Linear_equations_and_matrices.pdf

E-learning 03-03-2555

1

1. ija

11 12 1

21 22 2

1 2

... ...

...

n

n

m m mn

a a aa a a

A

a a a

A m n A ()

m n A ij m nA a ija i j 1 , 2 , ... , i m 1 , 2 , ... , j n

4 3 2 0

A

11a A 1 1 4 11( 4)a 12a A 1 2 3 12( 3)a 21a A 2 1 2 21( 2)a 22a A 2 2 0 22( 0)a 2. 1. (Zero matrix) 0 " 0 "

0 0

0 0 00 , 0 , 0 0 0

0 0 00 0

2. (Square matrix)

0 1 1

1 3 , 3 4 5

1 22 6 2

A B

1 2

m

1 2 n


2

2 0 0 02 0 0

3 1 0 00 1 0 ,

4 0 6 03 4 0

5 0 1 6

A B

2 1 0 0 1 0 2

0 5 1 7 0 4 3 ,

0 0 0 1 0 0 2

0 0 0 0

A B

0

0

3.

11 12 1

21 22 2

1 2

... ...

...

n

n

n n nn

a a aa a a

A

a a a

11 22 33 , , , ... , nna a a a 3.1 0

3.2 0

4. ( Diagonal matrix ) 0

1 0 00 2 00 0 5

A

(main diagonal)


3

5. (Sealar matrix)

1 0 0 0

2 0 0 0 1 0 0

0 2 0 , 0 0 1 0

0 0 2 0 0 0 1

A B

6. ( Indentity matrix )

( Unit matrix ) 1 n n " "nI

1 2 31 0 0

1 0 1 , , 0 1 0

0 1 0 0 1

I I I

3.

ij m nA a ij m nB b A B ( A B ) ij ija b i j A B

4 1 08 g 3

A

1 8 2 3x y

B

A B ,x y g 4 , 0x y 2g # 4. ( Transpose of matrix ) A m n A 1 A 1 2 A 2 m A m A

A " "tA ij m nA a

tji n m

A a


4

aA b

c

t

aA b

c

3 2

1 23 42 5

A

2 3

1 3 22 4 5

tA

3 2

1 2( ) 3 4

2 5

t tA A

5. ( Symmetric matrix ) A n n A tA A tA A A

3 5 7

5 1 07 0 2

A

3 5 7

5 1 0 7 0 2

tA A

6.

ij m nA a ij m nB b ij ij m nA B a b , ij ijm n m nkA k a ka k

1 2 3 4

A

1 2 3 4

B

0 0 0 0

A B

1 2 3 63 3 3 4 9 12

A

1 2 1 2( 1) ( 1) 3 4 3 4

A A


5

(1) ij m nA a ij m nA a (2) ( )A B A B

, ,A B C m n 0 1. A B A B 2. ( )A B B A 3. ( ) ( )A B C A B C 4. 0 0A A A 0 5. A ( ) 0 ( )A A A A A A

,A B m n ,c d 1. ( ) ( ) ( )cd A c dA d cA 2. ( )c A B cA cB 3. ( )c d A cA dA 4. 1A A ( 1)A A 5. 0 0A 6. 0 0c 7. 0cA 0c 0A

7. ( Skew symmetric matrix ) A n n A tA A 8.

ij m nA a ij rnB b


6

2c cx dy

ij m rAB c 1 1 2 2 ...ij i j i j in njc a b a b a b A B AB A B AB A B

11 12 1

21 22

1 2

1 2

... ... ...

...

n

in

i i in

m m mn m n

a a aa a aa a a

a a a

11 12 1 1

21 22 2 2

1 2

... ...

... ...

b ... ...

j r

j r

n n nj nr n r

b b b b

b b b b

b b b

1 2 2

a b

Ac d

2 1

xB

y

1

2 2 1

cAB

c

1c ax by

BA #

2 p q

Ar s

w x

By z

pw qy px qzAB

rw sy rx sz

w x p q

BAy z r s

wq yq

wp xr xsyp zr zs

#

AB BA

11 12 1

1 2

1 2

............. ............................

... ...

