Tài liệu Ma trận - Định thức - Hệ phương trình pdf
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Chu.o.ng I
. .
`
´.
ˆ
ˆ
- .
MA TRA N - DINH THU C - HE PHU O NG TRI NH
.
.
ˆ
§1. MA TRA N
.
-.
˜
1.1. Dinh ngh a
ã Ma tr n c p m ì n (d ˆi khi con goi la co. m × n) la mˆt bang hı
a a
¯o
`
` o ’
`nh
.
. ` ˜
.
’
˜ a `
˜
`
`
o ` ´
a ’ ’
a ¯ .
e
e
chu. nhˆ t gˆm m−hang, n−cˆt va ca c phˆn tu. cua ma trˆ n d u.o.c biˆ u diˆ n
. o
.
.
´i .
du.o. da ng sau:
a11 a12 a13 . . . a1n
a
21 a22 a23 . . . a2n
a31 a32 a33 . . . a3n
.
.
.
.
.
.
.
.
.
.
.
...
.
am2
am1
am3
...
amn
-e ¯
’
´
’
Dˆ d o.n gian ta kı hiˆ u ma trˆ n A cˆ p m × n nhu. sau: A = (aij )m×n ,
´ e
a
a
.
.
’
´
` o
´
a
` `
a ’ ’ `
trong d´ aij la phˆn tu. o. hang thu. i va cˆt thu. j cua ma trˆ n A.
¯o
.
.
˜ `
´
`
’ ’
• Nˆ u ca c phˆn tu. cua ma trˆ n A d` u nhˆ n gia tri thu.c, co nghı a la
e ´
a
´
a
¯ˆ
e
a
´ .
.
.
.
aij ∈ R, thı ma trˆ n A d u.o.c goi la ma trˆ n thu.c.
`
a
¯ . . `
a
.
.
.
*. Vı du :
´ .
´
A = ( 15 ) la ma trˆ n cˆ p 1 × 1.
`
a a
.
1 4
´
B = 2 7 la ma trˆ n cˆ p 3 × 2.
`
a a
.
5
A=
−3
cos x
sin x + cos x
ln x sin x
2
−3
´
la ma trˆ n cˆ p 2 × 3.
`
a a
.
’ ´
`
a `
• Ma trˆ n hang: Ma trˆ n co. 1 × n (chı co 1 hang) goi la ma trˆ n hang.
a
`
a ˜
. `
.
.
.
˜
*. Vı du : Ma trˆ n ( 1 2 3 4 ) la ma trˆ n hang (co. 1 × 4).
´ .
a
`
a `
.
.
’ ´
o
a o
• Ma trˆ n cˆt: Ma a n co. m × 1 (chı co 1 cˆt) goi la ma trˆ n cˆt.
a
o
trˆ ˜
.
. `
.
.
.
.
.
2
˜
*. Vı du : Ma trˆ n 3 la ma trˆ n cˆt (co. 3 × 1).
´ .
a
`
a o
.
.
.
4
1
´
`
`
’ a
o
a ’ ´
a
• Ma trˆ n thu.c gˆm tˆ t ca ca c phˆn tu. b˘ ng 0 d u.o.c goi la ma trˆ n
a
a
¯ .
. `
. `
.
.
khˆng.
o
´
´
a
o
a
• Ma trˆ n cˆ p n × n d u.o.c goi la ma trˆ n vuˆng cˆ p n.
a a
¯ . . `
.
.
`
´
`
• Ma trˆ n d .n vi: La ma trˆ n vuˆng cˆ p n co ca c phˆn tu. n˘ m trˆn
a ¯o
e
a
o
a
´ ´
a ’ a
.
. `
.
`
`
`
´nh a
` ´
a ’ a
´nh ¯ˆ
e
` ¯ `ng ´
d u.o. che o chı b˘ ng 1 va ca c phˆn tu. n˘ m ngoai d u.o. che o chı d` u
¯ `ng ´
1 0 ... 0
. la co da ng: I = 0 1 . . . 0 . Ky hiˆ u la: I (d ˆi khi ta
`
b˘ ng 0, tu ` ´ .
a
´c
.
´ e ` n ¯o
. .
.
. . ... .
. .
.
0 0 ...
1
con ky hiˆ u: I).
` ´ e
.
´
• Ma trˆ n con: Cho A la ma trˆ n cˆ p m × n, ta goi Mij la ma trˆ n
a
`
a a
`
a
.
.
.
.
.o.c tu. ma trˆ n A b˘ ng ca ch bo d i hang i va cˆt j, khi d´ M goi la
`
’ ¯ `
a
lˆ p d u . `
a ¯
a
´
` o
¯o ij . `
.
.
.
’
´ng
´i `
a ’
ma trˆ n con cua ma trˆ n A u. vo. phˆn tu. aij .
a
a
.
.
1 2 3
*. Vı du : Cho ma trˆ n A = 0 −1 4
´ .
a
.
3
−2 8
0 4
;
M12 =
Ta co : M11 =
´
−2 8
3 8
2 3
1 3
M21 =
;
M22 =
−2 8
3 8
2 3
1 3
;
M32 =
M31 =
−1 4
0 4
’
´
´
1.2. Ca c phe p biˆ n d o i so. cˆ p trˆn hang
´
´
e ¯ˆ
a
e
`
−1 4
;
M13 =
;
M23 =
;
M33 =
0 −1
3 −2
1 2
3
1
−2
2
0 −1
’
(cˆt) cua ma trˆ n
o
a
.
.
’
´
´
’
o
a ¯ .
• Ca c phe p biˆ n d ˆ i sau d ˆy d ˆ i vo. hang (cˆt) cua ma trˆ n d u.o.c goi
´
´
e ¯o
¯a ¯o ´i `
.
.
.
. cˆ p theo hang (cˆt) cua ma trˆ n:
’
´
´
’
la ca c phe p biˆ n d o i so a
` ´
´
e ¯ˆ
`
o
a
.
.
-o
’
’
’
(1). Dˆ i chˆ hai hang (cˆt) cua ma trˆ n cho nhau.
o
`
o
a
.
.
. cua mˆt hang (cˆt) cua ma trˆ n vo. mˆt
´
`
’
o `
o
a ´i o
(2). Nhˆn tˆ t ca ca c phˆn tu ’
a a ’ ´
a ’
.
.
.
.
´
sˆ λ = 0.
o
(3). Cˆng vao mˆt hang (cˆt) nao d´ cua ma trˆ n mˆt hang (cˆt) kha c
o
`
o `
o
` ¯o ’
a
o `
o
´
.
.
.
.
.
.
sau khi d˜ nhˆn vo. mˆt sˆ λ = 0.
¯a
a ´i o o
. ´
2
1
3
4 −2
*. Vı du : Cho ma trˆ n A = −1 2 0 1
´ .
a
.
2 −2 0 6
Khi d´ :
¯o
-o
’
˜
(1) Dˆ i chˆ hang 1 cho hang 2 (cˆt 1 cho cˆt 2)ta d u.o.c:
o `
`
o
o
¯
.
.
−1 2 0 1
3
1 4 2
B= 1
3 4 −2 ; B = 2 −1 0 1
2 −2 0 6
−2 2 0 6
´
`
’
¯ .
