ระบบสมการเชิงเส้น และเมทริกซ์ ·...

ระบบสมการเชงเสนและเมทรกซ อ.ภานวฒน เกยรตนฤมล

1 เจอคนเกง…จงเรยนรจากเขา…เจอคนงเงา…ใหเอามาสอนตวเอง…| ถาตงใจวาท าได…แมหนหนกกยงยกไหว…

ระบบสมการเชงเสนและเมทรกซ

1. ระบบสมการเชงเสน (System of Linear Equations)

บทนยาม ให a, b, c, d, e, f เปนจ ำนวนจรงใดๆท a, b ไมเปนศนยพรอมกน และ c, d ไมเปนศนยพรอมกน

เรยก

ax by e

cx dy f วาระบบสมการเชงเสนสองตวแปร

โดยท x และ y เปนตวแปร ค ำตอบของระบบสมกำรน คอ คำของ x และ y ทท ำใหสมกำรทงคเปนจรง

ค ำตอบของระบบสมกำรเชงเสนสองตวแปรม 3 ค าตอบ ไดแก

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

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

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

ตวอยาง จงแกระบบสมกำร 2x y 3+ =

3x 2y 8- =

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

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

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



ตวอยาง จงแกระบบสมกำร 4x 2y 5

8x 4y 10

- =

- =

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

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

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

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

ตวอยาง จงแกระบบสมกำร

4x 2y 1

6x 3y 4

+ =

+ =

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

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

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

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

บทนยามระบบสมการเชงเสน n ตวแปร ระบบสมกำรเชงเสน n ตวแปร โดยท n โดยท n 2

บทนยาม ให 1 2 na , a , ..., a , b เปนจ ำนวนจรงใดๆท 1 2 na , a , ..., a ไมเปนศนยพรอมกน เรยกสมกำร 1 1 2 2 n na x a x ... a x b+ + + = วา สมการเชงเสน n ตว แปรโดยท 1 2 nx , x ,..., x เปนตวแปร

บทนยาม ระบบสมกำรเชงเสนทม 1 2 nx , x ,..., x เปนตวแปร หมำยถง ชดของสมกำรเชงเสนท ประกอบดวยสมกำรเชงเสนทม 1 2 nx , x ,..., x เปนตวแปร จ ำนวน m สมกำร โดยท m เปนจ ำนวนเตมบวกทมคำมำกกวำ 1

ค ำตอบของระบบสมกำรน คอ คำของ 1 2 nx , x ,..., x ทท ำใหสมกำรทง m สมกำรเปนจรง




x y z+ + 6= 2x 2y 2z- + 4= 3x 3y 3z+ - 0=

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

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

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

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

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

แบบฝกหด 1.1

1. 2x + y = 3

3x 2y 8- =

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

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

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

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

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

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

2. 4x 2y 5- =

8x 4y 10- =

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

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

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

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

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

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



3. 4x 2y 1+ =

6x 3y 4+ =

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

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

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

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

................................................................................................................................................................................. 4. 3x y 1+ =

2x 3y 8- =

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

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

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

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

................................................................................................................................................................................. 5. x 2y z 1- + = -

3x y 2z 4+ - =

y z 1- = -

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

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

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

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

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

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



2. เมทรกซ (Matrix)

ในบำงครงเรำจะพบวำมกำรเขยนตวเลขเปนแถวหลำยๆแถว หลำยๆหลก เชน ปำย ประกำศจ ำนวนนกเรยนชนมธยมศกษำตอนปลำย แยกตำมสำขำของโรงเรยนแหงหนงเขยนดงน

ระดบชน วทย ศลป-คณต ศลปภาษา

ม.4 126 75 102

ม.5 132 61 124

ม.6 111 55 110

ถำเรำน ำตวเลขทปรำกฏในตำรำงมำเขยนใหมภำยในวงเลบ [ ] หรอ ( ) เปน

จะไดสงทในวชำคณตศำสตรเรยกวำ เมทรกซ บทนยาม เมตรกซ คอ ชดของจ ำนวน mn ตว ),( Inm ซงเขยนเรยงกน m แถว (row) n หลก