............................ .............

r

i i ij ir

m m mr m r

c c c

c c c c

c c c


7

1. , ,A B C

( ) ( )AB C A BC 2. n nA nI n nAI I A A nI 3. , ,A B C , , , A B B C AB AC BC ( )A B C AC BC ( )A B C AB AC

1. AB BA

2. ( ) ( )A B C D AC AD BC BD 3. 2 2 2( ) ( )( )A B A B A B A AB BA B 4. 2 2( ) ( )A B A B A AB BA B

5. 0AB 0A 0B 6. AB AC 0A B C 7. AB CB 0B A C

, ,A B C k 1. ( )t tA A 2. A B ( )t t tA B A B 3. A B ( )t t tA B A B 4. ( )t tkA kA 5. AB ( )t t tAB B A


8

6. A n I ( ) ( )t n n tA A 9. (Inverse) A n n n nAI A I A nI n n

0R 0 0

A n n B n n nAB BA I B A B 1A

1. n n B n n A AB BA I 2. n n A (non-singular matrix) A 3. n n A (Singular matrix) A 4. n n A

5. a b

Ac d

0ad bc

1 1 d b

Ac aac bc


9

,A B n n 1A 1B

1. 1 1( ) A A 2. 1 1 1( ) AB B A 3. 1 1( ) ( )t tA A 4. 1 1( ) ( )n nA A 5. 1 11( ) kA A

k k R 0k

1 2 2 1 , , 3 2 0 1x y y a

A B Cz y

AB C a 1. 29

36 2. 27

36

3. 1936

4. 1736

AB C

2 2 1 3 2 0 1x y y a

z y

2( ) 4 ( ) 2 1 6 2 3 0 1x y x y y y a

z y yz

2( ) 4 1x y .........(1) ( ) 2 x y y y a .........(2) 6 2 0z .........(3) 3 1y yz .........(4)

(3) 3z (4) 1

6y

(1) 5 2

x y x y y (2) 5 1 12

2 6 6a


10

1

( )

2 1 1 0 0 1

1 3 1 2 1 2

1 1 0 1 1 1

0 1 1 2 1 2

2 11 1 1( 1)(2) (1)( 1)

( 1)

X B C A

X

2 1 2 1

1 1 1 1

3 4

a

27 36

#

2 0 1 2 1 , 1 2 1 3

A B

1 0

1 2C

( )X B C A 1X X 1X

1. 2 1 1 1

2. 2 11 1

3. 1 11 0

4. 1 1 1 0

#

3 1 1 0 10 1 2 , 0 , X=3 0 1 2

xB C y

z

I A 3 3 2AB I AX C x y z (Ent. 1 2548) 1. 20 2. 24 3. 26 4. 30

2 AB I (2 ) A B I 1 AA I


11

1 2A B AX C

1 X A C 1 1 A AX A C 1 2A B (2 )X B C 2( )BC

1 1 0 1

2 0 1 2 03 0 1 2

1

2 45

2

810

X

2

810

xyz

2 , 8x y 10z 2 8 10 20x y z #

4 cos sin 1 0 ,

sin cos 0 1A I

2 1 2( ) 2B A A I

1 2( )A B 1. 2I 2. 4I

3. 4A 4. 8A 1AA I

2 1 2 ( ) 2B A A I 1 1 1 1 2 1 ( )( ) 2A B A AA A A A I 1 1 3 1 ( ) 2A B A A A 1 2 1 1 1 3 1 1( ) ( ) 2A B A A A A A A 1 2 1 4 1 2( ) ( ) 2( )A B I A A ...................(1)

cos sin sin cos

A


12

1 2 2cos sin1 sin coscos sin

A

cos sin ( 1)sin cos

cos sin sin cos

1 A A 1 4 1 1( ) A AA AA I 1 2 1( ) AA A I (1) 1 2( ) 2 4A B I I I I #

10. det det

A n n A det( )A A 1. A a a R det( )A a

2. a b

Ac d

det( ) a b

A ad bcc d

, , ,a b c d R

3. 11 12 13

21 22 23

31 32 33

a a aA a a a

a a a

det A 1 2 A 3

11 12 13

21 22 23

31 32 33

a a aa a aa a a

11 12

21 22

31 32

a aa aa a

A det( )A

det

11 22 33a a a 12 23 31a a a 13 21 32a a a

31 22 13a a a 32 23 11a a a 33 21 12a a a


13

h 11 22 33 12 23 31 13 21 32 a a a a a a a a a k 31 22 13 32 23 11 33 21 12 a a a a a a a a a det( )A h k

4. n n 2n ij n nA a ija R 4.1 (Minor) ija i j A ija ( )ijM A 4.2 (Co-factor) ija ( )ijC A ( ) ( 1) ( )i jij ijC A M A 4.3 1 1 2 2det( ) ( ) ( ) .... ( )i i i i in inA a C A a C A a C A

1

det( ) ( )n

ij ijj

A a C A

i 1, 2,3,.....,n

1

det( ) ( )n

ij iji

A a C A

j 1, 2,3,.....,n

11 12 1

21 22 2

1 2

... ...

... ... ... ...