(2) Nhˆn tˆ t ca ca c phˆn tu. cua hang 2 cua A cho λ = 2 ta d u.o.c:
a a ’ ´
a ’ ’ `
1
3 4 −2
1
3 4 −2
C = 2. −1 2 0 1 = −2 4 0 2
2
−2 0
6
2
−2 0 6
’
¯ .
(3) Cˆng hang 1 vao hang 2 sau khi d˜ nhˆn vo. λ = 2 cua A ta d u.o.c:
o
`
` `
¯a a ´i
.
1
3 4 −2
D = −1 7 4 0
2
−2 0
6
-.
˜
`
’
’
’
´
¯ˆ
a
e
o `
a
• Dinh nghı a: Phˆn tu. kha c 0 d` u tiˆn cua mˆt hang cua ma trˆ n
a
.
.
.o.c tı
. tra i sang phai) d u.o.c goi la phˆn tu. co. so. cua hang d´ .
’ ¯ . . ` `
’
’ ’ ` ¯o
`
a
(d u . ´nh tu ´
¯
-.
˜
a
a
e
• Dinh nghı a: Mˆt ma trˆ n d u.o.c goi la ma trˆ n bˆ c thang trˆn
o
a ¯ .
. `
.
.
.
.
´
’ ˜ ´ ¯`
nˆ u no thoa ma n ca c d iˆu kiˆ n sau:
e ´
e
e
.
`
’
´i ´ `
´
o
(1). Ca c hang b˘ ng khˆng o. du.o. ca c hang kha c khˆng.
´ `
a
o
`
`
’
’
’ ’ `
´i a
e
´i `
a ’
(2). Phˆn tu. co. so. cua hang phı du.o. n˘ m bˆn phai so vo. phˆn tu.
a
´a
’ ’ `
´a e
co. so. cua hang phı trˆn.
*. Vı du :
´ .
1 4
0 2
A=
0 0
0 0
8
1
7 −3
4 5
0
1 4 0
0 2 0
B=
0 0 4
;
0
0 0 0
1
5
−3 3
5 1
2
1
-.
’
` .
• Dinh ly : Moi ma trˆ n d` u co thˆ d u.a vˆ da ng ma trˆ n bˆ c thang
´
a ¯ˆ ´ e ¯
e
e
a
a
.
.
.
.
’
´
´
’
trˆn nho. ca c phe p biˆ n d ˆ i so. cˆ p theo hang cua ma trˆ n.
e
` ´
´
e ¯o
a
`
a
.
3
1
’
*. Vı du 1: Tı ma trˆ n bˆ c thang cua ma trˆ n A = 1
´ .
`m
a a
a
.
.
.
2
’
´n d ˆ i so. cˆ p ta co
´
Dung ca c phe p biˆ ¯o
`
´
´
e
a
´
1 2 1 7
1 2 1 7
A −→ 0 3 0 3 −→ 0 3 0 3
0 5 1
3
0 0
2 1
2
1
4
A=
4
3
6
2
1
2
0 1 −5
A −→
0 −1 −2
4
1
−2
2
4
9 3 17
1 −2
2
2
1
2
0
8
−→
0
−3
1
−5
0
−7
0 −5 −6 −10
0
2 1
2
4
2
0 1 −5 8
0
−→
−→
0 0 −7 5
0
0 0
−1 0
2 1
0
−31 30
4
0 0
0
55
7
1.3. Ca c phe p toa n ma trˆ n
´
´
´
a
.
`
• Hai ma trˆ n b˘ ng nhau:
a
a
.
´
a
Cho hai ma trˆ n A = (aij )m×n , B = (bij )p×q . Khi ˆ y:
a
.
´
o `
m = p (sˆ hang)
´ .
A = B ⇐⇒ n = q (sˆ cˆt)
o o
aij = bij
´
`
´c ` ´ `
a ` `ng `
a ’
´ng a
(Tu. la no cung cˆ p va tu. phˆn tu. tu.o.ng u. b˘ ng nhau.)
*. Vı du :
´ .
4
8
5
−5 −6 −10
1 2
4
1 −5 8
0 −7 5
5 1 10
` .
a
e
a a
*. Vı du 2: Du.a ma trˆ n sau vˆ da ng ma trˆ n bˆ c thang
´ .
.
.
.
7
1
A = 0
0
4 0
1
2 7 −5
1 4
1 4 0
1
B = 0 2 7
;
5
−5
0 1 4
5
• Phe p cˆng ma trˆ n:
´
o
a
.
.
’
˜
´
’
`
Tˆ ng cua hai ma trˆ n cung cˆ p A = (aij )m×n , B = (bij )m×n cu ng la
o
a `
a
.
´
’
ma trˆ n cˆ p m × n, ky hiˆ u la: A + B, d u.o.c xa c d inh bo.i:
a a
´ e `
¯ . ´ ¯.
.
.
A + B = (aij + bij )m×n
*. Vı du :
´ .
1 4
A = 0 2
0 1
0
1
7 −5
4
3 7 6
9
B = 0 8 7
;
5
2
1 0 2
4
´
Khi ˆ y
a
1+3
4+7 0+6
A + B = 0 +0
2+8 7+7
0+1
1+9
4
−5 + 2 = 0
1+0 4+2
5+4
1
11
6
10
10 14 −3
1
6
9
´
o
a
• Phe p nhˆn mˆt sˆ vo.i mˆt ma trˆ n:
´
a
o o ´
.
.
.
´
´
´
´i
Cho ma trˆ n A = (aij )m×n va sˆ λ = 0. Khi ˆ y tı cua sˆ λ vo. ma
a
` o
a ´ch ’ o
.
.o.c xa c d inh bo.i:
˜
´
’
trˆ n A cu ng la ma trˆ n cˆ p m × n, ky hiˆ u la: λ.A, d u . ´ ¯.
a
`
a a
´ e `
¯
.
.
.
λ.A = (λ.aij )m×n
*. Vı du :
´ .
´
Cho sˆ λ = 5 va ma trˆ n A =
o
`
a
.
λ.A =
5.1 5.4 5.0
1
4 0
0
2 7 −5
5.1
5.0 5.2 5.7 5.(−5)
=
1
´
. Khi ˆ y ta co :
a
´
5 20
0
5
0 10 35 −25
• Phe p nhˆn ma trˆ n:
´
a
a
.
Tı cua ma trˆ n A = (aij )m×n vo. ma trˆ n B = (bij )n×p la mˆt ma
´ch ’
a
´i
a
` o
.
.
.
5
´
’
trˆ n cˆ p m × p, ky hiˆ u la: A.B, d u.o.c xa c d inh bo.i:
a a
´ e `
¯ . ´ ¯.
.
.
n
A.B = cij =
aik .bkj
m×p
k=1
*. Vı du :
´ .
Cho hai ma trˆ n A =
a
.
1 2
5 6
9
13
10 14
´
va B =
`
a
´
. Khi ˆ y ta co :
11 15
7 8
3 4
12 16
A.B =
1.9 + 2.10 + 3.11 + 4.12 1.13 + 2.14 + 3.15 + 4.16
5.9 + 6.10 + 7.11 + 8.12 5.13 + 6.14 + 7.15 + 8.16
110 136
=
278 339
’
´ .