(column) ภำยในเครองหมำยวงเลบ ในรปแบบ

11 12 1na , a , ... , a แถวท 1

21 22 2na , a , ... , a แถวท 2 M

m1 m2 mna , a , ... , a แถวท n

หลกท1 หลกท2 ... หลกท n

เรยก ija วำเปนสมาชก (entry) ในแถวท i และหลกท j ของเมทรกซ หรอเรยกวำเปนสมาชกในต าแหนงท ij ของเมทรกซ เมอ },...,3,2,1{ mi และ },...,3,2,1{ nj เรยก เมทรกซทม m แถว และ n หลกวำเปน m n´ เมทรกซ (อำนวำ เอม คณ เอน เมทรกซ) และเรยก nm วำเปนมตของเมทรกซ (dimension of matrix)

ตวอยาง

51 ............................................................



1

5

17

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

1 6

5 2

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

1 6 14

5 2 21

8 9 15

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

1 6

5

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

เพอควำมสะดวกในกำรกลำวถงเมทรกซ จะใชตวอกษรภาษาองกฤษตวใหญ A, B,C,K แทน

เมทรกซ และใชตวอกษรภาษาองกฤษตวเลก kcba ,,, ทมตวเลขสองตวเขยนตอไวทำงขวำในระดบต ำลงไปเลกนอยแทนสมาชกของเมทรกซ A, B,C,K ตำมล ำดบ เชน

11 12 13

21 22 23

31 32 33

, ,

, ,

, ,

a a a

A a a a

a a a

11 12

21 22

31 32

,

,

,

b b

B b b

b b

ตวเลขทเขยนไวกบตวอกษร a, b เปนตวเลขทใชบอกต ำแหนงของสมำชกของเมทรกซ โดยทตวเลขตวแรกแสดงวำสมำชกตวนนอยในแถวใด และตวเลขตวหลงแสดงวำสมำชกตวนนอยในหลกใด เชน

A = 1 6 14

5 2 21

8 9 15

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

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

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



นอกจำกเขยนเมทรกซ A และสมำชกของเมทรกซ A ในรปแบบทกลำวมำแลว เรำอำจจะเขยน

เมทรกซใหสนลง ดงน

ij m n

A a หมำยถง เมทรกซ A เปน m n เมทรกซทมสมำชกในต ำแหนงท ij เปน ija

เมอ 1,2, ,i m และ 1,2, ,j n

ตวอยาง ก ำหนดให 0 1 2 3 8

2 3 4 5 7

6 6 1 4 8

A

จงหำ(1) มตของ A (2) 12 23 35 33 14 21, , , , ,a a a a a a ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................................................................ตวอยาง จงเขยนเมทรกซ

4 4 ijA a โดยท

1

0

1

ija

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

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

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

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


1. ก ำหนดให 2 4 24

6 5 4

A จงหำ

1) มตของ A 2) 13 21 22 11 23 12, , , , ,a a a a a a .................................................................................................................................................................................

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

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

เมอ i = j

เมอ i > j

เมอ i < j



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

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

2. ถำ 0 4 1 0

1 6 2 5

A และ 6 3

1 2

4 0

B จงหำ

1) มตของ A และ B 2) จ ำนวนสมำชกของ A และ B 3)

11 14 22 23 21 22 31 32, , , , , , ,a a a a b b b b ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ................................................................................................................................................................................. ................................................................................................................................................................................. .................................................................................................................................................................................3. ให

3 3 ijD d โดยท

2

0

2

ijd

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

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

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

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

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

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

4. ก ำหนดให 1 2 3 4 0 1 2 3

,3 0 5 1 1 0 5 6

A B

และ 3 4 1 0

2 0 0 1

C จงหำ

1) มตของ A, B และ C 2) 23 14 22 24a , a , a , a

3) 11 22 13 24b , b , b , b 4) 14 23 11 22c , c , c , c

5) ( )12 12 12a b c+ + 6) ( )23 14 122a 3b 5c- +

เมอ i > j เมอ i < j เมอ i = j เมอ i > j



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

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

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

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

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

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

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

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

3. การเทากนของเมทรกซ

บทนยาม ให

ij m nA a และ

ij m n

B b

A เทำกบ B กตอเมอ ij ija b ส ำหรบทก 1,2, ,i m และ 1,2, ,j n และเขยน A B= แทน A เทำกบ B

