เอกสารประกอบการสอน Math II.pdf

7/26/2019 Math II.pdf

1/93

Page 1of 93

Mr.Jaranawit Kongkaew : Mathermatics II

2

2554

@


2/93

Page 2of 93


(.)

.

20 2554


3/93

Page 3of 93


1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20


4/93

Page 4of 93


1. 1.1

1.2 1.3 1.4

2. a a a n

a n na aaaaa n ...

n

na a n

1)

=

2) =

3) () =

4)

() =

5)

=

6) =

7) =

8)1a = a

9)

n m

a =

1

2


5/93

Page 5of 93


Ex.1

1.

2.

2,187

-128

3x3x3x3x3x3x3

(-2)x(-2)x(-2)x(-2)x(-2)x(-2)x(-2)

37

(-2)7

3

-2

7

7

222

759 xxx =2

)759( x = 27x

3528 aaaa =

3528 a

=8

a

14

124

2

22 =

141242

=2

2


6/93

Page 6of 93


3.

I.

1.2.

3.

4.

5.

6.

7.

8.

9.

10.

125

27

-2.197

47.61

1331

64

7,776

0.4 x 0.4 x 0.4 x 0.4

..

..

105

..

..

..

..

..

..

..

-0.2

3

-1.3

-6.9

0.01

6

8

6

3

2

II.

.. 1. {53 + 24}

0

= 1

..2. 2 x 2m = 4

m

..3. 53m

= 5m

x 5mx 5

m

.. 4. 104n

= 104 x 10

m

.. 5. a x ak = a

k+1

..6.15

23

723

k

k

= 238

..7.a

a

77

1 = 7

0

.. 8. 286 = 4 x 7

6

.. 9. 295294 x 290 = 29

.. 10.32

32

yx

yx =

4

6

x

y

.. 11. (2m4n

2)

3 = 8m

12n

6


7/93

Page 7of 93


.. 12. (0.5x-1y)

2 = 2

2

4x

y

.. 13.

2

5342

428

ba

ba =

2

2

2b

a

.. 14. (25)

5 =

55

2

..15. (p + q)9 = p

9+ q

9

.. 16.

8

4

4

m

=18

2

8)4( m

.. 17.1

122

3

)(3

dc =

2

2

c

d

.. 18.3

3

323

17

51

yx =3

2yx

.. 19.5

107

14

25

132

735

= 2

.. 20.262

4312

)2()52()2()5(

= 2

III.

1)8

10

410

=

2) 7

4

316

24 =

3)5

2

1283

2 =

4) 10533 -- zyx =

5)

2

ab

a =


8/93

Page 8of 93


6) 23

77

a

ba =

7) baba 352 =

8)

2

225

3

ba

ab =

9)55

101010

25

842

bb

bbb

=

10)

544

4

26

xy

xyxy =

11) xxxx 634 353 =

12)1

23

a

aa =

13)32

23

yxy

yxx

=

14)

2

233

ab

baba

=

15)32

2543

mn

nmnm =

16)1

4

ma

ma

=

17)

21

44

)2()3( y =

18)2

5

13

32

n

n

n

n

x

x

x

x =

19)

41

22

3

12

16

8

yx

xy =

20)

nnn

nn1

112

1213

33

33

=


9/93

Page 9of 93


1. 2.1

2.2

2.

: 1

1

1)

n ma = n

m

a

2)

n a = na

1

3)

nn a

= a

4) n bn a = n ab

5)

n b

n a = n

b

a

6)mn a = nm a 2m

Ex. 6x

6x = 21

6x =

3x

Ex. 32

32 = 32 = 6

2

2


10/93

Page 10of 93


2425 = 29

6868 = 66686888 = 8 - 6

= 2

3. I.

1) 6x =

2) 8y =

3) 410yx =

=

4)

3 6x =

=

5) 3 93yx =

=

6) 4 8)( yx =

=

7)3

2

6

327

b

a =

=


11/93

Page 11of 93


II. 1) 32 =

=

=

2) 3 23 4 =

=

=

3)3 9

3 27

=

=

=

4)

4 4

4 324 =

=

=

5)3 6a =

=

=

6)5 3 30a =

=

=

=


12/93

Page 12of 93


III. 1) 3212372 =

=

2)