` o o ’
a
*.Chu ´ : Dˆ hai ma trˆ n nhˆn d u.o.c vo. nhau thı sˆ cˆt cua ma trˆ n
´ y -e
a
a ¯ . ´i
.
.
.o. phai b˘ ng sˆ hang cua ma trˆ n sau.
`
´
’ a
’
o `
a
tru ´c
.
’
• Phe p chuyˆ n vi ma trˆ n:
´
e
a
.
.
’
´
Cho ma trˆ n A = (aij )m×n . Khi ˆ y ma trˆ n chuyˆ n vi cua ma trˆ n A la
a
a
a
e . ’
a
`
.
.
.
’ `
’ .
`
a
´
e
`
o
e o
mˆt ma trˆ n co d u.o.c tu. A b˘ ng ca ch chuyˆ n hang thanh cˆt, chuyˆ n cˆt
o
a ´ ¯ . `
.
.
.
´ .
thanh hang theo d´ ng thu. tu..
`
`
¯u
Ky hiˆ u la: AT . Nhu. vˆ y ta co :
´ e `
a
´
.
.
a11 a21 . . . am1
a11 a12 . . . a1n
a12 a22 . . . am2
a21 a22 . . . a2n
T
; A = .
A= .
.
.
.
.
.
.
.
.
.
.
.
...
.
.
...
.
.
.
am1 am2 . . . amn m×n
a1n a2n . . . amn n×m
*. Vı du :
´ .
1
2
3
Cho ma trˆ n A = 5
a
.
6
4
7
1
2
´
8 . Khi ˆ y ta co : AT =
a
´
3
9 10 11 12
4
´ tı
´t cua phe p toa n ma trˆ n
’
1.4. Mˆt sˆ ´nh chˆ
o o
a
´
´
a
.
.
5
9
6 10
7 11
8 12
-.
´
• Dinh ly 1: Cho ca c ma trˆ n A, B, C va ca c sˆ α, β sao cho ca c phe p
´
´
a
` ´ o
´
´
.
˜ ´
´
`
a
toa n sau d ˆy d u.o.c ta o thanh. Khi ˆ y ta se co :
´
¯a ¯ . .
6
1. A + B = B + A
6. (α.β).A = α.(β.A)
2. (A + B) + C = A + (B + C)
7. α.(A.B) = (α.A).B = A.(α.B)
3. A.(B.C) = (A.B).C
8. α.(A + B) = α.A + α.B
4. (A + B).C = A.C + B.C
9. (α + β).A = α.A + β.A
5. A.(B + C) = A.B + A.C
10. No i chung, A.B = B.A
´
-.
´
• Dinh ly 2: Cho ca c ma trˆ n A, B. Khi ˆ y ta co :
´
´
a
a
´
.
1. (AT )T = A
2. (A + B)T = AT + B T
3. (A.B)T = B T .AT
4. (λ.A)T = λ.AT
7
´.
- .
§2. DINH THU C
a11
a12
...
a1n
a21 a22 . . . a2n
´
• Cho ma trˆ n vuˆng cˆ p n co da ng: A = .
a
o
a
´ .
.
. . Ta
.
.
.
.
.
...
.
.
an1 an2 . . . ann
a
¯a ’ ¯ `
´
`
a
o
a `
ky hiˆ u Mij la ma trˆ n vuˆng lˆ p tu. ma trˆ n A sau khi d˜ bo d i hang thu.
´ e
.
.
.
.
’
’
´
a
`
¯ .
i va cˆt thu. j cua ma trˆ n A va Mij d u.o.c goi la ma trˆ n con cua ma trˆ n
` o
a
a
.
.
. `
.
.
´ng ´i `
a ’
A u. vo. phˆn tu. aij .
1 −2 3
*. Vı du : Cho ma trˆ n A = −5 2
´ .
a
7 . Khi d´ ta co :
¯o
´
.
M11 =
2
−2
2
7
1
−3
, M21 =
1
3
1
−3
−3
, M32 =
1
3
−5 7
...
-.
˜
2.1. Dinh nghı a
-.
´c ’
a
` o o ´ e `
• Dinh thu. cua ma trˆ n A = (aij )n×n la mˆt sˆ , ky hiˆ u la
.
. ´
.
a11
a12
a13 . . .
a1n
a21
a22
a23 . . .
a2n
det(A) = a31
.
.
.
a32
.
.
.
a33 . . .
...
a3n
.
.
.
an1
an2
...
ann
va d u.o.c xa c d inh nhu. sau:
` ¯ . ´ ¯.
´
(1). A la ma trˆ n cˆ p 1(n = 1):
`
a a
.
`
A = ( a11 ) thı
det(A) = a11
´
(2). A la ma trˆ n cˆ p 2 (n = 2):
`
a a
.
a11 a12
= a11 .a22 − a12 .a21
det(A) =
a21 a22
= a11 . det(M11 ) − a21 . det(M21 )
`
`
`
’ a
’ `
’
`
` ´
a
(Chu ´ r˘ ng a11 va a12 la ca c phˆn tu. n˘ m cung o. hang 1 cua ma trˆ n
´ y a
`
a
.
’
A), vˆn vˆn, va mˆt ca ch tˆ ng qua t,
a a
` o ´
o
´
.
´
(3). A la ma trˆ n cˆ p n (n ≥ 3) thı
`
a a
`:
.
8
det(A) = a11 . det(M11 ) − a21 . det(M21 ) + a31 . det(M31 ) − . . . +
+ (−1)i+j .aij . det(Mij ) + . . . + (−1)n+1 .an1 . det(Mn1 )
’
`i
e
`
(Ngu.o. ta goi la phe p khai triˆ n theo hang 1).
. ` ´
*. Vı du :
´ .
1 2 3
4 5 6 = 1.
7 8 9
5 6
− 4.
8 9
2
3
8
9
+ 7.
2 3
5 6
= 1.(45 − 48) − 4.(18 − 24) + 7.(12 − 15) = 0.
’
’ ¯.
´ o
´c
e
´c
`
`
• Tu.o.ng tu. ta co cˆng thu. khai triˆ n cua d inh thu. theo hang k nao
.
d´ :
¯o
det(A) = (−1)k+1 [ak1 det(Mk1 )−ak2 det(Mk2 )+. . .+(−1)n+1 akn det(Mkn )]
’
`
´c
a
´
e
`
*. Vı du 1: Tı
´ .
´nh d inh thu. sau b˘ ng ca ch khai triˆ n theo hang 3.
¯.
1 −2
0
2
4
−1 = (−1)3+1 2
2
3
−2
−5
4
0
−1
−3
1
0
2 −1
+ (−5)
1 −2
2
4
= 2.(2 − 0) − 3.(−1 − 0) + 7.(4 − 4) = −1.
’
`
´c
a
´
e
`
*. Vı du 2: Tı
´ .
´nh d inh thu. sau b˘ ng ca ch khai triˆ n theo hang 4
¯.
2
1 1
1
1
2 1
1
1
1 2
1
a
b
d
c
1
= (−1)4+1 a. 2
1
2 1 1
1 1
2 1
1 1 − b. 1 1 1 + c. 1 2
1 2 1
2 1
1 1
1
1 − d. 1 2
1
= −a − b − c + 4d
’
´
• Chu ´ : Trong tru.o. ho.p n = 3 ta co thˆ dung quy t˘c Sarrus sau
´ y
`ng .