...

n

n

n n nn

a a aa a a

A

a a a

A A

1. ( ) ( )tij jiM A M A 2. 1( ) ( )nij ijM kA k M A k R A n n

,A B n n k 1. A () 0 () det( ) 0A 2. A () det( ) 0A 3. B A () det( ) det( )B A 4. B A () k det( ) det( )B k A 5. B A () () det( ) det( )B A


14

6. 6.1 2

1 1 2 1 1 2 5 3 2 (3 2) (1 2) (1 ( 3)) 2 4 3 2 4 3

1 1 2 1 1 2

3 1 1 2 2 3 2 4 3 2 4 3

6.2 3 1 1 2 1 1 (2 + 0) 5 3 2 5 3 (2 4) 2 4 3 2 4 (1 + 2)

1 1 2 1 1 0

5 3 2 5 3 4 2 4 1 2 4 2

7. det( ) 1nI 8. det(0) 0 9. A det( )A 10. det( ) det( )tA A 11. det( ) (det( ))n nA A 12. det( ) det( )nkA k A A n n 13. det( ) det( )det( )AB A B 14. A det( ) 0A 15. 1 1det( )

det( )A

A


15

1 1 1 3 1

A

3det( 2 ( ))t tA A A A

1. 768 2. 768 3. 384 4. 384

det( ) ( 1)( 1) (3)(1) 1 3 2A

3 3det( 2 ( )) det( 2 )det( )det( )t t t tA A A A A A A A

2 3 ( 2) det( ) det( )det( )tA A A A 3 4 det( ) det( )det( )tA A A A

4 4 det( ) det( )tA A A 4 4( 2) det( )tA A

tA A 1 1 1 3 2 4 3 1 1 1 4 2

det( ) ( 2)( 2) (4)(4) 4 16 12tA A 3 4det( 2 ( )) 4( 2) ( 12)t tA A A A 4(16) ( 12) 768 #

2 , ,a b c 1 0 1 1 1 1

aA b

c

( )ijC A i j A 12 ( ) 1C A det( ) 5A a 1. 5 2. 1 3. 2 4. 3 12 ( ) 1C A det( ) 5A 11 11 12 12 13 13det( ) ( ) ( ) ( )A a c A a c A a c A .(1)

1 111 1 1

( ) ( 1) 1 1

c A

2

11 11det( ) 5 , ( ) ( 2) , A c A a a 12 12( ) 1 , ( 1)c A a 13 0a (1)

135 ( )( 2) ( 1)(1) (0) ( )a c A


16

5 2 1a 2a # 3 A B

2 2 2 2 4

, 2 02 2

x xA B

x

x

det(2 ) 76A C det( )BC ( 100 , 50) ( A-NET 2551 )

1. 1 11 2

C

2. 1 2 1 1

C

3. 2 11 4

C 4. 2 1

3 1C

2 2 2

2 2

xA

x

2 3det( ) ( ) ( 2 2)(2 2) 8A x x x .(1) det(2 ) 76A 22 det( ) 76A det( ) 19A .(2) (1) (2) 3 8 19x 3 27x 3x

2 4 2 0

xB

3x

2 12 2 0

B

det( ) ( 2)(0) (12)(2) 24B

1. 1 11 2

C

det( ) 2 ( 1) 3C

det( ) det( )det( ) ( 24)(3) 72 ( 100) , 50)BC B C #


17

4 2 1

( ) det 0 1 2 1 1

x xf x

x

,a b

( ) 2f x a b (Ent. 1 2544) 1. 1

3 2. 2

3

3. 43

4. 53

2 1

( ) det 0 1 2 1 1

x xf x

x

2 1 0 1 2 1 1

x x

x

2 0 1

1

x x

x

2 2 2 2( ) 2 2 3f x x x x x x x ( ) 2f x 23 2x x 23 2 0x x (3 2)( 1) 0x x 2 1 ,

3x

,a b 2 1 , 3

a b

2 5 1 3 3

a b #

11. ij n nA a 2n ( )ijC A ija A

A ( )ijC C A (Adjoint matrix) A A ( )adj A ( ) ( ) tt ijadj A C C A

ij n nA a 2n

1. det( ) 0A 1 1 ( )det( )

A adj AA

1( ) det( )adj A A A


18

2. ( )( ( )) ( ( )( ) det( )A adj A adj A A A I I 3. 1det( ( )) (det( ))nadj A A

1 ( )adj A A , 1( ) ,det( )

8tadj A B B A

1 1det( ) tC A B B AB , A B 2 2 1det(2 )tC 1( ) det( )adj A A A ( ) tadj A B B 1det( ) tA A B B 11

8tA B B

11 8

tA A BA B A

1 1 1 118

tA AB BAB B AB

1 1 118

tB BAB B AB

1 1 11 B8

tB AB BAB ..(1)