333 5425016 =

=

3) 54335332 =

=

=

4)3263

23

=

=

=

5)xy

yx28 =

=

6)13

2

=

=

=

7) 243 812 baba =

=

8)8

5 =

=


13/93

Page 13of 93


1. 1.1

1.2

2

(A x10n) 1A < 10

n 1

A < 10 A 1.09.999 A

Ex. (A x10n) 1A < 10 n 1. 3,490,000 = 3.49 x 10

6

2. 0.00078 = 7.8 x 10- 4

3. 42 x 1011

= 4.2 x 1012

4.12

9

107

1035.0

= 3107

35.0

= 0.05 x 10- 3

= 5.0 x 10- 2

x 10- 3

= 5.0 x 10- 5

6.002.063

1031042 68

=3

2

10263

10342

= 1.0 x 10

5

(0.002 = 2 x 10

- 3 ,

263

342

= 1.0)

7. 7.4 x 105+ 1.6 x 10

5 = (7.4 + 1.6) x 10

5

= 9.0 x 105

8. 7 x 1074.2 x 10

6 = 70 x 10

64.2 x 10

6

= (70- 4.2) x 106

= 65.8 x 106

3

2


14/93

Page 14of 93


3. I.

1.) 32,000,000 = .

2.) 138,830 = .

3.) 711,000,000 = .

4.) 4,040,000 = .

5.) 99,990,000 = .

6.)

123,000 = .

7.) 1,010,000 = .

8.) 543,210,000 = .

9.) 22,222,000 = .

10.)789,000 = .

11.)

0.000202 = .

12.)0.00123 = .

13.)

0.7890 = .

14.)0.0123 = .

15.)

0.9876 = .

16.)0.000011 = .

17.)0.0009 = .

18.)0.00000099 = .

19.)0.000501 = .

20.)

0.0707 = .


15/93

Page 15of 93


II.

1.) 3.0 x 108 = .

2.)

1 x 108 = .

3.) 9.99 x 109 = .

4.) 3.45 x 106 = .

5.) 4.44 x 104 = .

6.) 110 x 1010

= .

7.) 501 x 105 = .

8.) 7.65 x 104 = .

9.)

2 x 103 = .

10.)2.0 x 105 = .

11.)3.0 x 10-8 = .

12.)7.05 x 10-4 = .

13.)9.99 x 10-3 = .

14.)3.45 x 10-6 = .

15.)

4.44 x 10-4

= .

16.)11.0 x 10-1 = .

17.)5.01 x 10-2 = .

18.)7.65 x 10-3 = .

19.)2 x 10-3 = .

20.)2.0 x 10-5 = .


16/93

Page 16of 93


III. 1. 7.54 x 10-6 5 4

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

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

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

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

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

2. 3.0 x 108 0.5 x 10-2

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

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

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

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

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

3. 0.928

0.6 .................................................................................................................................

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

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

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

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

4.

25 x 1010

25 % .................................................................................................................................

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

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

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

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


17/93

Page 17of 93


1. 1.1

1.2 1.3

2.

:

2 1

1) 2)

2

Ex.

1) 11

21

xx

1x 2)1( x = 1x

1x = 1x

2)1( x = 1x

122 xx = 1x

xx 32 = 0

)3( xx = 0

x = 0 , 3

x = 3

4

2


18/93

Page 18of 93


2) 06442 x

x24 = 64

x24 = 34

x2 = 3

x =2

3

3) 11 68 xx

0

1x = 0

x = 1

3. I.

1.) 3x + 8 = 2x10

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

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

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

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

2.) 4x2+ 7x - 2 = 0

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

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

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

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

3.) 6x2+ 13x - 5 = 0

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

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

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

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

4.) 8x + 2

= 4x

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

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

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

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

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


19/93

Page 19of 93


5.) 43x - 1

= x16

1

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

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

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

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

6.) 22x

2x + 1

+ 1 = 0

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

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

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

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

7.) 32x

(4)3x + 1

+ 27 = 0

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

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

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

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

8.) 4x =

16

1

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

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

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

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

9.) 3x + 2

= 81

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

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

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

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

10.) 25x + 2

=

x

125

1

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

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

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

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

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


20/93

Page 20of 93


11.) 23 - x

= 8x + 2

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

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

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

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

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

12.) 32x

- 3x + 2

= 0

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

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

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

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

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

13.) 32x

- 3x + 2

= 0

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

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

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

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

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

14.) 33

33

x

x

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

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

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

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

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

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

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

15.) 23

14

x

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

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

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

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

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

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


21/93

Page 21of 93


1. 1.1

1.2 1.3

3.

Exponential :

1,0,),( aaxayRRyxf

Log :

1,0,),(1 aayaxRRyxf

: 1,0,log),( aaxayRRyxf

1,0,),( aaxayRRyxf

xay yax " xalog " f () 11

11

5

2

Log1.7

x

Logex

Log10

x


22/93

Page 22of 93


1. MNalog =

NM aa loglog

2.

