เอกสารประกอบการเรียน เรื่อง เมทริกซ์

52

6 44 300 1 0 0
A 1 A 10 8 0 0 4 0 0 2 0

6 4 2 005 0 0 3

2 x1 x
A 1 x2 2

4 x3 2

A, B 3 3 AB 3I I adjB 1 A
3

det A

102 MAAR 53

aR A 030 a 10 det(adjA) 225 a

40a

a, b R A a2 A b adjA ab
2b 3

12 0 x0 det(2adjA) 288 (x 10)2
A1 3 1 1

x0 2

54 AB 22 A 2B 54 AB 21
8 16 15
det(2 A 1B)

A 11 B B 2A 1 det(3adjB)
21

A 1 2 , B 2 0 , C 20 CXA 2I B X
1 4 3 0 21

21 3 03 1

A 1 3 1 ,B 1 1 2 (adjB)X A X

25 2 02 1

A, B, I 33 Monkman 55
1 01
1 A 2 10
B1 0 101

AB I 1

B 7 8 ,C 6 10 , D 05 AB 2I AC D A
6 12 4 13 21

3 12 25 2

A 2 2 1 ,C 1 3 1 B(adjA) C B

010 213

56 3 12
1. M 32 ( A) A 21 2

010

2. M 23( A)

3. M32 (A 1) 4. det(adjA)

5. det Cij ( A)

A, B, C nn

Mij ( A) Mij ( At ) 2. Mij ( A) M ji ( At )

3. Mij ( A 1) 1 4. Mij ( A 1) 1
Mij ( A) M ji ( A)

5. det(adjAn n ) (det A)n 6. det(adjAn n ) (det A)n 1

7. det Cij ( An n ) det(adjAn n ) 8. det Cij ( An n ) (det A)n 1

9. adjI AA 1 10. adj(kA) (k n 1 det A) A 1

kR

momma 57
A 55 M 53 ( A) 2 M35 (2At )

03 1
A 1 12

02 1

1. det(2adjA) 2. det 3Cij ( A)

An n det(adj(adjA))
A n2 2n
2. A n2 2n 3. A (n 1)2 4. A (n 1)2

A, B 3 3 , B At A 1 , det(adjA) 9, 3(C B) adjA

det A 0 det C 2. 3 3. 9 4. 27
1

3

58

AB 22 I 22

A(adjA) BA I det B 4 det A 3.1 4. 2
1
2. 0

245 A BC B Bt , C Ct BC Dt
A 536
-
3 11

D dij 3 3 d12 d22 d32
3
2.11 3. 22 4.

12.

1. AX B A
1.1 B
X
1.2
X A 1B

59

1. xy 1
2x y 4

2. 3x 2y 1
y 2x 3

2x y z 5

3. 3x 2y 2z 3

x 3y 3z 2

xyz6

4. x y z 2

x yz 0

60 (Row operation)
2.
2.1 AX B
2.2 A:B

2.3 I :C

c1, c2 , c3, , cn

1. xy 1 2. 3x 2y 1
2x y 4 y 2x 3

2x y z 5 xyz6

3. 3x 2y 2z 3 4. x y z 2

x 3y 3z 2 x yz 0

3. egg 61

A n n det A 0 B AX B

x1, x2 , x3, , xn b1, b2 , b3, , bn
x1 b1
X x2 , B b2

xn bn

x1 det( A1) , x2 det( A2 ) , , xn det( An )
det A det A det A

Ai i A

D det A

1, 2, 3 x, y, z
Dx det( A1)
Dy det( A2 )
Dz det( A3)

1. xy 1 2. 3x 2y 1
2x y 4 y 2x 3

2x y z 5 xyz6

3. 3x 2y 2z 3 4. x y z 2

x 3y 3z 2 x yz 0

62

1. x 3y 4
6y 2x 2

x 3y 0

2. y 5z 3

2x z 1

x 2y 4z 2

3. 2x 3y 7z 3

3x y 5z 1

2x y 3z w 3

4. 3x 2y z w 13
x 3y z 2w 4

x y 4z 3w 0

63

10 1 p adjA 2 A3 X 18
A 0 13 Xq 45
9
3 20 r

p q 5r

A 33 AX i Bi i 1, 2, 3

1 1 11 00

X1 0 , X 2 2 , X3 3 , B1 0 , B2 1 , B3 0 det A
5 5 1 0 01

A a 1a a I 10
1a a 01

det A 2I A 3I A 5I A 7I

64

A 4a a, b, c A I 3 3A I det A 3I 0
bc

A

310 1 10

A 2 43 B 0 21

54 2 123

A 1B C Bt A t det 2A I C1

A 1 2 , B I A A2 A9 C I A A2 A9
11
det B k det C
k

common 65

A B 22 AB BA 20
02

A ab A A2 0
ca

1 23
A1
0k k det A2 2A 96
2 13

23x5 23x3 20kx2 46x 20k 0

A, B, C 33

221 x yz x, y, z Bt AB 80 0
A 2 5 2,B 0 y 2z 0 27 0
x yz 0 0 42
123
x y z det adjC
C AB

66 51 1

det A aI3 0 A 13 1 a

1 13

x x0

y A aI3 y 0 x2 y2 z2 1
z z0

1 2 13

A 1 11 3 A1
120 3

2 4 27

a, b, c 1 2a 12 1
3 1b 0 57
10 c 10 2

R2 3R1 abc