นนคอ A = B กตอเมอ 1) มตของ A และ B เทำกน

2) สมำชกของ A และ B ทอยในต ำแหนงเดยวกน เทำกนทกตว

ดงนน 1. 3

6

กบ 6

3

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

2. 1 3 กบ 1

3

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

3. 16 กบ 1 6 .......................................................................................................

4. 1 0

1 0

กบ 0 6

3 1

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



ตวอยาง จงพจำรณำวำเมทรกซ A กบ B ทก ำหนดใหเทำกนหรอไม เพรำะเหตใด

ถำ 2 3 0

4 1 2

A และ

6

3 5 53

42 2 2

4

B

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

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

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

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

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

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

ตวอยาง ถำ 1 5 1

2 0 2

x

y แลวจงหำคำ x และ y

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

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

................................................................................................................................................................................. ตวอยาง จงหำคำ x และ y ทท ำใหเมทรกซตอไปนเทำกนเปนจรง

1. 2 4

3 2

x y

x y

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

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

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

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


1. จงหำคำ x และ y ทท ำให 1 3 5 1

0 5 0 5

x y

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

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

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



2.

01x

34

=

y5

34

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

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

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

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

3.

50

31x

=

50

1-y5

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

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

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

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

4. 1

3 8

x y

x y

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

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

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

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

5.

1

1

2

2

yx

yx

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

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



4. ชนดของเมทรกซ

ท ชอ นยาม ตวอยาง 1

เมทรกซแถว (Row matrix)

เมทรกซทมมต 1 n´ (เมทรกซซงมเพยง 1 แถว)

2

เมทรกซหลก (Column matrix)

เมทรกซทมมต n 1´ (เมทรกซซงมเพยง 1 หลก)

3 เมทรกซศนย (Zero matrix)

เมทรกซทมสมำชกทกตวเปนศนย

4 เมทรกซจตรส (Square matrix)

เมทรกซทมจ ำนวนแถวเทำกบจ ำนวนหลก

5 เมทรกซเฉยง (Diagonal matrix)

เมทรกซจตรสทมสมำชกใน เสนทแยงมมหลก เปนจ ำนวนจรงใด ๆ และสมำชกทเหลอเปนศนย

6 เมทรกซหนงหนวยหรอ เมทรกซเอกลกษณ (unit matrix, Identity matrix)

เมทรกซจตรสทมสมำชกใน เสนทแยงมมหลก เปนเลข 1 และสมำชกทเหลอเปนศนย

7 เมทรกซสมมำตร (Symmetric matrix)

เมทรกซจตรสทมคณสมบต

ij jia a

8 เมทรกซเสมอนสมมำตร (Skew Symmetric matrix)

เมทรกซจตรสทมสมำชกใน เสนทแยงมมหลกเปนเลข 0 และมคณสมบต ij jia a

9 เมทรกซสำมเหลยม (Triangle matrix)

เมทรกซจตรสทมสมำชกทอยดำนบนหรอดำนลำงของแนวเสนทแยงมมหลกเปนศนยทงหมด



ท ชอ นยาม ตวอยาง 10 เมทรกซสเกลำร

(scalar matrix) เมทรกซทมสมำชกใน เสนทแยงมมหลกมคำเทำกนทกตว สมำชกทเหลอเปนศนย


ก ำหนดเมทรกซตอไปน

A 0 ,

B 0 0 ,

C 1 1 1 ,

0D

0,

1 0E

0 1

,

0 1 1F

0 1 0,

0

G 0

0

,

1 0

H 0 1

0 1

, 3I I , 1 1 12 3 4

0 1 2

J

1 1 0

,

0 0 0 0

0 0 0 0K

0 0 0 0

0 0 0 0

, 1 0 0

L 0 1 0

0 0 1

,

3 1053

2 3 54 9 20

22

0

M

1 0

1) จงหำมตของเมทรกซทก ำหนดให ..........................................................................................................................................................................

