เอกสารประกอบการเรียน เรื่อง เมทริกซ์

เอกสารประกอบการเรยี น

วิชา คณิตศาสตร์ ค32201
ช้นั มธั ยมศึกษาปที ่ี 5

เรอื่ ง เมทริกซ์

ครูผ้สู อน
ครูเสฎฐวุฒิ เพง็ เจรญิ

4

angora5

1.

mn (m I n I) m (row)

n (column) a1n
a2n
a11 a12
a21 a22 amn m

am1 am2

n

aij i j ij
mn
i 1, 2, 3, , m j 1, 2, 3, , n

mn mn

(dimension of matrix)

A, B, C,...

a, b, c,...

A, B, C,...

a11 a12 a13 B b11 b12 b13
A a21 a22 a23 , b21 b22 b23

a31 a32 a33

a, b

a23

23

b12

aij i j
ij

A aij m n A mn ij aij
i 1, 2, 3, , m j 1, 2, 3, , n

6
1

14

A 02

23

B 52
13

C 420 1

D 4
2

E0

02 5

F 0 21

03 8

2 A 123 40
123 B 22

a11 b21 a13 04

a11b11 a12b21 a13b31

a11 b32 b22 a23b21 a13b12

A aij 3 3 aij 2(i j) B bij 2 2 bij 2i j2

BABA 7

A aij 3 3 aij i 2 j C cij 4 4 cij i j

A aij 3 3 A aij 3 3
0;i j
aij 2 ;i j
1 ;i j aij 1 ; i j
1;i j

A aij 3 3 i ;i j

aij 0 ; i j
a ji ; i j

aij A

022 1 3 45
A 20 2
A 34 5 6
22 0 4 59 7

5 6 7 16

8

2.

row matrix

1 ,2 1 3 ,0 1 8 0
11 13 14

column matrix

2 ,0 , 1
01 1 2
4
21
31

zero matrix

0 0 , 0 0 0, 0 0 0
0 0 13 0 0 0 23
22

0 0
AB
00
A 0,B 0 0

00

AB0

AB

square matrix

1 2 2 , 0 24
0 1 1 15
2 2 3 0 33

(diagonal elements)

12 4
A 20 5

423

A 1, 0 H2

a11, a22 , a33, , ann
1, 2, 3, 4, , n

(Unit identity matrix) Boga 9

1 (n n)
22
aij 1;i j
0;i j 33

In n

I2 10
01

100

I3 0 1 0
001

(scalar matrix)

aij k;i j k
0;i j

A 3 0 ,B 200
0 3 020
002

1) (triangular matrix)
2) upper triangular matrix
2.4.4
aij 0 ij

A 450 3 25 1
0 4 1 ,B 0 167
003 0 04 4
0 00 2

Wang lower triangular matrix

aij 0 ij

A 10 0 1 00 0
1 3 0 ,B 4 40 0
24 2 1 120
2 01 3

diagonal matrix

aij 0 ij

A 400 400 0
0 2 0 ,B 0 10 0
001 002 0
000 1

Mama

10

3.

A aij m n B aij m n i 1, 2, 3, , m j 1, 2, 3, , n
A B aij bij
AB
AB

A 1 2 , B 4 3 ,C 12 2 ,D x22 x3
4 3 1 2 22 3 4x

32 32 2. x2 1 1 01
x1 y 4x x y2 y1

3. xy 3 13 2x 1 1 3x 1
2 2x 8 2 3y 21
4. y2 1 1 17

1 z1

x x2 2x 1 0 A 2x 1 x 1 ,B x2 x x2
AB x2 1 x2 x 2x 3x 1

Masaoka 11
4.

A aij m n B aij m n cij cil ail bil
A B

i 1, 2, 3, , m j 1, 2, 3, , n AB

A aij m n B aij m n

A B cij cil ail bil

i 1, 2, 3, , m j 1, 2, 3, , n AB

A 2 1 ,B 2 2 ,C 0 0 , D 1 0 ,E 02 3
3 4 1 1 0 0 0 1 31 4

AB 2. B A

3. A C 4. A B

BE 6. C D
8. A B D
7. A B D

12

X

42 41 X 20
10 25 43

2. 23 X 24 12
10 51 44

1. AB
AB BA
AB
(A B) C A (B C)
2. 0 A0 A 0 A
0 A ( A) 0 ( A) A
AB

3.

A, B C

4.

A

5.

A

5.

