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# 3.1 矩陣的行列式

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## 第三章 行列式

### 3.4 特徵值介紹 3.5 行列式的應用

Elementary Linear Algebra 投影片設計製作者

R. Larsen et al. (6 Edition) 淡江大學 電機系 翁慶昌 教授

(2)



### 2 × 2 矩陣的行列式 (determinant)



 

= 

22 21

12 11

a a

a A a

12 21 22

| 11

| )

det(A = A = a aa a



 =

 

22 21

12 11

a a

a a

22 21

12 11

a a

a a

(3)



2 1

3 2 −

2 4

1 2

) 3 ( 1 ) 2 (

2 − −

= = 4 + 3 = 7

) 1 ( 4 )

2 (

2 −

= = 4 − 4 = 0 2

4

4 2

3 0



) 3 ( 2 )

4 (

0 −

= = 0− 6 = −6

(4)





n i j

i j

i i

n j

j

a a

a a

a a

a a

a

M

L L

M M

M

L L

) 1 ( )

1 )(

1 ( )

1 )(

1 ( 1

) 1 (

1 )

1 ( 1 )

1 ( 1 12

11

+

+

= aij



ij j

i

ij M

C = (−1) + aij

nn j

n j

n n

n i j

i j

i i

n i j

i j

i i

ij

a a

a a

a a

a M a

L L

M M

M M

L L

) 1 ( )

1 ( 1

) 1 ( )

1 )(

1 ( )

1 )(

1 ( 1

) 1 (

) 1 ( )

1 )(

1 ( )

1 )(

1 ( 1

) 1 (

+

+ +

+

+ +

+

=

(5)



=

33 32

31

23 22

21

13 12

11

a a

a

a a

a

a a

a A

13 12

21 a a

a M = a

11 13

22 a a

a M = a

33 32

21 a a

M =

21 21

1 2

21 ( 1) M M

C = − = −

+

33 31

22 a a

M =

22 22

2 2

22 ( 1) M M

C = − + =

(6)



+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

M M M M M

L L L L L

+

+

+

+

+

+

+

+

M M M M M



(7)



2

−1

1 3 −

=

1 0

4

2 1 3

1 2

0 A

12

,

1 1

0

2 1

11 − = −

=

⇒ M 4

0 4

1 3

13 − =

= M

21

4,

1 4

1 0

22 = = −

M 8

1 4

2 0

23 = = −

M

, 2 5

1 1 2

31 =

= −

M 3,

2 3

1 0

32= = −

M 6

1 3

2 0

33 = −

= − M

(8)

, 1 5

4

2 3

12 = − =

C , 1 1

0

2 1

11 − = −

+

=

⇒ C 4

0 4

1 3

13 − =

+

= C

ij j

i

ij M

C = (−1) +

0 1 0 2

21

4,

1 4

1 0

22 = + = −

C 8

1 4

2 0

23 = − =

C

, 2 5

1 1 2

31 =

+ −

=

C 3,

2 3

1 0

32= − =

C 6

1 3

2 0

33 = −

+ −

= C

(9)



=

+ +

+

=

=

= n

j

in in i

i i

i ij

ijC a C a C a C

a A

A a

1

2 2 1

| 1

| ) det(

)

(

=

+ +

+

=

=

= n

i

nj nj j

j j

j ij

ijC a C a C a C

a A

A b

1

2 2 1

| 1

| ) det(

)

( L

(10)



### 範例：3階矩陣的行列式





=

33 32

31

23 22

21

13 12

11

a a

a

a a

a

a a

a A

13 13 12

12 11

) 11

det(A = a C + a C + a C

33 33 23

23 13

13

32 32 22

22 12

12

31 31 21

21 11

11

33 33 32

32 31

31

23 23 22

22 21

21

13 13 12

12 11

11

C a C

a C

a

C a C

a C

a

C a C

a C

a

C a C

a C

a

C a C

a C

a

+ +

=

+ +

=

+ +

=

+ +

=

+ +

=

(11)

### 0 A

6

, 3

, 5

8 ,

4 ,

2

4

, 5 ,

1

33 32

31

23 22

21

13 12

11 2 E

=

=

=

=

=

=

=

=

=

C C

C

C C

C

C C

C

x

14 )

4 )(

1 ( ) 5 )(

2 ( ) 1 )(

0 ( )

det( = + + = − + + =

A a C a C a C



14 )

6 )(

1 ( ) 8 )(

2 ( ) 4 )(

1 (

14 )

3 )(

0 ( ) 4 )(

1 ( ) 5 )(

2 (

14 )