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

2) เมทรกซทก ำหนดใหเมทรกซใดบำงทเปนเมทรกซแถว

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

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

3) เมทรกซทก ำหนดใหเมทรกซใดบำงทเปนเมทรกซหลก ..........................................................................................................................................................................

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

4) เมทรกซทก ำหนดใหเมทรกซใดบำงทเปนเมทรกซจตรส

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

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

5) เมทรกซทก ำหนดใหเมทรกซใดบำงทเปนเมทรกซศนย ..........................................................................................................................................................................

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



5. การบวกเมทรกซและการลบเมทรกซ บทนยาม ให

ij m n

A a และ

ij m nB b

เมทรกซ A บวกกบเมทรกซ B คอเมทรกซ

ij m nc เมอ ij ij ijc a b

ส ำหรบทก 1,2, ,i m และ 1,2, ,j n เขยนแทน A บวกกบ B ดวย A B+

ตวอยาง ก ำหนดให

0 1 3

2 0 4

A 1 0 1

5 0 2

B

1 0 1

3 2 4

5 3 0

C 7 1 1

2 4 0

1 2 4

D

จงหำ 1. A 2B+ 2. C D+

3. A D+ ................................................................................................................................................................................. .................................................................................................................................................................................................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ตวอยาง ก ำหนดให

3 1 4 1 2 4,

2 5 0 1 1 2

A B และ

2 5 5

1 4 4

C

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

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

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

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

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

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



สรปการบวก ลบเมทรกซ

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

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


1. จงพจำรณำวำเมทรกซทก ำหนดใหบวกหรอลบกนไดหรอไม

3 2 1 3

1 ,0 5 6 2

A B

5 2 1 4 2 6

3 0 4 3 , 0 1 2

5 5 2 4 0 6

A B

3 5 0 0

2 ,2 0 0 0

A B

1 33 2 5

4 , 2 22 3 4

4 4

A B

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

2. ก ำหนดให 1 2 2 2

,3 1 1 2

A B และ

1 0

0 1

C จงหำค ำตอบในแตละขอตอไปน

2.1 A B+

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

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

................................................................................................................................................................................. 2.2 A B-

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

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

................................................................................................................................................................................. 2.3 (A B) C+ +

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

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

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



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

................................................................................................................................................................................. 2.4 A (B C)- -

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

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

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

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

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

................................................................................................................................................................................. 2.5 (A B) C- -

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

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

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

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

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

................................................................................................................................................................................. 2.6 (A B) C+ -

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

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

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

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

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

................................................................................................................................................................................. 2.7 A (B C)- +

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

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

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

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

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

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



6. การคณเมทรกซดวยสเกลาร บทนยาม ให

ij m n

A a และ c เปนคำคงตว

ผลคณของ c กบเมทรกซ A คอ เมทรกซ

ij m nb เมอ ij ijb ca ส ำหรบทก

1,2, ,i m และ 1,2, ,j n เขยนแทนผลคณของ c กบเมทรกซ A ดวย cA

ตวอยาง จงหำเมทรกซของผลลพธตอไปน

1.1 2 3

5 3 1 4

0 5 2

2. 4 6 81

10 12 42

3. 3 1 4 5 7 4.

1

2

2 7

3 3

4

5

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

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

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

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

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

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

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

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

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

เมทรกซทมมต m n และสมำชกทกต ำแหนงเปน 0 เรยกวำ เมทรกซศนย จะแทนดวย 0 m n หรอ 0



7. การคณเมทรกซดวยเมทรกซ หลกการคณ

หมายเหต 1. ผลคณ AB จะคณกนไดกตอเมอ จ านวนหลกของตวตง (A) = จ านวนแถวของตวคณ (B)

2. ผลคณ AB ทเกดขน จะมจ ำนวนแถวเทำกบจ ำนวนแถวของ A และมจ ำนวน หลก เทำกบจ ำนวนหลกของ B นนคอ

m n n k m k

ตวอยาง ก ำหนดให 3 4

,1 0

A

1 0 7,

2 3 5

B 1

2

3

C

จงหำ , , ,AB BC A BC AB C ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. .................................................................................................................................................................................


