mt = −6
;
x + y +(1− m)z = m +2
(1 + m)x − y +2z =0
2x − my +3z = m +2
c. vˆo d¯i
.
nh:
mx
1
+ x
2
+ x
3
+ ···+ x
n
=0
x
1
+ mx
2
+ x +3+···+ x
n
=0
x
1
+ x
2
+ mx
3
+ ···+ x
n
=0
x
1
+ x
2
+ x
3
+ ···+ mx
n
=0
3x +2y + z + t =1
2x +3y + z + t =1
x +2y +3z − t =1
5x +5y +2z =2m +1
;
3x +2y + z =3
mx + y +2z =3
mx − 3y + z = −2
;
x + my − z +2t =0
2x − y + mz +5t =0
x +10y − 6z + t =0
22
d. c´o nghiˆe
.
m duy nhˆa
´
t:
x +3y − z + t =1
3x +3y − z + mt =2
2x +2y + z + t =3
5x +3y +2t =1
;
x + y + z + mt =1
x + my + z + t =1
mx + y + z + t =1
x + y + mz + t =1
;
x +4y +3z +6t =0
−x + z + t =0
2x + y − z =0
2y + mx =0
2x +5y +3z +7t =0
2.5. Ch´u
.
ng minh hˆe
.
sau c´o nghiˆe
.
m duy nhˆa
´
t, t`ım nghiˆe
.
m d¯´o:
x
2
+ x
3
+ x
4
+ ···+ x
n−1
+ x
n
=1
x
1
+ x
3
+ x
4
+ ···+ x
n−1
+ x
n
=2
x
1
+ x
2
+ x
4
+ ···+ x
n−1
+ x
n
=3
x
1
+ x
2
+ x
3
+ x
4
+ ···+ x
n−1
= n
2.6. T`ım d¯iˆe
`
ukiˆe
.
n theo a d¯ ˆe
’
hˆe
.
sau c´o nghiˆe
.
m duy nhˆa
´
t
x
1
+ ax
2
=0
x
1
+(1+a)x
2
+ ax
3
=0
x
2
+(1+a)x
3
+ ax
4
=0
x
3
+(1+a)x
4
+ ax
5
=0
x
4
+(1+a)x
5
=0
2.7. Biˆe
.
n luˆa
.
n theo a sˆo
´
nghiˆe
.
mcu
’
ahˆe
.
phu
.
o
.
ng tr`ınh:
(a − 3)x + y + z =0
x +(a − 3)y + z =0
x + y +(a − 3)z =0
;
ax + ay + z = a
ax + y + az =1
x + ay + az =1
;
ax + ay +(a +1)z = a
ax + ay +(a − 1)z = a
(a +1)x + ay +(2a +3)z =1
;
x − y + az + t = a
x + ay − z + t = −1
ax + ay − z − t = −1
x + y + z + t = −a
2.8. T`ım nghiˆe
.
m nguyˆen du
.
o
.
ng (nˆe
´
u c´o) cu
’
ahˆe
.
phu
.
o
.
ng tr`ınh sau:
x + y + z = 100
x +15y +25z = 500
;
x +2y +3z =14
2x +3y − z =5
;
x +3y − 3z =1
3x − 3y +4z =4
;
x − y + z + t =2
2x + y − 3z +2t =2
3x − 2y + z + t =3
2.9. T`ım c´ac d¯a th´u
.
cbˆa
.
c3f(x)biˆe
´
t:
a. f(−1) = 0; f(1) = 4; f(2) = 3; f(3) = 16;
b. f(−1) = 5; f(1) = 5; f(3) = 45; f(−4) = −25.
2.10. T`ım nghiˆe
.
mtˆo
’
ng qu´at v`a hˆe
.
nghiˆe
.
mco
.
ba
’
ncu
’
ahˆe
.
phu
.
o
.
ng tr`ınh sau:
23
2x − y +5z +7t =0
4x − 2y +7z +5t =0
2x − y + z − 5t =0
;
x + y − 4z =0
2x +9y +6z =0
3x +5y +2z =0
4x +7y +5z =0
;
x +2y +4z − 3t =0
3x +5y +6z − 4t =0
4x +5y − 2z +3t =0
3x +8y +24z − 19t =0
;
x +8z +7t =0
2x + y +4z + t =0
3x +2y − z − 6t =0
7x +4y +6z − 5t =0
2.11.
a. Trong mˆo
.
t x´ı nghiˆe
.
psa
’
n xuˆa
´
t, c´o 15 cˆong nhˆan d¯u
.
o
.
.
c chia l`am 3 bˆa
.
c (I,II,III),
hu
.
o
.
’
ng lu
.
o
.
ng th´ang lˆa
`
nlu
.
o
.
.
t l`a: 600.000, 500.000, 400.000 d¯ˆo
`
ng. Mˆo
˜
i th´ang x´ı
nghiˆe
.
p ph´at 7,7 triˆe
.
ud¯ˆo
`
ng tiˆe
`
nlu
.
o
.
ng. Ho
’
i trong x´ı nghiˆe
.
pˆa
´
y, sˆo
´
cˆong cˆong mˆo
˜
i
bˆa
.
cc´othˆe
’
l`a bao nhiˆeu?
b. Mˆo
.
tho
.