N

Malog =

NM aa loglog

3. aalog =

1

4. naMlog =

Man log

5. 1loga =

0

6. balog = a

b

log

log

7. xaa log =

x

8.

N1loga =

Nalog

Ex. 1) log

23 + log

24 + log

26

= log2( 3 x 4 x 6 )

= log272

2) log2 5 - log

210

= log2

10

5

= log2

2

1

Ex. 12log39

=

12log2 33

=

212log33

=

212log33

= 122

= 144

Ex. log 1.358 = 0.1329 log 1358log 1358 = log (1.358 )

= log 1.358 + log

= 0.1329 + 3

= 3.1329


23/93

Page 23of 93


3. I.

1)

log 100 + log 10 + log 1

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

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

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

2) ( log 1000) ( log 105)

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

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

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

3) log232 + log

525 + log

381

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

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

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

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

4)

log3log

2log

3log

2512

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

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

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

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

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

5) log3 3 33

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

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

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

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


24/93

Page 24of 93


6) log216

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

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

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

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

7)

81log

3

1

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

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

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

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

8)

81

1log

9

1

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

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

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

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

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

9) log7343

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

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

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

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

10) log16

2

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

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

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

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


25/93

Page 25of 93


11) log832

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

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

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

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

12) log12

4 + log12

3

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

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

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

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

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

13) 3log2 33

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

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

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

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

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

14) )251

log(01.0

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

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

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

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

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

15)

3010.02log , 4771.03log , 6990.05log 8451.07log 125.0log420log

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

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

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

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

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

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

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


26/93

Page 26of 93


1. 1.1

1.2 1.3

2.

: 1,0,log),( aaxayRRyxf

1,0,),( aaxayRRyxf

3. MNa

log =

NM

aa loglog

4.

N

Malog =

NM aa loglog

5.

aalog =

1

6. naMlog =

Man log

7. 1loga =

0

8. balog = a

b

log

log

9. xaa log =

x

10.

N

1loga =

Nalog

6

2


27/93

Page 27of 93


Ex.

1) log2 )3( x - log2 )2( x = 3

log2

2

3

x

x = 3

2

3

x

x = 32

2

3

x

x = 8

3x = 8 )2( x

3x = 168 x

x7 = -19

x =7

19

2)

log3 )4( x + log3 )4( x = 2

log3 )4()4( xx = 2

)4()4( xx = 23

162 x = 9

2x = 25

x = 5


28/93

Page 28of 93


3. I.

1)

log3 )8(x log3x = 2

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

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

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

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

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

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

2)

log5 x log5 )4( x = 1

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

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

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

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

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

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

3)

log49

16

1x

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

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

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

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

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

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

4) log2log

2log

2x = 0

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

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

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

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

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

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


29/93

Page 29of 93


5) ln2ln xe = xe 4lnln

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

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

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

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

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

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

6) log x log )8( x = log )3(x log )4( x

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

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

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

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

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

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

7) log 2 = 0.3010 02.0)2.0( x

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

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

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

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

8) log8x + log

8 )2( x = log

8 )32( x

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

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

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

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

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

9) ln (2 ln x + 3 ) = ln 3

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

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

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

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

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


30/93

Page 30of 93


10) ln 102xe = )2

2ln( xxe

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

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

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

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

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

11) x2log3 = 5log1 3

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

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

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

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

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

12) )13(log6 x = 2log10log 66

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

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

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

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

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

13) 5loglog2

x = x2log7log

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

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

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

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

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

14)

zyx zy 27logloglog = 5log3

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

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

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

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

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


31/93

Page 31of 93


1.

1.1 1.2 1.3

2.

n !n

1 n n !n = n )1n( )2n( )3n( . . . . 123

n = 0 0! = 1 !n = n )!1n(

)!1n( = nn!

n = 1

)!11( =1

!1

0! = 1

Ex. !3

!6

!3

!6

= 123

123456

= 456

Ex. 2526272829

2526272829 =!24

!242526272829

=

!24

!29

7

2


32/93

Page 32of 93


rn r

n n

r + 1 2

2 n,r 0 r n

r

n=

)!rn(!r

!n

Ex.

1)

5

9 2)

4

9

1)

5

9 =

)!59(!5

!9

=!4!5

!56789

= 126

2)

4

9 =

)!49(!4

!9

=!5!4

!56789

= 126


33/93

Page 33of 93


3. I.

1) 54321 =

2) 321 =

3) 7654 =

4) 3231302928 =

5) ( 4321 ) + ( 654 ) =

II.