,1 1

A

2 1,

1 2

B 1 0

0 1

I

จงหำ AB, BA, AI

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

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

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

11 12 1k11 12 1n 11 12 1k

21 22 2n 21 22 2k21 22 2k

m1 m2 mn m1 m2 mkn1 n2 nk

b b ba a a c c c

a a a c c ab b bAB

a a a c c ab b b

มตของ AB มตของ B มตของ A



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


2 3

A จงหำ 2 ,A 3A

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

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

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

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


1. จงหำผลคณตอไปน ถำหำไมไดใหบอกดวยวำเพรำะเหตใด

2 1

(1) 5 43 0

1 2

1 3 1 2 2 1(2)

0 1 2 1 3 2

1 2

4

(3) 1 3 5 8

0

1 2 3 1 0

(4)1 0 1 2 0

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

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

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

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

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

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

2. ก ำหนดให 1 1

1 2 1, 2 2

0 1 30 1

A B จงหำ AB และ BA

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

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

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

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

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

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



3. ให 3 5 1 0 0 0

, , 00 2 0 1 0 0

A I จงหำ

(1) AI และ IA ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. .................................................................................................................................................................................

(2) A0 และ 0A ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. .................................................................................................................................................................................

(3) 2 3 4I , I , I ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. (4) 2A ................................................................................................................................................................................. ................................................................................................................................................................................. .................................................................................................................................................................................

(5) I0 ................................................................................................................................................................................. ................................................................................................................................................................................. .................................................................................................................................................................................



(6) 2A I ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. .................................................................................................................................................................................

4. ให 3 0 0

A 0 3 0

0 0 3

และ 1 0 0

B 0 2 0

0 0 3

จงหำคำของ 2 3 2 2A , B , A B, B A

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

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

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

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

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

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

สมบตของเมทรกซ

เรำสำมำรถแสดงไดวำ ส ำหรบ A, B, C, 0 ทมมต m n มสมบตตอไปน

ก ำหนดให เมทรกซ A , B และ C เปนเมทรกซมต m n มสมบตดงน 1. สมบตปดของการบวก

A B+ เปนเมทรกซมต m n 2. สมบตการสลบทของการบวก

A B B A+ = + 3. สมบตการเปลยนกลมของการบวก

( ) ( )A B C A B C+ + = + + 4. สมบตการมเอกลกษณของการบวก

มเมทรกซ 0 ซงท ำให A 0 A 0 A+ = = + 5. สมบตการมอนเวอรสของการบวก

ม A- ทท ำให ( ) ( )A A 0 A A+ - = = - + เรยก 0 วำเปนเมทรกซเอกลกษณของการบวก เรยก A- วำเปนตวผกผนการบวกของ A



6. ( )c A B cA cB+ = + เมอ c เปนคำคงตว

7. ( )c d A cA dA+ = + เมอ c, d เปนคำคงตว

8. ( ) ( )cd A c dA= เมอ c, d เปนคำคงตว

9. 1A A , 0A 0= =

8. เมทรกซสลบเปลยน หรอ ทรานสโพส (Transpose)

บทนยาม ให

ij m nA A ถำ

ij n m

B b มสมบตวำ ij jib a ทก 1,2, ,i n

และ 1,2, ,j m และเรยก B วำเปนเมทรกซสลบเปลยนของ A และแทนดวย tA

ตวอยาง ก ำหนดใหA 1 7

4 5

, B 5 7 3

2 5 9

และ C

3 1 2

6 1 8

3 4 1

จงหำ t tA , B และ tC

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

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

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

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

สมบตของเมทรกซสลบเปลยน (ส าคญมากกกกกกกกกกกกกกกกกก!!!!) ก ำหนด A, B, C เปนเมทรกซ k เปนจ ำนวนจรงใดๆ จะได 1.