A aij m n c bij m n bij caij
c A

i 1, 2, 3, , m j 1, 2, 3, , n

c A cA

2 5 32 13 sooooooo 13
1 0 10 24
A ,B C

3A B 2C

2. 2A 3 B 2C A

A 2 5 , B 32 C 13 X
1 0 10 24

1 X A 3X B 3C
2

2. A 2X 4 2X 3X B 2C

3 A 2X 5 C 1 A 5 2X 3B
2 3 3 6

14

3 x y 2 2 x 2y 1 1 1 x(2y z)
2x y xz x y z 4 25 5
4
4

A, B m n c, d

1. cd A c dA d cA

2. 2(A B) cA cB

3. (c d)A cA dA

4. 1A A 1A A

5. 0A 0

6. c0 0

6.

A aij m n B bij n p

AB C C cij m p cij ai1b1 j ai2b2 j ainbnj
1, 2, 3, , n
i 1, 2, 3, , m j

A 2 1 0 ,B 13
3 1 2 20

1. AB 2. BA

uoaoaaaaaaaaom 15

1. A 1 14 144 2. A 5 4 1 ,B 6
3 1 2,B 302 3 2 0 0
002 1

A 1 3 ,B 24 2 ,C 3
2 4 31 0 1
2

1. AB 2. BC

3. A BC 4. AB C

16 A 1 0 , B 1 3 ,C 2 3 ,I 10
1. AB 2 1 4 5 0 1 01

2. BA

3. AI 4. IA

5. A B C 6. AC BC

A 1 1 , B 2 1 ,I 10
1 1 1 2 01

1. AB 2. BA

17

mam3. AI 4. IA

AB BA A AI A BI AB AI A
A2
AB AB BA

AB BA AB BA

AA AA
1
Ak AAk 1 k

A 01
23

1. A2 2. A3

A2 A AA2

A 1 2 , B 10
3 4 21

1. A B 2 2. A B A B

18 4. A2 B2
3. A2 2AB B2

A, B n n ( A B)2 A2 2 AB B2 AB BA

x y
1
1. 13 x 5 2. 20 x 6
21 y 01 y 4

MADAM19

1.

A, B, C
A BC AB C

2. In AI IA A

An n
I

3. A B, B C, AB, AC BC

A, B, C
(A B)C AC BC
A(B C) AB AC

1. AB BA

2. ( A B)2 A2 2 AB B2 AB BA
AB BA
3. ( A B)2 A2 2 AB B2 AB BA
B0
4. ( A B)( A B) A2 B2 BC

5. AB 0 A0

6. AB AC A 0

A 11 B 1 1 a (A B)2 4I
a1 2 1

A 1a B 15 a AB I
01 01

20

A 1a ab A2 1
b1

A 12 n An 1 20
01 01

A 2a a A2 2A I 0
10

A xy 2 ,B 2 y ,C 1a AB C a
3 z 2 y 01

IMAM 21

7. (transpose of a matrix)

A aij m n B bij n m
, m j 1, 2, 3, , n
bij aji i 1, 2, 3,
transpose of a matrix
A

B At

14 At
A0 2 Bt
Ct
23
Dt
B 12 5
50 1 Et

1 40 Ft
C 123
(Symmetric matrix)
2 14 (Skew-Symmetric matrix)

2
D5

7

5. 7 4 0

E 423
0 34

6. 0 5 1

F 503
1 30

5 A At
6 A At

22 21 3 3 42
1. At A 4 2 3 ,B 213
0 12
26 1
2. 3At

3. 3A 4. 3A t

5. At Bt 6. ( A B)t

7. AB t 8. At Bt

9. Bt At 10. At t

mmmm. 23

1. A ( At )t A ( A B)t At Bt AB t Bt At
2. A kA t kAt np
3. A B
4. A mn
mn B

A, B, C A(B C)
3A(B 2C) t 6Ct At 3Bt At

AB AB A BA A At 2 At

A 0 a ,B 22 C 01
1 2 11 01

a At t B Bt A t 4C t

24

A a 1 ,B 10 C 31
1 0 51 10

a At B AB 2C t

A ba 1 ,B 12 C 1 3
a, b b 2a 0 21 3 1

At B 2BA B Ct t t

8. Determinant)

(Determinant) A det A A

A

Aa 11
det A a

A5 det A B2 det B
C0 det C D 1.2 det D

22 Momoa25

A ab det A a b ad bc
cd c d

A 41 B 3 1
32 2 2

C 34 D 61
25 33

33
a1 b1 c1
A a2 b2 c2
a3 b3 c3

det A a1 b1 c1 a1 b1 a1b2c3 b1c2a3 c1a2b3 a3b2a1 b3c2a1 c3a2b1
a2 b2 c 2 a2 b2
a3 b3 c3 a3 b3