5 )(

4 ( ) 2 )(

3 ( ) 1 )(

0 (

14 )

6 )(

1 ( ) 3 )(

0 ( ) 5 )(

4 (

14 )

8 )(

2 ( ) 4 )(

1 ( ) 2 )(

3 (

14 )

4 )(

1 ( ) 5 )(

2 ( ) 1 )(

0 ( )

det(

33 33 23

23 13

13

32 32 22

22 12

12

31 31 21

21 11

11

33 33 32

32 31

31

23 23 22

22 21

21

13 13 12

12 11

11

=

− +

+

= +

+

=

= +

− +

= +

+

=

= +

− +

= +

+

=

=

− +

+

= +

+

=

= +

− +

= +

+

=

= +

+

= +

+

=

C a C

a C

a

C a C

a C

a

C a C

a C

a

C a C

a C

a

C a C

a C

a

C a C

a C

a A

(12)



=

1 4

4

2 1

3

1 2

0 A

### 解：

2 9 ) 1

1

( 1 1 − = −

= +

C C = (1)1+2 3 2 = (1)(5) = 5

1 9 4

2 ) 1

1 ( 1 1

11 − = −

= +

C ( 1)( 5) 5

1 4

2 ) 3

1 ( 1 2

12 = − + = − − =

C

4 -8 - 4

1 ) 3

1 ( 1 3

13 − =

= +

C

2

) -8 )(

1 ( ) 5 )(

2 ( ) 9 )(

0 ( )

det( 11 11 12 12 13 13

=

+ +

=

+ +

=

A a C a C a C

(13)













=

2 0

4 3

3 0

2 0

2 0

1 1

0 3

2 1

A

(14)

### 解：

) )(

0 ( ) )(

0 ( ) )(

0 ( ) )(

3 ( )

det(A = C13 + C23 + C33 + C43

2 4

3

3 2

0

2 1

1 )

1 (

3 1 3

= +

3C13

=

2 4

3 −

39

) 13 )(

3 (

) 7 )(

1 )(

3 ( ) 4 )(

1 )(

2 ( 0 3

4 3

1 ) 1

1 )(

3 2 (

3

2 ) 1

1 )(

2 2 (

4

2 ) 1

1 )(

0 (

3 2 1 2 2 2 3

=

=

− +

− +

=



 

 −

− +

− −

− +

= + + +

(15)



=

33 32

31

23 22

21

13 12

11

a a

a

a a

a

a a

a A

32 31

33 32

31

22 21

23 22

21

12 11

13 12

11

a a

a a

a

a a

a a

a

a a

a a

a

12 21 33 11

23 32

13 22 31 32

21 13 31

23 12 33

22 11

|

| ) det(

a a a a

a a

a a a a

a a a

a a a

a a A

A

− +

+

=

=

(16)



### 範例 5：





=

1 4

4

2 1

3

1 2

0 A

4 4

1 3

2 0

− -4

0 -12

0 6

0

2 6

0 ) 4 ( 12 16

0

|

| )

det( = = + − − − − − =

A A

16 -12

(17)









(18)

33 23 22

13 12

11

a 0

0

a a

0

a a

a

33 32

31

22 21

11

a a

a

0 a

a

0 0

a



33 22

11

a 0

0

0 a

0

0 0

a

(19)



ann

a a a A

A

33 22

| 11

| )

det( = =

(20)



(a)

= −

3 3

5 1

0 1

6 5

0 0

2 4

0 0

0 2

A (b)

=

0 4

0 0

0

0 0

2 0

0

0 0

0 3

0

0 0

0 0

1

B

1 5 3 3

### 

0 0 0 0 − 2 0 4

0 0

0

|A|=(2)(-2)(1)(3)=-12

|B|=(-1)(3)(2)(4)(-2)=48 (a)

(b)

(21)

### 摘要與複習 (3.1節之關鍵詞)

 determinant : 行列式

 minor : 子行列式

 cofactor : 餘因子

 expansion by cofactors : 餘因子展開

 upper triangular matrix: 上三角矩陣

 lower triangular matrix: 下三角矩陣

 diagonal matrix: 對角矩陣

(22)



### 令 A 和 B 是方形矩陣

)

( )

(a B = rij A ⇒ det(B) = −det(A) )

( )

(b B = ri(k) A ⇒ det(B) = k det(A)

) )

(

(i.e. rij A = − A ) )

(

(i.e. ri(k) A = k A

) ( )

(b B = ri A ⇒ det(B) = k det(A) )

( )

(c B = rij(k) A

### ⇒

det(B) = det(A)

) )