1) 4! =

2) 7! =

3)!3

!6 =

4)!3!8

!11 =

5) 5! + (2! + 3!) =

6)!3

!6 =

7)!3!8

!11 =

8) 5! + (2! + 3!) =

9)!4

!9+ 3! =

10) 3! +

!7!3

!5 =


34/93

Page 34of 93


III.

1)

3

6 =

=

=

2)

5

9 =

=

=

3)

0

5 =

=

=

4) 58 =

=

=

5)

4

7 =

=

=

6)

3

12 =

=

=


35/93

Page 35of 93


IV.

1)

2

4

2

6 =

=

=

2)

4

5

4

7 =

=

=

3)!4

!9+

2

5 =

=

=

4) )!1n()!3n(

n = 4 =

=

=

5) 563

n

n =

=

=

6)

6

10+

!2

!3

!5 =

=

=


36/93

Page 36of 93


1.

1.1 1.2 1.3

2.

n)ba( a , b n

0)ba( = 11)ba( = a + b2)ba( = 22 bab2a 3)ba( = 3223 bab3ba3a 4)ba( = 432234 bab4ba6ba4a 5)ba( = 54322345 bab5ba10ba10ba5a

.

.

.

=

.

.

.

nba )(

1 1 2 1 1 3 1 2 1 4 1 3 3 1 5 1 4 6 4 1 6 1 5 10 10 5 1 7 1 6 15 20 15 6 1

8 2


37/93

Page 37of 93


Ex. 6)ba(

6 1 5 10 10 5 1

7 1 6 15 20 15 6 16)ba( 7

6)ba( =

60514233241506

ba)1(ba)6(ba)15(ba)20(ba)15(ba)6(ba)1(

=6542332456

bab6ba15ba20ba15ba6a

n r 0 r n

n)ba( =

n

0r

rrnba

r

n

=n1nrrn1nn

bn

nab

1n

n...ba

r

n...ba

1

na

0

n

1. r + 1 rrn

bar

n

2. r + 1

r

n=

)!rn(!r

!n

Ex. 5)ba(

5)( ba =

54322345b

5

5ab

4

5ba

3

5ba

2

5ba

1

5a

0

5

0

5 =

5

5 =

!5!0

!5 = 1

1

5 =

4

5 =

!4!1

!5 = 5

2

5 =

3

5 =

!3!2

!5 = 10

5)ba(

=

54322345510105 babbababaa


38/93

Page 38of 93


3. I.

1) 2

ba =

=

=

2) 3ba =

=

=

3) 4ba =

=

=

4) 6ba =

=

=

5) 52 y3x =

= =

=

6) 52yx =

=

=

=

7) 5

2x = =

=

=

8) 432 yx =

=

=

=


39/93

Page 39of 93


=

9) 3x21 =

=

=

=

10) 24x =

=

=

=

II.

1) 5ba =

=

=

=

2) 52yx =

=

=

=

3) 52 3yx =

=

=

=

4) 52x = =

=

=

5) 331 x =

=

=

=


40/93

Page 40of 93


III.

1) 5 103y2x .................................................................................................................................

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

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2) 6 102

y3x .................................................................................................................................

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3) 8 153x .................................................................................................................................

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3) 5 532 x .................................................................................................................................

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41/93

Page 41of 93


1. 1.1

1.2 1.3

2.

A =

342

301

2 3 2 3

A M X N A

A =

mnmm

n

aaa

naaa

aaa

.. .

.

.. .

2.. .

.. .

21

2221

11211

A = ija m x n

1. 2.

9

2

A =[a1j]

mn B = [bij]mn

A + B = [a1j+ b

ij]

mn

A - B = [a1j - b

ij]

mn

A M n A M N ( )


42/93

Page 42of 93


Ex. A =

35

12

B =

62

43

A+B =

6325

4132

=

97

35

S M x N A,B,C S

1. A + B = S2. A + B = B + A3. ( A + B ) + C = A + (B + C )4. A + 0 = A 0 5. A+ (-A )= 0 -A A

(Transpose)

Ex. A =

540

221 At

At =

53

42

01

1. A m x n (At )t = A2. A m x n k ( kA) t = kA t

3. A B m x n (A + B)t = At + Bt

4.