´ e `
a
d ˆy:
¯a
a11 a12 a13 a11 a12
a21
a22
a23
a21
a22
a31
a32
a33
a31
a32
9
2 1
1 1
1
1
2
` ¯o
´
Tu. d´ ta co :
a11
a12
a13
a21
a22
a23
a31
a32
a33
=
a11 .a22 .a33 + a12 .a23 .a31 + a13 .a21 .a32
− a13 .a22 .a31 − a11 .a23 .a32 − a12 .a21 .a33
´
´
’ ¯i
´
2.2. Mˆt sˆ tı
o o ´nh chˆ t cua d .nh thu.c
a
.
´
• Tı
´nh chˆ t 1: A = AT
a
*. Vı du :
´ .
1 2
3 4
= −2 ,
1 3
2 4
= −2
’
´
• Tı
´nh chˆ t 2: Khi d ˆ i vi trı cua hai hang (hai cˆt) cho nhau thı d .nh
a
¯o . ´ ’
`
o
` ¯i
.
’ ´
´c ¯o a
thu. d ˆ i dˆ u.
*. vı du : Ta co :
´ .
´
1 3
2 5
= 1.5 − 2.3 = −1
-o
’
’
˜
Dˆ i chˆ hang 1 cho hang 2 ta cu ng d u.o.c:
o `
`
¯ .
2 5
1 3
= 2.3 − 1.5 = 1 = −(−1)
-.
´
´
´c ´ o `
o o
` ¯o `
o
` o
• Tı
´nh chˆ t 3: Dinh thu. co mˆt hang (mˆt cˆt) nao d´ gˆm toan sˆ
a
.
. .
`
0 thı b˘ ng 0.
` a
*. Vı du :
´ .
1 2
0 0
0 =0
2 7
6
9
hay
9 0
1
0 0 =0
2
3
3 0
-.
´
`
’ e
• Tı
´nh chˆ t 4: Dinh thu. co hai hang (hai cˆt) ty lˆ nhau thı b˘ ng 0.
a
´c ´
`
o
` a
.
.
10
*. Vı du 1:
´ .
m.a
n.a
x
y
a
b
hay
c
*. Vı du 2:
´ .
1
1 2
1
2 4
9
y
x =0
u
t
2.1 1
3 = 2
3 6
a
n.t
=0
z
b
n.x
m.b m.c
2.2 3 = 0
3
2.3 9
´
´
• Tı
´nh chˆ t 5: Nˆ u nhˆn mˆt hang (mˆt cˆt) nao d´ cua d inh thu.
a
e
a
o `
o o
` ¯o ’ ¯.
´c
.
. .
´
´i o o
` ¯.
´c ¯ .
a e ´i o ¯o
vo. mˆt sˆ λ = 0 thı d inh thu. d u.o.c nhˆn lˆn vo. sˆ λ d´ .
. ´
*. Vı du :
´ .
m.a
m.b
x
y
*. Vı du :
´ .
= m.
2
3
4
8
=
a
b
x y
2
hay
3
n.a
t
n.x m
= 4.
2
= n.
a
t
x m
3
=4
4.1 4.2
1 2
-.
’ ´
´
´c o
¯o e
o
`
o `
• Tı
´nh chˆ t 6: Dinh thu. khˆng thay d ˆ i nˆ u ta cˆng vao mˆt hang
a
.
.
.p tuyˆ n tı
’
´
’
e ´nh cua mˆt sˆ hang (cˆt) kha c.
o o `
o
´
(mˆt cˆt) nao d´ mˆt tˆ ho
o o
` ¯o o o .
. ´
. .
.
.
*. Vı du 1:
´ .
a1
a2
a3
(a1 + α.a2 − β.a3 )
a2
a3
b1
b2
b3 = (b1 + α.b2 − β.b3 )
b2
b3
c1
c2
c3
(c1 + α.c2 − β.c3 )
c2
c3
1
3
*. Vı du 2:
´ .
2
2 1
3
4 5
7 = 4 + (−2).2 5 + (−2).1 7 + (−2).3 = −20
6 1
5
6
1
*. Vı du 3:
´ .
11
5
a
1
1
1
a
1
1
1
1
a
1
1
1
1
a+3
a
=
1
1
1
a+3 a
1
1
a+3
1
a
1
a+3
1
1 1 a
1
1
0 a−1
= (a + 3).
0
0
1
1
1
1
1 a
1
1
1
1
a
1
1
1
a
1
1
1
0
0
a−1
0
= (a + 3).
= (a + 3).(a − 1)3
0
0
0
a−1
-.
´
• Tı
´nh chˆ t 7: Dinh thu. cua ma trˆ n tam gia c co da ng du.o. d ˆy
a
´c ’
a
´ ´ .
´i ¯a
.
d u.o.c xa c d inh la:
¯ . ´ ¯.
`
a11
a12
a13
...
a1n
0
a22
a23
...
a2n
0
.
.
.
0
.
.
.
a33
.
.
.
...
...
a3n = a11 .a22 .a33 . . . ann
.
.
.
0
0
0
...
ann
a11
0
0
...
0
a21
a22
0
...
0
a31
.
.
.
a32
.
.
.
a33
.
.
.
...
...
0
.
.
.
an1
an2
an3
...
ann
1 2 3
4
0 5 6
−2
0 0 3
1
0 0 0
−2
= a11 .a22 .a33 . . . ann
*. Vı du 1:
´ .
= 1.5.3.(−2) = −30
12
*. Vı du 2:
´ .
1
0
0
0 0
4
3
0
0 0
3
2 −2 0 0 = 1.3.(−2).4.5 = −120
1
0
2
4 0
1
2
5
0 5
´
´
`
• Tı
´nh chˆ t 8: Khi tˆ t ca ca c phˆn tu. cua mˆt hang (hay mˆt cˆt) co
a
a ’ ´
a ’ ’
o `
o o ´
.
. .
’ a ´ch `
’
’
´
’
’
´c ´ e
o
da ng tˆ ng cua hai sˆ ha ng thı d .nh thu. co thˆ phˆn tı thanh tˆ ng cua
o
o .
` ¯i
.
˜ `
´c. ´
hai d inh thu. Co nghı a la:
¯.
a1
a2
a3 + a3
a1
a2
a3
a1
a2
a3
b1
b2
b3 + b3 = b1
b2
b3 + b1
b2
b3
c1
c2
c3 + c3
c2
c3
c2
c3
a1
a2
a3
a1
a2
a3
c1
c1
a1 + a1
a2 + a2
a3 + a3
b1
b2
b3
= b1
b2
b3 + b1
b2
b3
c1
c2
c3
c1
c2
c3
c2
c3
2 1
x+y
2
1
x
2 1
y
0 5
x +y
= 0
5
x
+ 0 5
y
c1
*. Vı du :
´ .
3 2 x +y
3 2 x
3 2 y
= 15(x + y) + 7(x + y ) + 10(x + y )
ˆ
- ’
§3. MA TRA N NGHICH DAO
.
.
-.