.
p t´ac x˜a nˆong nghiˆe
.
p c´o 300 ha d¯ˆa
´
t, 850 cˆong lao d¯ˆo
.
ng v`a 65 triˆe
.
ud¯ˆo
`
ng
tiˆe
`
nvˆo
´
n d`anh cho sa
’
n xuˆa
´
tvu
.
h`e thu v´o
.
idu
.
.
d¯ i
.
nh trˆo
`
ng c´ac loa
.
i cˆay I,II,III c´o chi
ph´ı sa
’
n xuˆa
´
t cho mˆo
˜
i ha giao trˆo
`
ng nhu
.
sau:
Loa
.
i cˆay Vˆo
´
nb˘a
`
ng tiˆe
`
n (d¯ˆo
`
ng) Lao d¯ˆo
.
ng (cˆong)
I 200.000 2
II 150.000 3
III 400.000 5
-ooOoo-
24
Chu
.
o
.
ng 3
H
`
AM NHI
ˆ
E
`
UBI
ˆ
E
´
N&T
´
ICH PH
ˆ
AN K
´
EP
I. H`am nhiˆe
`
ubiˆe
´
n
1. Kh´ai niˆe
.
m
* Cho D ⊂ R
2
.Mˆo
.
t ´anh xa
.
f : D → R
(x, y) → f(x, y)=z ∈ R
d¯ u
.
o
.
.
cgo
.
il`ah`am hai biˆe
´
n x´ac d¯i
.
nh trˆen D, D d¯ u
.
o
.
.
cgo
.
il`amiˆe
`
n x´ac d¯i
.
nh
cu
’
a h`am hai biˆe
´
n f(x, y).
V´ıdu
.
.
+Miˆe
`
n x´ac d¯i
.
nh cu
’
a h`am z = f(x, y)=
1 −x
2
− y
2
l`a tˆa
.
p
D =
(x, y) ∈ R
2
: x
2
+ y
2
≤ 1
(h`ınh tr`on tˆam O b´an k´ınh 1).
+Miˆe
`
n x´ac d¯i
.
nh cu
’
a h`am z = f(x, y)=ln(x+y) l`a tˆa
.
p D =
(x, y) ∈ R
2
: x + y>0
(nu
.
’
am˘a
.
t ph˘a
’
ng n˘a
`
m ph´ıa trˆen d¯u
.
`o
.
ng th˘a
’
ng y = −x trˆen m˘a
.
t ph˘a
’
ng xOy.
* Cho h`am hai biˆe
´
n z = f(x, y). Trˆen m˘a
.
t ph˘a
’
ng Oxy,mˆo
˜
ic˘a
.
p(x, y)d¯u
.
o
.
.
cbiˆe
’
udiˆe
˜
n
bo
.
’
imo
.
td¯iˆe
’
m M(x, y), nˆen ta c´o thˆe
’
xem z = f(x, y) l`a h`am c´ac d¯iˆe
’
m M(x, y), k´y
hiˆe
`
u z = f(M).
* Cho h`am hai biˆe
´
n z = f(x, y) c´o miˆe
`
n x´ac d¯i
.
nh D. Trong khˆong gian Oxyz,x´et
c´ac d¯iˆe
’
m P (x, y, z) tho
’
a m˜an (x, y) ∈ D v`a z = f(x, y). Khi M cha
.
y trˆen miˆe
`
n D,
c´ac d¯iˆe
’
m P va
.
ch trong khˆong gian mˆo
.
tm˘a
.
t cong d¯u
.
o
.
.
cgo
.
il`ad¯ ˆo
`
thi
.
cu
’
a h`am
hai biˆe
´
n x = f(x, y).
* Cho D ⊂ R
n
= {(x
1
,x
2
, ,x
n
):x
i
∈ R,i =1, , n}.Mˆo
.
t ´anh xa
.
f : D → R
(x
1
,x
2
, ,x
n
) → f(x
1
,x
2
, ,x
n
)=z ∈ R
d¯ u
.
o
.
.
cgo
.
il`ah`am n biˆe
´
n f(x
1
,x
2
, ,x
n
) x´ac d¯i
.
nh trˆen D (D d¯ u
.
o
.
.
cgo
.
il`amiˆe
`
n
x´ac d¯i
.
nh).
* Cho h`am hai biˆe
´
n z = f(x, y) x´ac d¯i
.
nh trong khoa
’
ng ho
.
’
U cu
’
a M
o
(x
o
,y
o
) (khˆong
cˆa
`
n x´ac d¯i
.
nh ta
.
i M
o
). Sˆo
´
L d¯ u
.
o
.
.
cgo
.
il`agi´o
.
iha
.
ncu
’
a f(x, y) khi M(x, y) dˆa
`
n
d¯ ˆe
´
n M
o
(x
o
,y
o
)nˆe
´
uv´o
.
imo
.
i d˜ay d¯iˆe
’
m M
n
(x
n
,y
n
) thuˆo
.
c U dˆa
`
nd¯ˆe
´
n M
o
(x
o
,y
o
), ta
d¯ ˆe
`
u c´o: lim
n→∞
f(x
n
,y
n
) → L.Tak´yhiˆe
.
u:
lim
x→x
o
y→y
o
f(x, y)=L.