A m x n B n x k (AB)t

= Bt

At

A = ia m x n A A1 m x n 1 1 A


43/93

Page 43of 93


3. I.

1. A =

021

411 B =

426

301

1) A+B 2) 2A-B 3) 2B+A 4) 2B-2A

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2. A =

02

30

11

,B =

30

12

21

C =

42

10

35

1) 2(A+B)-C 2) 3A-B 3) 2A+(2B-3C)

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

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44/93

Page 44of 93


3. A =

323

421

245

,B =

214

503

410

2

1A - 4B

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4. A =

24

32

03

,B =

12

30

12

-3A +2

1B

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5.

wzyx

wzyx 2

=

41

53

x + y + z + w

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

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45/93

Page 45of 93


1. 1.1

1.2 1.3

2.

k

kA =

gc

hb

da

k =

kgkc

khkb

kdka

Ex. A =

24

32

03

2A

2A = 2

24

32

03

=

2242

)3222

0232

(

=

48

64

06

10

2


46/93

Page 46of 93


A , B ,C A mn

B np AB = C C mp

Ex. A =

43

21

B =

34

12

AB =

43

21

34

12

=

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

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

=

)12()3()16()6(

)6()1()8()2(

=

922

610

Am x n

Bn x p

= Cm x p


47/93

Page 47of 93


3. I.

1. A =

3

2

1

,B = 321 C =

4231

1) AB 2) 2AB 3) 2A 2C.................................................................................................................................

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2. A =

75

14 ,B =

574

242 AB BA

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48/93

Page 48of 93


3. A =

32

10

12

,B =

05

43

12

,C =

352

701D =

1

0

2

1) (AC)2D A 3(C D)2) 3C 2(A+B) 2(C A) + (C B)3) 3(A+B) 2C (A C) + 2(B C).................................................................................................................................

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49/93

Page 49of 93


1.

1.1 1.2 1.3

2.

(Minor)A = ija

, n > 2 Minor Mij Determinant Matrix 1 j Matrix

Ex. A =

333231

232221

131211

aaa

aaa

aaa

M11

M11

=

333231

232221

131211

aaa

aaa

aaa

; 1 1

=

3332

2322

aa

aa

(Cofactor)A = ija

n x n , n > 2 Cofactor aij Cij (-1)

i+ jM

ij

C11 = (-1)2

M11

= M11

C12

= (-1)3 M12

= -M12

11

2


50/93

Page 50of 93


Ex. A =

323

421

245

22

M22

=

33

25

= (53)(32) = 156 = 9

3. I.

1) A =

25

31

22

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

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

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

2) A =

1210

2152

3423

3211

12 , 13 , 22 , 31

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

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51/93

Page 51of 93


3) A =

22

12 12 , 21

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II.

1) A =

734

225

123

C12 , C 21

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

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

3) A =

570

684

312

C12 , C 23 , C 32

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

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

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52/93

Page 52of 93


4) A =

1210

2152

3423

3211

C12 , C13 , C 22 , C 32 ,

(C33

+C11

)

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

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

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

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

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53/93

Page 53of 93


1. 1.1

1.2 1.3

2.

(Determinant) A nn

A det(A) A

det 1 2

*** 1. det(A) 33 2 2 det(A

t) =det(A)

3. det(An) = (det(A))

n

4. det(AB) = det(A)det(B)

A =

333231

232221

131211

aaa

aaa

aaa

det A =

3231

2221

1211

333231

232221

131211

aa

aa

aa

aaa

aaa

aaa

det A =

)( 322113312312332211 aaaaaaaaa - )( 122133112332132231 aaaaaaaaa

12

2


54/93

Page 54of 93


Ex. A =

734

225

123

det A

1

det A =

34

25

23

734

225

123

det A = )3)(5)(1()4)(2)(2()7)(2)(3( - )2)(5)(7()3)(2)(3()1)(2)(4(

=

151642 70188

=

11 80

=

91

2

det A = 131312121111 CaCaCa

=

34

25

)1)(1(74

25

)1)(2(73

22

)1)(3( 312111

=

)2)(4()3)(5()1)(1()2)(4()7)(5()1)(2()2)(3()7)(2()1)(3(

=

)23)(1()27)(2()20)(3(

=

91


55/93

Page 55of 93


3. I.

1) A =

323

421

245

2) B =

570

684

312

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

3) K =

1210

2152

3423

3211

4) G =

1212

1

2152

3443

2

1211

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56/93

Page 56of 93


5) H =

3211

1153

2

1483

2

121

2

1

P =

62

43

det Hdet P

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

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

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

5) H =

574

043

0020

D =

920

710

5120

HD = B det B

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

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เอกสารประกอบการสอน Math II.pdf