˜
3.1. Dinh nghı a
´
a
• Cho A la mˆt ma trˆ n vuˆng cˆ p n. Ma trˆ n B d u.o.c goi la ma trˆ n
` o
a
o
a
a
¯ .
. `
.
.
.
.
−1
´
’
’ ˜
nghich d ao cua ma trˆ n A (ky hiˆ u la: A ) nˆ u thoa ma n:
a
´ e `
e
. ¯’
.
.
A.B = In
va
`
13
B.A = In
*. Vı du : Cho ma trˆ n
´ .
a
.
2
3
9
−3
9
Xe t mˆt ma trˆ n B =
´
o
a
.
.
Ta co :
´
A.B =
B.A =
2 −1
3
3
9
−3
9
3
1
9
2
9
−1
3
A=
3
1
9
2
9
3
9
−3
9
.
.
1
9
2
9
2 −1
3
3
=
=
1 0
0 1
1 0
0 1
= I2
= I2
’
Vˆ y B chı
a
´nh la ma trˆ n nghich d ao cua ma trˆ n A.
`
a
a
.
.
. ¯’
.
´
` .
’
• Nˆ u A tˆn ta i ma trˆ n nghich d ao thı ta no i A la ma trˆ n kha nghich.
e
o
a
`
´
`
a
.
. ¯’
.
.
-.
• Dinh ly (d iˆu kiˆ n tˆn ta i ma trˆ n nghich d ao)
´ ¯`
e
e `
a
. o .
.
. ¯’
- `
’ .
´
`
Diˆu kiˆ n cˆn va d u dˆ mˆt ma trˆ n A vuˆng cˆ p n tˆn ta i ma trˆ n
e
e `
a ` ¯ ’ ¯e o
a
o
a
o .
a
.
.
.
nghich d ao la: det(A) = 0.
. ¯’ `
´ `m
a
¯a
3.2. Ca c phu.o.ng pha p tı ma trˆ n nghich d ’ o
´
.
.
a11
a21
` `m
’
’ ’
a
a
Gia su. ta cˆn tı ma trˆ n nghich d ao cua A = .
.
. ¯’
.
.
an1
• Phu.o.ng pha p 1
´
a12
...
a22
.
.
.
...
an2
...
...
a1n
a2n
.
.
.
ann
´
Ta ky hiˆ u Cij = (−1)i+j . det(Mij ) va d u.o.c goi la phˆn phu d a i sˆ cua
´ e
` ¯ . . ` `
a
.
. ¯. o ’
´
`
’
’
´c ´ ¯i
a
a
´ o
phˆn tu. aij . Khi ˆ y ta co cˆng thu. xa c d .nh ma trˆ n nghich d ao cua ma
a
.
. ¯’
trˆ n A nhu. sau:
a
.
T
C11 C12 . . . C1n
1 C21 C22 . . . C2n
−1
.
A =
.
.
.
.
det(A) .
.
...
.
.
Cn1 Cn2 . . . Cnn
14
1 2
’ ’
Gia su. cho ma trˆ n A = 2 5
a
.
1 0
−1 = 0. Ngoai ra ta co :
`
´
*. Vı du 1:
´ .
C11 = 40
3
3 . Ta co : det(A) =
´
8
C12 = −13 C13 = −5
C21 = −16
C22 = 5
C23 = 2
C31 = −9
C32 = 3
C33 = 1
Do d´ ta co ma trˆ n nghich d ao
¯o
´
a
.
. ¯’
40 −13
1
−1
5
A =
−16
−1
−9
3
’
cua ma trˆ n A nhu. sau:
a
.
T
−5
40 16
9
2 = 13 −5 −3
1
5
1
−2 −1
−3 4
’ ’
a
1 1
*. Vı du 2: Gia su. cho ma trˆ n A = 2
´ .
.
−1 −2 1
Ta co : det(A) = 0, nˆn ma trˆ n A khˆng co ma trˆ n nghich d ao.
´
e
a
o
´
a
.
.
. ¯’
´
• Phu.o.ng pha p 2
-a `
’
´
´
’
´
`
´
´
e ¯ˆ
a
`
Dˆy la phu.o.ng pha p dung ca c phe p biˆ n d o i so. cˆ p theo hang cua ma
’
’
´
` ` ´
trˆ n dˆ tı ma trˆ n nghich d ao. Nˆi dung cua phu.o.ng pha p nay la chu ng
a ¯e `m
a
o
.
.
. ¯’
.
´
´
’ ’
a
`
ta viˆ t vao bˆn phai cua ma trˆ n A mˆt ma trˆ n d o.n vi cung cˆ p. Dung
e ` e
a
o
a ¯
. `
.
.
.
’ ´
’
´
´
’
’
a
`
`
a ¯e e
ca c phe p biˆ n d ˆ i so. cˆ p theo hang (chı theo hang) cua ma trˆ n dˆ biˆ n
´
´
e ¯o
.
´
` o
ma trˆ n sau khi d˜ ghe p (co cˆ p la: n × 2n) vˆ mˆt ma trˆ n sao cho ma
a
¯a ´
´ a `
e .
a
.
.
.n vi n˘ m vˆ phı bˆn tra i va khi ˆ y phı bˆn phai cua ma trˆ n nay
`
` ´a e
´
’ ’
e
´ `
a
´a e
a `
trˆ n d o . a
a ¯
.
.
15
’
tı
`m. Cu thˆ chu ng ta mˆ ta nhu. sau:
o ’
. e ´
. . . a1n 1 0 . . . 0
. . . a2n 0 1 . . . 0
. . .
.
. . . ... .
.
...
. . .
an1 an2 . . . ann 0 0 . . . 1
’
´
−→ . . . . . . . . . biˆ n d ˆ i . . . . . . . . .
e ¯o
chı
´nh la ma trˆ n nghich d ao cˆn
`
a
a
.
. ¯’ `
a11 a12
a21 a22
A/I = .
.
.
.
.
.
’
´
−→ . . . . . . . . . biˆ n d ˆ i . . . . . . . . .
e ¯o
1 0 . . . 0 x11 x12 . . .
0 1 . . . 0 x21 x22 . . .
−→ . .
.
. . ... . .
. .
.
. .
.
...
. .
0 0 . . . 1 xn1 xn2 . . .
´
Khi ˆ y ta co :
a
´
−1
A
x12
...
xn2
x21
= .
.
.
xn1
...
x22
.
.
.
x11
x1n
...
x1n
x2n
.
.
.
xnn
x2n
.
.
.
xnn
...
’
*. Vı du : Tı ma trˆ n nghich d ao cua ma trˆ n sau:
´ .
`m
a
a
.
. ¯’
.
1 1 −3
A = −1 0 2
−3 5
0
Ta co :
´
1
1
1
0 0
1
1 −3 1 0
0
2
0
1 −1 1 1
0
0
1 0 −→ 0
0
A/I = −1 0
−3 5
−3
0 1
0
8 −9 3 0
1
1
1 −3
1
0
0
1
1
0
16
24
−3
−→ 0
1 −1
1
1
0 −→ 0
1
0
6
9
−1
0
0 −1 −5 −8 1
0
16
0 −1 −5 −8
1
1
0
0
10
15
−2
−→ 0
1
0
6
9
−1 −→ 0 1
0
0 −1 −5 −8
Vˆ y:
a
.