3 22 2 13
A 01 1 B 060

21 3 3 21

26 123
D 5 30
03 1
C 2 13 212

0 22

A 2 2 , B 12
2 3 12

det A det B

det AB det Adet B

5 det At det Bt

det A 1 A1 3 1 det B 1 11
2 B1 2 2

11 11
44

oaammmamAAsa 27

det A2 det A 2
det A B
det A det B
13 det A B det A det B
2det A
det 2A 3det A
det 3A 4det A
det 4A

28 2det A
3det A
101
A 2 11

302
det A

det 2A

det 3A

A, B nn kR

1. det(AB) (det A)(det B)
2. det( An ) (det A)n
3. det At det A

4. det(A 1) 1

det A

5. det(kAn n ) kn det A

6. det(A B) det A det B

A 3 1 , B 20 33 ,C 23
2 1 3 5 47

1 det(ABC) det( A3BtC 1)

det 2A 3B t Mamm.amomma 29

5 det(2 At B) 1 det 3A 2C 1 t

det(10A B)

det A C 1 det BA BCt t

9 det(B 2I )

30

302 1 11

A 2 1 1 ,B 0 2 0

101 311

1 det(2A) det( A 1)t
det 2B 1 t
det 2Bt 1 det (3At )(2B) 1

5 det (2A) 3B

det A B det At 2B

Mom 31

302 302 10 2
0 11
A 2 1 1 ,B 2 1 1 ,C 10 1

101 3 12

ab c det c k (k R)
de f ab
gh i kb a b c
ka d kb e kc f ke d e f
kh g h i
gh i
kC2 C1
(kR1 R2 )

302 101 32 0

A 2 1 1 ,B 3 1 1 ,C 3 1 1

101 302 11 0

det det

de f ba c ab c n det = det
ab c ed f de f det = det
gh i hg i gh i det = det
det = ( 1)n det

32

A 0 0 ,B 30 0 10
50 100 10 0 20
20 0 30

det

2 10 2 12 3 135 74 12
A 4 20 4 , B 1 2 3 ,C 10 20 30 , D 62 6
5 10 20 2 6 10 81 3
6 30 6

1 det

det

MADAM 33

A 3 2 ,B 3 4 ,C 32
1 3 1 6 39

302 306 12 0 4

D 2 1 1 ,E 2 1 3 ,F 4 1 1

20 2 103 2 01

det

ka kb det a kb k a b
cd ka b a b c kd c d

kc d kc kd

14 9
A 2 5 12

3 6 15

A

A1 A2 A, A1, A2
det A det A1 det A2
axbycz ab c x y z
f
def def de i

ghi gh i gh

abx c ab c a x c

dey f def dyf

ghz i gh i gz i

34

A 2 0 , B 0 3 , C 100 D 126
0 4 1 0 320, 540
463 200

A det

a11 0 0 0

det A 0 a22 0 0 a11 a22 ann
det B 00 0

0 0 0 ann

B det

0 0 0 a1n

00 0 an1 a(n 1)2 a1n

0 a(n 1)2 0 0
an1 0 0 0

34 5 32 1
6 8 10
1 00 0 32 1

6 7 10 50 120 84
30 50 37
21 6 10 20 10
4 5 12
17 3

200 BAMBAATAA 35

5 1 30 5 21
340
5 84 20 0

00 5 20 0
0 20 0 30
30 0 00 3

2 12 3 00 0 2
00 2 1
9 02 5 7 0 32 1
00 38 3 5 73

00 0 1 2 3 10
4 6 78
10 52 2 3 10
20 3 4 10 5 7 3
30 16
407 7

220 4 3 21 7
51 1 11
13 1 35 2 12 2 0
35 76 04 2 2

5 1 3 10

0 2 35 14 1 2
2 3 47 2 14 1
2 1 5 10 1 2 14
2 11 2 4 121

36

ab c xyz

det A d e f 2 , det B d e f 3

gh i gh i

acb ed f
D hg i
1 C dfe
ba c
gih

axbycz 2d 2e 2 f
Eg h i F axbycz

def ghi

3d 3 f 3e 2i h g
H 6 f 3e 3d
5 G 2g 2i 2h
2c 2z b y a x
axczby

ab c det A 4 det 2B 1 3g 3h 3i
A def B 2a 2b 2c

gh i de f

9. AMBAR 37

Submatrix A ij

A aij n n Aij A

A n1n1 3. A13

123 A23
A 456 A33

789

A11 A12

4. A21 A22

7. A31 A32

minor

A aij n n n2 (minor) aij

i jA

aij Mij ( A)