1 1det( ) tC A B B AB 1 11

8tC B B AB ......................(2)

(1) (2) 1C BAB 1det( ) det( ) det( ) det( )C B A B 1 1 det( )

8 det( )B

B

1det( ) 8

C

1 1det(2 ) det(2 )

ttC C

21

2 det( )tC

1 4det( )C

1 148

1det(2 ) 2tC #


19

2 2 2 3

1 1 0 0 1 4

tA

2 3 1A

( PAT 1 2552 ) 1. 2

3 2. 2

3. 23

4. 2

2 2 3

1 1 0 0 1 4

tA

2 1 0

2 1 1 3 0 4

A

det( ) 2 1 4 2 0 0 1 1 3 0 1 3 2 1 4 2 0 1A 8 3 8 3

( ) ( 1) ( )i jij ijc A M A

3 2322 0

( ) ( 1) 2 1

c A

( 2) 2 1 1 ( )

det( )A adj A

A

11 12 13

21 22 23

13 32 33

( ) ( ) ( )1 ( ) ( ) ( )3

( ) ( ) ( )

c A c A c Ac A c A c Ac A c A c A

t

11 21 31

12 22 32

13 23 33

( ) ( ) ( )1 ( ) ( ) ( )3

( ) ( ) ( )

c A c A c Ac A c A c Ac A c A c A

2 3 1A 321 ( ( ))3 c A

1 (2)3

2 3

#


20

3 A 3 3 ijA i j A

2 5 1

( ) 28 10 1 17 5 1

adj A

111 2

5 8

A

321 1

3 2

A

det( )A (Ent. 1 2547) 1. 92 2. 15 3. 15 4. 92

11 12 13

21 22 23

31 32 33

a a aA a a a

a a a

ijA i j A

111 2

5 8

A

321 1

3 2

A

12

31

1 1 3 1 2

5 8

aA

a

( )ijC A A ( )adj A A

11 21 31

12 22 32

13 23 33

( ) ( ) ( )( ) ( ) C ( ) ( )

( ) ( ) ( )

C A C A C Aadj A C A A C A

C A C A C A

2 5 1

( ) 28 10 1 17 5 1

adj A

31( ) 1C A 13( ) 17C A

A 123 131 1

( ) ( 1) 1 1 2a

C A

122 1 1a 12 0a

1 31331

3 1 ( ) ( 1) 17

5C A

a

3115 17a 31 2a


21

1 2 31 1 1

12 0a 31 0a A

1 0 1

3 1 2 2 5 8

A

det( ) ( 8 0 15) (2 10 0)A 23 ( 8) 15 #

12. 2 3x y 1x y

1 2 3 1 1 1

xy

.............(*)

1 2 , 1 1

xA X

y

3

1B

(*) = AX B A X B :A B

3 1. AX B A 1X A B 2.


22

3. (Row operation)

A B X AX B :A B (row- operation) :I C X C 1. 2 ijR i j 2. icR

i c i icR R 3. i jR cR i c j j i icR R R

( )Z ( )Y Z Y Z Y

() n n AX B

1

2

n

xx

X

x

1

2

n

bb

B

b

det( ) 0A 1 2

1 2det( )det( ) det( ) , , ... ,

det( ) det( ) det( )n

nAA Ax x x

A A A

iA i A B 1, 2,...,i n


23

1 1x 1 2 32 0x x x

1 2 33 2 5x x x 1 2 32 3 3 9x x x

1 1 2 3 x y x

Ay

y A

1. 0 2. 1 3. 2 4. 3

1

2

3

1 2 1 03 1 2 52 3 3 9

xxx

1

0 2 15 1 29 3 3

1 2 13 1 22 3 3

x

30 310

1 1 2 3 x y x

Ay

3 6 3

yy

A det( ) 0A 23 18 0y y ( 6)( 3) 0y y 6 , 3y y 3


24

2 1 2 1

2 3 , , 11 0 0

a xA b X y B

c z

, ,a b c

AX B 2 1 1 2 3 0 1 1 2

1 0 2A R R

x

1. 1 2. 23

3. 34

4. 2

AX B

1 2 a 1 2 3 1

1 0 0

xb yc z

1 2 a

2 3 1 0

A bc

2 1 1 2

2 0 1 21 0

aR R b a

c

1 2 1 2 3 0 1 2 0 1 1

1 0 1 0 2

ab ac

3 , 2a c 2 1 5b a b

1 2 3 1 2 3 5 1

1 0 2 0

xyz

1 2 31 3 50 0 2 2 1 2 3 32 3 5

1 0 2

x

#

Linear_equations_and_matrices.pdf

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