1
1 0
0 10 15 −2
0 6
1 5
0 0
10 15 −2
A−1 = 6
8
−1
−1
−1
9
5
9
8
−1
’
3.3. Ha ng cua ma trˆ n
a
.
.
-.
˜
´
´
’
´c
• Dinh nghı a: Ha ng cua mˆt ma trˆ n A la cˆ p cao nhˆ t cua d .nh thu.
o
a
` a
a ’ ¯i
.
.
.
a
´ e `
con kha c 0 lˆ p tu. ma trˆ n A. Ky hiˆ u la: rank(A) hay r(A)
´
a `
.
.
.
*. Vı du : Xe t ma trˆ n sau
´ .
´
a
.
1
A= 2
−3 4
1
1
−1 −2 1
2
4
−2
´
’
´c
a
`
Ca c d inh thu. con cˆ p ba cua A la
´ ¯.
1
1
2
1 =0
1
1
1
−2 1
1
4
1
=0
−3
=0
2
2
2
4
2
−1 1 −2
−1 −2 1
−3 4
4
2
−3 4
1
4
−1 −2 −2
−2
´
’
Ta co d inh thu. con cˆ p hai cua A la
´ ¯.
´c
a
`
1
−3
2
1
=7
Vˆ y r(A) = 2
a
.
’
´ `m .
a
• Phu.o.ng pha p tı ha ng cua ma trˆ n
.
17
=0
’
’
˜ ¯e `m .
’
Chu ng ta co thˆ dung d inh nghı a dˆ tı ha ng cua ma trˆ n, tuy nhiˆn
´
´ e ` ¯.
a
e
.
´
´
´
´ ’
´
´
` a .
e
a `
a
a a ´n. ` e
phu.o.ng pha p nay rˆ t ha n chˆ , nhˆ t la khi cˆ p cua ma trˆ n rˆ t lo. Vı thˆ
. ´
’
’
´
´
’ .
’
chu ng ta su. du ng phu.o.ng pha p biˆ n d ˆ i so. cˆ p cua ma trˆ n dˆ tı ha ng
´
´
e ¯o
a
a ¯e `m .
.
’
’
´
` ` ´
`
´
´
cua ma trˆ n, nˆi dung cua phu.o.ng pha p nay la chu ng ta dung ca c phe p
a
o
.
.
’
’
´
´
’
a
`
a o
a ’
a ¯e ¯
biˆ n d ˆ i so. cˆ p theo hang (ho˘ c cˆt, ho˘ c ca hai) cua ma trˆ n dˆ d u.a ma
e ¯o
. .
.
.
´
´
’
trˆ n d´ vˆ da ng ma trˆ n bˆ c thang thu gon nhˆ t. Khi ˆ y ha ng cua
a ¯o ` .
e
a
a
a
a
.
.
.
.
.
´
´
ma trˆ n chı la sˆ ca c hang kha c khˆng (ho˘ c sˆ ca c cˆt kha c khˆng, nˆ u
a
´nh ` o ´ `
´
o
a o ´ o
´
o
e
.
. ´
.
´
’
’
a
o `
nho ho.n) cua ma trˆ n cuˆ i cung.
.
*. Vı du 1 :
´
.
1
0
1 −2
1
Cho ma trˆ n A =
a
.
2
1
3 −2
’
´
´
a
`
´
e ¯ˆ
Dung phe p biˆ n d o i so. cˆ p theo
5 −1
’
hang cua ma
`
1
0
A −→
0
1
1 −1 1 4
trˆ n ta co :
a
´
.
0 1 −2
1 0
0 1
1 2 0
−→
0 0
1 3 3
0 −1 0
6
1 −2
Vˆ y ta co : rank(A) = 3.
a
´
.
1
1
2
2
1 0
0 1
0
−→
0 0
3
2
0 0
6
1
0 0
’
*. Vı du 2: Tı λ dˆ A = 2 λ −2 co ha ng la 2
´ .
`m ¯e
´ .
`
3 −6 −3
Ta co :
´
1 2
1
1 1
2
2 λ −2 −→ 3 −3 −6
3
−6 −3
1 1
−→ 0 −6
2 −2 λ
2
1 1
2
−12 −→ 0 −6 −12
0 −4 λ − 2
0
-e
’
`
Dˆ r(A) = 2 thı λ + 6 = 0 ⇒ λ = −6
thı
`
r(A) = 2
Vˆ y vo. λ = −6
a ´i
.
18
0
λ+6
1 −2
2
1
0
0
3
0
. .
`
´
´
ˆ
ˆ
§4. HE PHU O NG TRI NH TUYEN TI NH
.
-.
˜
4.1. Dinh nghı a
’ ´
`nh co da ng:
´ .
• Hˆ gˆm n−ˆ n sˆ {x1 , x2 , x3 , . . . , xn } va m−phu.o.ng trı
e `
a o
`
. o
a11 x1 + a12 x2 + . . . + a1n xn = b1
a21 x1 + a22 x2 + . . . + a2n xn = b2
(4.1)
...............
am1 x1 + am2 x2 + . . . + amn xn = bm
´
`nh tuyˆ n tı
e ´nh.
d u.o.c goi la hˆ phu.o.ng trı
¯ . . ` e
.
´
• Nˆ u chu ng ta d ˘ t
e
´
¯a
.
a11 a12 . . .
a21 a22 . . .
A= .
.
.
.
.
...
.
am1
am2
...
´ e
`nh
thı khi ˆ y hˆ phu.o.ng trı
`
a
.
sau:
a11 a12
a21 a22
.
.
.
.
.
.
am1
am2
a1n
x1
b1
a2n
x2
b2
. ; X = . ; B= .
.
.
.
.
.
.
amn
xn
bm
˜ ¯ .
´
(4.1) se d u.o.c viˆ t la i theo da ng ma trˆ n nhu.
e .
a
.
.
...
...
...
...
a1n
x1
b1
a2n x2 b2
. = .
. . .
.
. . .
xn
bm
amn
’ ´
hay co thˆ viˆ t gon la:
´ e e . `
A.X = B
a e o
a
¯ . . `
a
• Ma trˆ n A d u.o.c goi la ma trˆ n hˆ sˆ , ma trˆ n B d u.o.c goi la ma trˆ n
a
¯ . . `
.
. ´
.
.
.
. do va X d u.o.c goi la ma trˆ n nghiˆ m sˆ o. da ng cˆt.
´
`
¯ . . `
a
e
o ’ .
o
hˆ sˆ tu
e o .
.
.
.
. ´
´
’
e
e
¯ .
• Bˆ n−sˆ co da ng X = (α1 , α2 , . . . , αn ) d u.o.c goi la nghiˆ m cua hˆ
o
o ´ .
. `
.
.
.
19
´
`nh
e
phu.o.ng trı (4.1) nˆ u khi thay
´
´c.
ca c d` ng nhˆ t thu.
´ ¯ˆ
o
a
x1 = α1
x2 = α2
...
xn = αn
vao hˆ (4.1) thı chu ng ta d u.o.c
` e
` ´
¯ .
.
´ ´ ´
• Hˆ (4.1) d u.o.c goi la tu.o.ng thı
e
¯ .