* H`am sˆo
´
z = f(x, y) x´ac d¯i
.
nh trong miˆe
`
n D d¯ u
.
o
.
.
cgo
.
il`aliˆen tu
.
cta
.
i M
o
(x
o
,y
o
) ∈ D
nˆe
´
u:
lim
x→x
o
y→y
o
f(x, y)=f(x
o
,y
o
).
25
2. D
-
a
.
o h`am v`a vi phˆan h`am nhiˆe
`
ubiˆe
´
n
2.1. D
-
a
.
o h`am riˆeng
* Cho h`am sˆo
´
z = f(x, y) x´ac d¯i
.
nh trˆen khoa
’
ng ho
.
’
U cu
’
a M
o
(x
o
,y
o
), khi d¯´o ∆x =
x−x
o
v`a ∆y = y−y
o
d¯ u
.
o
.
.
cgo
.
ilˆa
`
nlu
.
o
.
.
tl`asˆo
´
gia cu
’
abiˆe
´
nsˆo
´
x v`a y,∆
x
z = f(x
o
+
∆x, y
o
)−f(x
o
,y
o
)v`a∆
y
z = f(x
o
,y
o
+∆y)d¯u
.
o
.
.
cgo
.
ilˆa
`
nlu
.
o
.
.
tl`asˆo
´
gia riˆeng cu
’
a h`am
z = f(x, y) theo x v`a theo y ta
.
i M
o
(x
o
,y
o
), c`on ∆z = f(x
o
+∆x, y
o
+∆y)−f(x
o
,y
o
)
d¯ u
.
o
.
.
cgo
.
il`asˆo
´
gia to`an phˆa
`
ncu
’
a h`am z = f(x, y) ta
.
i M
o
(x
o
,y
o
).
*Nˆe
´
u lim
∆x→0
∆
x
z
∆x
v`a lim
∆y→0
∆
y
z
∆y
tˆo
`
nta
.
ih˜u
.
uha
.
n th`ı c´ac gi´o
.
iha
.
n d¯´o d¯u
.
o
.
.
cgo
.
i l`a c´ac
d¯ a
.
o h`am riˆeng cu
’
a h`am x = f(x, y) ta
.
i (x
o
,y
o
) cu
’
abiˆe
´
n x v`a biˆe
´
n y,k´y
hiˆe
.
ulˆa
`
nlu
.
o
.
.
t l`a:
z
x
(x
o
,y
o
)=f
x
(x
o
,y
o
)=
∂z
∂x
(x
o
,y
o
) = lim
∆x→0
∆
x
z
∆x
z
y
(x
o
,y
o
)=f
y
(x
o
,y
o
)=
∂z
∂y
(x
o
,y
o
) = lim
∆y→0
∆
y
z
∆y
*Nˆe
´
u h`am z = f(x, y) c´o c´ac d¯a
.
o h`am riˆeng theo biˆe
´
n x v`a biˆe
´
n y ta
.
i ∀(x, y) ∈ D,
ta n´oi z = f(x, y) c´o c´ac d¯a
.
o h`am riˆeng theo biˆe
´
n x v`a theo biˆe
´
n y trong
miˆe
`
n D,k´yhiˆe
.
u l`a:
f
x
(x, y)=z
x
=
∂z
∂x
; f
y
(x, y)=z
y
=
∂z
∂y
V´ıdu
.
. T´ınh c´ac d¯a
.
o h`am riˆeng cu
’
a
+ z = x
y
,x>0:
z
x
=(x
y
)
x
= yx
y−1
;
z
y
=(x
y
)
x
= x
y
. ln x
+ z = e
x
y
:
z
x
=
e
x
y
x
= e
x
y
.
x
y
x
= e
x
y
.
1
y
;
z
y
=
e
x
y
y
= e
x
y
.
x
y
y
= e
x
y
.
−
x
y
2
= −
x
y
2
.e
x
y
+ z = Arctg xy;
z
x
= (Arctg xy)
x
=
(xy)
x
1+(xy)
2
=
y
1+x
2
y
2
;
z
y
=
(xy)
y
1+(xy)
2
=
x
1+x
2
y
2
2.2. Vi phˆan
* Vi phˆan to`an phˆa
`
ncu
’
a h`am hai biˆe
´
n z = f(x, y) l`a: dz = z
x
dx + z
y
dy,c´othˆe
’
´u
.
ng
du
.
ng d¯ˆe
’
t´ınh gˆa
`
nd¯´ung gi´a tri
.
cu
’
a h`am sˆo
´
ph´u
.
cta
.
p theo cˆong th´u
.
csˆo
´
gia h˜u
.
uha
.
n
nhu
.
sau:
f(x
o
+∆x, y
o
+∆x) f
x
(x
o
,y
o
) · ∆x + f
y
(x
o
,y
o
) · ∆y + f(x
o
,y
o
)