ttA A

2. ถำ A B หำได t t tA B A B

3. ถำ A B หำได t t tA B A B

4. t tkA kA

5. ถำ AB หำได t t tAB B A

6. t tA A

7. A เปนเมทรกซจตรส และ n I จะได n tt nA A



ตวอยาง ก ำหนดให 1 2 2 3 0 2,

1 1 0 4 1 0

A B และ1 4

2 3

0 3

C

จงหำ (1) ( )t

A B+ ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. (2)

tAC

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

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

................................................................................................................................................................................. (3) t( 2)B ................................................................................................................................................................................. ................................................................................................................................................................................. ................................................................................................................................................................................. (4)

tA B C

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

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

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

ตวอยาง ก ำหนด A 1 3 4

,4 0 3

B

1 2

2 1

จงหำ ,t tA B

tBA

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

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

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

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

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

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




1. ก ำหนดให A 1 1,

2 3

B = 4 1

2 2

จงหำเมทรกซในแตละขอตอไปน

1) tB 2) -2 tA 3) 4 tA 4)

t tA B+ 5) t(A B)+ 6) 3

tA B+

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

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

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

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

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

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

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

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

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

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

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

2. ก ำหนดให 1 2 2 3 0 2

A , B1 1 0 4 1 0

และ 1 4

C 2 3

0 3

จงหำ 1.1 tA 1.2 ttA 1.3

tttA 1.4 t tB C 1.5 tt

2B 3A

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

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

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

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

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

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

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



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

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

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

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

9. ดเทอรมแนนต (Determinemt)

คอ ฟงกชนทมโดเมนเปนเซตของเมตรกซจตรส เรนจเปนเซตของจ ำนวนเชงซอน และดเทอรมแนนตเปน

ฟงกชนแบบ many to one

สญลกษณ ถำ A เปนเมตรกซจตรสทมขนำด n x n ดเทอรมแนนตของ A เขยนแทนดวยสญลกษณ det A หรอ

|A|

เมตรกซขนาด 1 x 1

ถำ A = [a] จะไดวำ det A = |a| = a

ตวอยาง 1. A = [3] จะได det A = 3

2. B = [0] จะได det B = .......

3. C = [-2] จะได det C = .......


ก ำหนดให

dc

baA จะได det A = ad – bc

ตวอยาง จงหา det

1.

21

31A

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

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



2.

24

51B

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

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

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

3. 2 1

C4 2

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

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

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



ihg

fed

cba

A

จะไดวำ hg

ed

ba

ihg

fed

cba

A det idbhfageccdhbfgaei

ตวอยาง จงหา det

1.

987

654

321

A

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

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

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



2.

2 0 1

B 1 2 3

1 4 0

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

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

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


1. จงหำดเทอรมนนตของเมทรกซทก ำหนดใหตอไปน

1.1 B 2 1.2 5 0

C2 3

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

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

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

1.3

4 1F

2 5

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

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

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

1.4 2 3 4

E 0 5 7

1 6 5

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

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

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

1.5

1 2 3

H 2 1 2

1 2 3

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

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

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



เมตรกซขนาด n x n กอนอน ตองรเรองนกอน....

Minor (Mij) คอ det ของเลขทเหลอจำกกำรตดแถวท i และหลกท j ทง

Cofactor(Cij) คอ Minor คดเครองหมำย

132

211

402

A , Minor ของเลข 1 คอ M11 = ……………………………

Cofactor ของเลข 1 คอ C11 = ……………………………..

Minor ของเลข 2 คอ M23 = ……………………………..

Cofactor ของเลข 2 คอ C23 = ……………………………...