´ch nˆ u no co nghiˆ m, d u.o.c goi la
e
e
¯ .
.
. `
. `
.
´
´
´ch nˆ u nhu. no vˆ nghiˆ m, va d u.o.c goi la vˆ d .nh nˆ u
e
´ o
e
` ¯ .
khˆng tu.o.ng thı
o
e
.
. ` o ¯i
´ ´
o
e
nhu. no co ho.n mˆt nghiˆ m.
.
.
a11 a12 . . . a1n b1
a21 a22 . . . a2n b2 . .
• Ma trˆ n A co da ng: A = .
a
´ .
.
.
. d u o c goi la
.
. `
.
.
.
. ¯ .
.
...
.
.
.
am1 am2 . . . amn bm
’
ma trˆ n hˆ sˆ bˆ sung cua ma trˆ n A.
a
e o o
a
.
. ´ ’
.
-.
’
o
e
e
`nh
4.2. Dinh ly vˆ su. tˆn ta i nghiˆm cua hˆ phu.o.ng trı
´ ` . `
e
.
.
.
-.
• Dinh ly : Hˆ (4.1) la tu.o.ng thı khi va chı khi rank(A) = rank(A).
´
e
`
´ch
` ’
.
• Nhˆ n xe t:
a
´
.
´
` e
o
e
(1). Nˆ u rank(A) = rank(A) thı hˆ (4.1) vˆ nghiˆ m.
e
.
.
´
´
` e
´
e
a
(2). Nˆ u rank(A) = rank(A) = n thı hˆ (4.1) co nghiˆ m duy nhˆ t.
e
.
.
´
´
` e
´ o o
e
(3). Nˆ u rank(A) = rank(A) < n thı hˆ (4.1) co vˆ sˆ nghiˆ m.
e
.
.
’ .
*. Vı du 1: Tı m dˆ hˆ sau co nghiˆ m.
´ .
`m
¯e e
´
e
.
x1 + 2x2 − x3 + 4x4 = 2
2x1 − x2 + x3 + x4 = 1
x1 + 7x2 − 4x3 + 11x4 = m
Ta co :
´
20
1
A =2
1
1
−→ 0
0
2
−1
4
2
1
2
−1
4
2
1 −→ 0 −5 3 −7 −3
7 −4 11 m
1 7 −4 11 m
2 −1 4
2
1 2 −1 4
2
−5 3 −7 −3 −→ 0 −5 3 −7 −3
−1
1
1
5
−3
7
m−2
0
0
0
0
m−5
-e e ´
’ .
Dˆ hˆ co nghiˆ m thı r(A) = r(A)
e
`:
.
Ma theo trˆn thı r(A) = 2 ⇒ r(A) = 2
`
e
`
⇐⇒ m − 5 = 0 ⇒ m = 5
Vˆ y vo. m = 5 thı hˆ phu.o.ng trı
a ´i
` e
`nh trˆn co nghiˆ m.
e ´
e
.
.
.
.o.ng trı
´
´
’
`nh theo tham sˆ a:
o
*. Vı du 2: Biˆ n luˆ n sˆ nghiˆ m cua phu
´ .
e
a o
e
.
.
.
ax1 + x2 + x3 = 1
x1 + ax2 + x3 = 1
x1 + x2 + ax3 = 1
Ta co :
´
a
1
1 1
1
1
a 1
A =1
a
1 1 −→ 1
a
1 1
1
1
−→ 0
a
1
a 1
1 1
1
a
1
1
1
a − 1 1 − a 0 −→ 0 a − 1
1
a
1
1
a
0
0
a
1
1−a
0
2 − a − a2 1 − a
´
• Nˆ u: 2 − a − a2 = 0 =⇒ a = 1, a = −2
e
Khi a =1 thı
`:
1
A = 0
0
1 1 1
0 0 0 =⇒ r(A) = r(A) = 1 < 3
0 0 0
´
Nˆn hˆ vˆ d .nh (co vˆ sˆ nghiˆ m).
e e o ¯i
´ o o
e
.
.
21
Khi a = -2 thı
`:
1
1
A = 0 −3
0
0
−2 1
3
0
0 =⇒ r(A) = 2, r(A) = 3
3
=⇒ r(A) = r(A) nˆn hˆ vˆ nghiˆ m.
e e o
e
.
.
´
`
• Nˆ u: 2 − a − a2 = 0 =⇒ a = 1 va a = −2
e
´
e e ´
e
a
=⇒ r(A) = r(A) = 3. Nˆn hˆ co 1 nghiˆ m duy nhˆ t.
.
.
Vˆ y: - Vo. a = 1 thı hˆ vˆ d .nh.
a
´i
` e o ¯i
.
.
. a = -2 thı hˆ vˆ nghiˆ m.
´i
` e o
e
- Vo
.
.
. a = 1 va a = −2 thı hˆ co nghiˆ m duy nhˆ t.
´
´i
`
` e ´
e
a
- Vo
.
.
’
’ e
4.3. Phu.o.ng pha p giai hˆ phu.o.ng trı
´
`nh tˆ ng qua t
o
´
.
’
• Phu.o.ng pha p Gauss: Nˆi dung cua phu.o.ng pha p nay la chu ng ta
´
o
´
` ` ´
.
’ ´
’
’
´
´
’
a
e
`nh dˆ biˆ n d ˆ i va loa i
¯e e ¯o ` .
dung ca c phe p biˆ n d ˆ i so. cˆ p cua hˆ phu.o.ng trı
`
´
´
e ¯o
.
’
´
` ˆ o
´
`nh cuˆ i cung dˆ dang thu d u.o.c nghiˆ m
o `
e `
¯ .
e
dˆn a n sˆ sao cho hˆ phu.o.ng trı
a ’
e
.
.
.n. Ca c phe p biˆ n d ˆ i so. cˆ p cua hˆ phu.o.ng trı
’
´
´
`
’
ho
´
´
e ¯o
a
e
`nh gˆm:
o
.
-o . ´
’
`nh cho nhau.
(1). Dˆ i vi trı hai phu.o.ng trı
`nh.
(2). Nhˆn mˆt sˆ λ = 0 vao mˆt phu.o.ng trı
a
o o
`
o
. ´
.
’
(3). Cˆng vao mˆt phu.o.ng trı
o
`
o
`nh cua hˆ mˆt phu.o.ng trı
e o
`nh kha c sau
´
.
.
. .
khi d˜ nhˆn vo. mˆt sˆ kha c 0.
¯a a ´i o o ´
. ´
’
´
´ ’ e
´
• Nhˆ n xe t: Vı ca c phe p biˆ n d ˆ i so. cˆ p cua hˆ phu.o.ng trı
a
´
` ´
´
e ¯o
a
`nh giˆ ng
o
.
.
’
´
´
’
´
´
e ¯o
a
`
a
a
´
´
nhu. ca c phe p biˆ n d ˆ i so. cˆ p theo hang cua ma trˆ n. Do vˆ y chu ng ta co
.
.
’
’
’
´
’
’
thˆ dung ca c phe p biˆ n d ˆ i theo hang (chı theo hang) cua ma trˆ n dˆ
e `
´
´
e ¯o
`
`
a ¯e
.