123 A
A 456

789

M11( A) M12 ( A) 3. M13( A)

4. M 21( A) M 22 ( A) M 23 ( A)

7. M31( A) M32 ( A) M33 ( A)

38

cofactor

A aij n n n 2 (cofactor) aij
Cij ( A)
( 1)i j Mij ( A)

aij

Cij ( A) ( 1)i j Mij ( A) A
123

A 456
789

1 35 B
B 04 1

6 43

MAAG 39

A aij n n n2 A
a11C11( A) a12C12 ( A)
a1nC1n ( A)
a11 a12 A det A
a21 a22 a1n
a2n

an1 an2 ann

a11 a12 a13 a13a21a32 a31a22a13 a32a23a11 a33a21a12
A a21 a22 a23

a31 a32 a33

det A a11a22a33 a12a23a31

1 12 det A
A 324

0 13

40

20 40

A 11 12 det A
2 324

1 0 13

1 15 2

A 2 012 det A
1 38 0

1 12 1

134 1 MARANO 41

A 1551 det A
2 6 9 10

1 3 44

0 21 2 31
0 43 0 52
512 1 23

418 34 5
1 25 12 3
2 34 25 4

42 2014
4128
28 25 38 1208
42 38 65 4324
56 47 83
6 336
31 5 3 6 11 1
24 2 2 116 1
24 2 1 1 11 6
10 1 2

bc ca ab a2 a 1
ab c b2 b 1
111 c2 c 1

2 120 MADAM 43

A 0 216 Cij ( A) Mij ( A)
ij 6 0 01

4 3 42

M 34 ( A) C43 ( A)

A aij n n x y4 Cij ( A) ij
38 0
x y1

C21 6 C23 4 a11C11 a22C22 a33C33

10.

A x2 x x det A 0

2x3 3x 1

44

2 1ab 4 2 23 ab

3 2 cd 1 2 21 cd

A, B, C 2 2 det A 1 det 2I , det(Bt ) 8
det C
( At B2C 1) 80
14

6 2 3x x det A 0
A 21 4

4 0 2x

ab 4 1 6 4 21 ab
cd 3 2 2 320 cd

7 4 a b 6 5 10 6 8 5 monsoon 45
ab

2 4cd 2 5 84 68 cd

A2 5 1 det A
54

A 2x x , B 42 x det A det B
5 x 31

A 2x 1 2 x x det A 0
2x 1 2 3x

46 12 y1 2 4 u A x 4 y 2 2u
34 31 2 20 1 z3 y5
det A3 x1 4
4 2z 1 4

A, B, C 2 2 det 2 A 20 , det(Ct ) 1 det 3I
det B
2A 1BtC2 5 3
3 3

A a2 x det 2AAt det 16A 1 t
11

A a 3 , B 21 C 2AB B a det C 24
1 2 11

A 2a 1 momma 47
1a
a det(3A2 ) 2a2 1 3 det A 1 t 40

11.

A1 A ab ad bc 0
cd

A ab ad bc 0 A A1 1 d b
cd ad bc c a

A 31 B 45
52 23

48 A 2 1 , B 31
3 2 52
( AB) 1
A 1B 1

3. B 1A 1 4. 2A 1

5. 2A 1 6. 1 A 1

2

A B non-singular matrix

1. ( A 1) 1 A

2. ( AB) 1 B 1A 1

3. ( A 1)t ( At ) 1

4. ( An ) 1 ( A 1)n nI

5. (kA) 1 1 A 1 k0

k

A 22 A1

A

nn n2 A1
A1 A nn n2

n2

MADAM 49

A nn
A
A (singular matrix) det A 0
(non-singular matrix) det A 0

A nn n2 (adjoint matrix) A

Cij ( A) t 10 1
A3 12
A adjA
25 8
det A, adjA, A adjA , adjA A

A adjA A(adjA) (adjA) A (det A)I3 A nn
adjA A det A I3 det A 0

A nn n2

(1) A adjA adjA A det A In

(2) A A

A 1 1 adjA
det A

50 1 32
B 254
124
A 38 0 39 6

12 1

a00 100
C 0b0 D 010

00c 001

nstsssssssssssssarosssso 51

00a 001
E 0b0 F 010

c00 100

1234

G 0123
0012

0001