การหา det

1. เลอกแถวหรอหลกมำ 1 แถว หรอ 1 หลก (อะไรกได..ตำมใจฉน) 2. หำ Cofactor ของสมำชกแตละตวในแถวหรอหลกทเลอกมำ 3. เอำสมำชกในต ำแหนงนน คณกบ Cofactor ของสมำชกทคดมำจำก 2. แลวจบบวกกน ผลทไดคอ det ท

ตองกำร นนคอ

i1 i1 i2 i2 in indet A a C A a C A a C A (คดแถวใดแถวหนงเปนหลกในกำรกระจำยโคแฟกเตอร)

หรอ 1j 1j 2j 2j nj njdet A a C A a C A a C A

(คดหลกใดหลกหนงเปนหลกในกำรกระจำยโคแฟกเตอร) ตวอยาง

1.

132

211

402

A จงหำคำของ det A



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

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

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

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

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

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

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

2.

0321

1010

2031

2006

B จงหำคำ det B

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

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

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

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

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

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

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


1. จงหำไมเนอรและโคแฟกเตอรของสมำชกทกตวของเมทรกซ

1 2 1

3 1 5

1 6 3

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

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

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

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

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



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

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

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

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

2. จงหำดเทอรมนนตของเมทรกซทก ำหนดใหตอไปน

2.1)

1 2 1

W 3 1 5

1 6 3

2.2)

1 3 2 0

1 1 2 1B

3 1 2 4

1 2 3 2

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

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

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

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

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

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

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

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

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

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

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

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

10. อนเวอรสการคณของเมตรกซ

เรองหนกสมอง แตตองร...

A เปน Singular Matrix (เมตรกซเอกฐำน) เมอ A ไมสำมำรถหำ A-1 ได (detA 0)

A เปน Non-Singular Matrix (เมตรกซไมเอกฐำน) เมอ A ไมสำมำรถหำ A-1 ได (detA ≠ 0)




ถา A = [a] จะได 0;11

a

aA

ตวอยาง 1. A = [3] จะได 1A …………………………………..

2.

5

2B จะได 1B …………………………………..

3. C 5 จะได 1C …………………………………..



dc

baA จะได 0det;

det

11

A

ac

bd

AA

ตวอยาง จงหำอนเวอรสกำรคณของ

1. 1 3

A1 2

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

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

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

2. 3 1

B6 2

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

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

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



เมตรกซขนาด n x n

บทนยาม t

ijadj A [C (A)]

และ t1

ij

1 1A adjA C (A)

detA detA เมอ det A 0

ตวอยาง

1.

132

211

402

A จงหำคำของ A-1

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

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

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

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

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

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

2.

2 0 1

B 1 2 3

1 4 0

จงหำคำของ B-1

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

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

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

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

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

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

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




จงหำวำเมทรกซตอไปนมอนเวอรสกำรคณหรอไม ถำมจงหำอนเวอรสกำรคณของเมทรกซนน

1.1)

2 5A 5

12

1.2) 3 1

B4 2

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

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

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

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

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

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

1.3) 1 3

M2 5

1.4)

1 1 2

T 1 3 2

1 3 1

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

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

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

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

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

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

1.5)

1 3 7

C 2 3 5

1 4 2

1.6)

1 2 3

D 2 3 4

1 2 1



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

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

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

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

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

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

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

11. คณสมบตของดเทอรมแนนต

คณสมบตทส าคญ (มากกกกกกกกกกก++) ของดเทอรมแนนต

1. det(AB) = detA.detB

2. det(A-1) = 0det;det

1A

A

3. det(At) = det A 4. det(kA) = kndetA ; n คอ มตของเมตรกซ A 5. det(An) = (det A)n 6. ถำ A = B แลว det A = det B

ตวอยาง ถำ A และ B เปนเมทรกซมต 2 2 โดย det A 3 และ det B 2 จงหำ

(1) det AB (2) 2det A

(3) tdet 5B .................................................................................................................................................................................

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

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

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

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



ตวอยาง ถำ 1 1

A3 1

แลว 3 t tdet( 2A A (A A )) เทำกบเทำใด (รบตรง ม.อ.)