’
’
´
´
’ e
`nh. Cu thˆ : Dung ca c phe p biˆ n d ˆ i so. cˆ p
`
´
´
e ¯o
a
tı nghiˆ m cua hˆ phu.o.ng trı
`m
e
. e
.
.
’
` .
’
a e o o
theo hang cua ma trˆ n dˆ d u.a ma trˆ n hˆ sˆ bˆ sung vˆ da ng ma trˆ n bˆ c
`
a ¯e ¯
e
a a
.
. ´ ’
.
.
.
.o.ng trı
˜
´
´
´
`nh
thang thu gon nhˆ t, khi ˆ y ma trˆ n cuˆ i cung se cho ta hˆ phu
a
a
a
o `
e
.
.
.
’
¯
´i e
`nh ban d` u va do d´ ta dˆ dang co d u.o.c
¯ˆ `
a
¯o
e `
´ ¯ .
tu.o.ng d u.o.ng vo. hˆ phu.o.ng trı
.
’ e
nghiˆ m cua hˆ ban d` u.
e
¯ˆ
a
.
.
22
’ e
`nh:
*. Vı du 1: Giai hˆ phu.o.ng trı
´ .
.
2x1 + 4x2 + 3x3 = 4
3x1 + x2 − 2x3 = −2
4x1 + 11x2 + 7x3 = 7
Ta co :
´
2
4
A =3
1
4
2
4
−2 −2 −→ 3
1
7
0
4
2
−→ 0 10 13 16 −→ 0
0 29 29 29
0
4 11
2 4
3
7
3
3
4
−2 −2
29 29 29
4
3 4
10 13 16
0 87 174
a
´
Nhu. vˆ y ta co :
.
x1 = 1
2x1 + 4x2 + 3x3 = 4
⇐⇒ x2 = −1
10x2 + 13x3 = 16
87x3 = 174
x3 = 2
’ e
`nh sau:
*. Vı du 2: Giai hˆ phu.o.ng trı
´ .
.
3x1 − 5x2 + 2x3 + 4x4 = 2
7x1 − 4x2 + x3 + 3x4 = 5
5x1 − 7x2 − 4x3 − 6x4 = 3
Ta co :
´
3
−5
2
4
2
A =7
−4
1
3
5 −→ 7 −4
5
1
−→ 0
0
1
6
−3 −5 −1
1
3
5
−7 −4 −6 3
5 7 −4 −6 3
6
−3 −5 −1
1
6
−3 −5 −1
−46 22 38 12 −→ 0 −46 22 38 12
−23
11
19
8
0
Co : r(A) = 2 va r(A) = 3
´
`
=⇒ r(A) = r(A)
23
0
0
0
−4
`nh d˜ cho trˆn la vˆ nghiˆ m.
¯a
e ` o
e
Vˆ y hˆ phu.o.ng trı
a e
.
.
.
’ e
`nh sau:
*. Vı du 3: Giai hˆ phu.o.ng trı
´ .
.
x1 + x2 − 3x3 − 2x4 + 3x5 = 1
2x + 2x + 4x − x + 3x = 1
1
2
3
4
5
3x1 + 3x2 + 5x3 − 2x4 + 3x5 = 1
2x1 + 2x2 + 8x3 − 3x4 + 9x5 = 6
Ta co :
´
1 1 3
2 2 4
A =
3 3 5
−2 3 1
1 1
3
0 0 −2
−1 3 1
−→
0 0 −4
−2 3 1
2 2 8 −3 9 6
1 1 3 −2 3
1
0 0
2
1 1
−2
3
3
−3
4
1
3
1
0
−6 −2
3 4
−2 3
1
0 0 −2 3 −3 0
0 0 −2 3 −3 0
−→
−→
0 0 0 −2 0 −2
0 0 0 −2 0 −2
0 0
0
4
0
4
0 0
0
0
0
0
´
´
’
´c
a
Tˆ t ca ca c d .nh thu. con cˆ p 4 cua A = 0
a ’ ´ ¯i
1 3 −2
Ta co : 0 −2
´
3
= 4 = 0 =⇒ r(A) = r(A) = 3
0 0 −2
´ ’
´
Vˆ y r(A) = r(A) = 3 < 5 = n (sˆ ˆ n). Nˆn hˆ co vˆ sˆ nghiˆ m.
a
o a
e e ´ o o
e
.
.
.
x1 + x2 + 3x3 − 2x4 + 3x5 = 1 (1)
˜ cho tu.o.ng d u.o.ng vo.
¯
´i:
Hˆ d a
e ¯
−2x3 + 3x4 − 3x5 = 0 (2)
.
−2x4 = −2 (3)
(3)=⇒ x4 = 1
3 − 3x5
2
−2x2 + 3x5 − 3
(1)=⇒ x1 =
2
-a
D˘ t : x2 = s, x5 = t; s, t ∈ R
.
3
3 3
3
Ta co : x1 = −s + t − ; x3 = − t;
´
2
2
2 2
(2)=⇒ x3 =
24
x4 = 1
’
`nh la:
`
Vˆ y nghiˆ m tˆ ng qua t cua hˆ phu.o.ng trı
a
e
o
´ ’
e
.
.
.
3
3
3 3
(−s + t − , s, − t, 1, t); ∀s, t ∈ R
2
2
2 2
4.4. Hˆ Cramer
e
.
-.
˜
´
`nh tuyˆ n tı
e ´nh co da ng:
´ .
• Dinh nghı a: Hˆ phu.o.ng trı
e
.
a11 x1 + a12 x2 + . . . + a1n xn = b1
a21 x1 + a22 x2 + . . . + a2n xn = b2
(4.4)
...............
an1 x1 + an2 x2 + . . . + ann xn = bn
’ ˜ ¯`
va thoa ma n d iˆu kiˆ n:
`
e
e
.
a11
...
a1n
a21
.
.
.
a22
.
.
.
...
...
a2n
. =0
.
.
an1
det(A) =
a12
an2
...
ann
d u.o.c goi la hˆ Cramer.
¯ . . ` e
.
-.
´
• Dinh ly : Hˆ Cramer co nghiˆ m duy nhˆ t va d u.o.c xa c d .nh nhu. sau:
´
e
´
e
a ` ¯ . ´ ¯i
.
.
xj =
det(Aj )
det(A)
`
Trong d´ A la ma trˆ n ca c hˆ sˆ cua hˆ , Aj la ma trˆ n suy tu. A b˘ ng
¯o
`
a ´ e o ’
e
`
a
`
a
.
. ´
.
.
’ o e o .
´
ca ch thay cˆt thu. j bo.i cˆt hˆ sˆ tu. do.
´
o
.
. ´
.
’
• Hˆ qua:
e
.
´
+ Nˆ u det(A) = det(Aj ) = 0,
e
´
+ Nˆ u
e
∀j = 1, n
thı hˆ vˆ d .nh
` e o ¯i
.
det(A) = 0
thı hˆ vˆ nghiˆ m.
` e o
e
.
.
∃j, det(Aj ) = 0
’ e
*. Vı du 1: Giai hˆ phu.o.ng trı
´ .
`nh:
.
2x1 + 4x2 + 3x3 = 4
3x1 + x2 − 2x3 = −2
4x1 + 11x2 + 7x3 = 7
25