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

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

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

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


1. ถำ A และ B เปนเมทรกซมต 2 2 โดย 2)det( A และ 5)det( A จงหำ

1.1 )det(AB

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

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

1.2 )det( 2B

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

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

1.3 )2det( tA

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

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

1.4 )5det( tt BA

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

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

1.4 )3det( 3 tBBA

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

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



12. การใชเมตรกซแกระบบสมการเชงเสน

การหาค าตอบของระบบสมการเชงเสน

1. วธกำรก ำจดตวแปร (แบบทเรำคนเคย) 2. ใชอนเวอรสกำรคณของเมตรกซ 3. ใชดเทอรมแนนต หรอกฎของเครเมอร (Cramer’s rule) 4. วธกำรก ำจดตวแปรของเกำสและจอรแดน (Gauss-Jordan Elimination)

ระบบสมการเชงเสนในรปเมตรกซ

ถำมระบบสมกำรเชงเสน m ตวแปร n สมกำร ดงน

mnmnmm

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

2211

22222121

11212111

เขยนใหอยในรปสมกำรเมตรกซไดดงน

mnmnmm

n

n

b

b

b

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211

หรอ AX = B

วธท 2 การใชอนเวอรสการคณของเมตรกซ

AX = B จะได X = A-1B

1. เขยนระบบสมกำรใหอยในรป AX =B 2. หำ A-1 3. ไดค ำตอบเปน X = A-1B

หมายเหต ถำ 0det A จะไดวำระบบสมกำรเชงเสนมค ำตอบเดยว

ถำ 0det A จะไดวำระบบสมกำรเชงเสนมค ำตอบหลำยชด หรอ ไมมค ำตอบ




x + 2y = 5

2x + 3y = 8

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

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

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

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

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

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

วธท 3 การใชดเทอรมแนนต หรอกฎของเครเมอร

1. เขยนระบบสมกำรใหอยในรป AX =B เมอ

nx

x

x

X2

1

2. จะไดค ำตอบเปนดงน

A

Ax

A

Ax

A

Ax

n

ndet

det

det

det

det

det

22

11

ตวอยาง จงหำคำ x และ y จำกก ำหนด

x + 2y = 5

2x + 3y = 8.

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

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

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



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

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

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


2x + y + 2z = 3

x + y – z = 2

1

3x + 2y - 2z = -2

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

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

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

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

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

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

แบบฝกหดท 1.12

จงแกระบบสมการเชงเสนตอไปนโดยใชเมทรกซ

1. 2x 3y 7

x 2y 4

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

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

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

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

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



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

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

2. y x 4

2x 3y 22 .................................................................................................................................................................................

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

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

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

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

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

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

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

3. x y z 6

x y z 2

x y z 0

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

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

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

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

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

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



ตารางการสงงานเรองความนาจะเปน

วน/เดอน/ป งำน ลำยเซนผปกครอง ลำยเซนคร แบบฝกหดท 1.1 แบบฝกหดท 1.2 แบบฝกหดท 1.3 แบบฝกหดท 1.4 แบบฝกหดท 1.5 แบบฝกหดท 1.6 แบบฝกหดท 1.7 แบบฝกหดท 1.8 แบบฝกหดท 1.9 แบบฝกหดท 1.10 แบบฝกหดท 1.11 แบบฝกหดท 1.12

ประโยชนของการตรงตอเวลา 1.ท ำใหเรำมนสยขยนขนแขงเอำกำรเอำงำนอยำงจรงจง

2.ฝกใหเรำเปนคนกระตอรอรนมชวตชวำ 3.ท ำใหเรำมควำมซอตรงตอตวเองรกษำเกยรตยศของตนเอง

4.ท ำใหเรำท ำงำนไดสะดวกรวดเรวเรยบรอยและมผลด 5.หนำทกำรงำนประสบควำมส ำเรจชวตกำวหนำ

6.สำมำรถก ำหนดกจกรรมตำงๆ ทเรำจะกระท ำไดในแตละวนท ำใหชวตมระเบยบและมวนยกบตนเอง 7.เปนทเชอถอและไวใจของคน


41 เจอคนเกง…จงเรยนรจากเขา…เจอคนงเงา…ใหเอามาสอนตวเอง…| | ถาตงใจวาท าได…แมหนหนกกยงยกไหว…

ระบบสมการเชิงเส้น และเมทริกซ์ ·...

Documents