(

(i.e. ri A = k A ) )

(

(i.e. rij(k) A = A

(23)







=

1 2

1

4 1

0

3 2

1 A

= 0 1 4 12 8

4

A1

= 1 2 3 4 1

0

A2

= 2 3 2

3 2

1 A3

2 )

det(A = −

=

1 2

1

4 1

0 A1

=

1 2

1

3 2

1 A2

=

1 2

1

2 3

2 A3

8 )

2 )(

4 ( ) det(

4 ))

( det(

) det(

)

( 1 1(4)

) 4 (

1 = r1 A

### ⇒

A = r A = A = − = −

A

2 )

2 ( )

det(

)) ( det(

) det(

)

( 2 12

2 = r12 A

### ⇒

A = r A = − A = − − =

A

2 )

det(

)) ( det(

) det(

)

( 3 12( 2)

) 2 (

3 = r12 A

A = r A = A = −

A

(24)



### 注意：

)) ( det(

) det(

) det(

)) (

det(rij A = − AA = − rij A

)) ( 1 det(

) det(

) det(

)) (

det( ( ) r( ) A

A k A

k A

ri k = ⇒ = i k

)) (

det(

) det(

) det(

)) (

det(rij(k) A = AA = rij(k) A

)) (

det(

) det(

) det(

)) (

det(rij A = AA = rij A

(25)

2 −3 10





−−

= −

3 1

0

2 2

1

10 3

2

A

det(A) = ?

3 1

0

10 3

2

2 2

1

3 1

0

2 2

1

10 3

2 )

det(

12

=

= r

A

(26)

7 )

1 )(

1 )(

1 )(

7 ( 1

0 0

2 1

0

2 2

1 7

) 1 (

23 = − = −

=

r

3 1

0

2 1

0

2 2

1 1 ) )(

1 ( 3

1 0

14 7

0

2 2

1

7 1

7) ( 1 2 )

2 ( 12

=

=

r

r

1 0

0 −

(27)



A E EA =

Rij

E =

) 1

( ⇒ E = Rij = −1

### ( )

A A R A E A

r

EA = ij = − = ij =

)

(

) 2

( E = Ri kE = Ri(k) = k

### ( )

A k A R A E A

r

EA = i k = = i k =

( ) ( )

)

(

) 3

( E = Rijk

E = Rij(k) =1

### ( )

A A R A E A

r

EA = ijk = = ijk =

( ) 1 ( )

(28)



) ( )

(a B = cij A

### 令 A 和 B 是方形矩陣

) det(

)

det(B = − A

⇒ (i.e. cij(A) = − A)



### 定理： (基本列運算與行列式)

) ( )

(b B = ci(k) Adet(B) = k det(A) )

( )

(c B = cij(k) A ⇒ det(B) = det(A)

) )

(

(i.e. ci(k) A = k A ) )

(

(i.e. cij(k) A = A

(29)





 −

=

2 0

0

1 0

4

3 1

2 A



### 範例：





 −

= 0 4 1 3 2

1

A





 −

= 2 0 1 3 1

1

A 



= 4 0 1 0 1

2 A3

8 )

det(A = −



 



=

2 0

0

1 4

2 0 A



 



=

2 0

0

1 0

1 2 A



 



=

2 0

0

1 0

4 A3

4 )

8 2)(

(1 )

2 det(

)) 1 ( det(

) det(

)

( 1 1(4)

2) (1

1 = c1 AA = c A = A = − = −

A

8 )

8 ( )

det(

)) ( det(

) det(

)

( 2 12

2 = c12 A

### ⇒

A = c A = − A = − − =

A

8 )

det(

)) ( det(

) det(

)

( 3 23(3)

) 3 (

3 = c23 A

A = c A = A = −

A

(30)



### 定理 3.4： 產生零行列式的條件

(a) 一整列(或一整行)全為零 (b) 兩列(或行)是相等的

### 若A是方陣並且下列任何的條件是成立的，則det (A) = 0

(c) 某一列(或行)是另一列(或行)的倍數

(31)



0 6

5 4

0 0

0

3 2

1

= 0

0 6

3

0 5

2

0 4

1

= 0

6 5

4

2 2

2

1 1

1

=

0 2

6 1

2 5

1

2 4

1

= 0

6 4

2

6 5

4

3 2

1

=

0 6

12 3

5 10

2

4 8

1

=

(32)

3 5 9 5 10



### 注意：

5 119 205 30 45

10 3628799 6235300 285 339

(33)



=

6 0

3

1 4

2

2 5

3 A

3 ) 1 )(

3 3 (

4

4 ) 5

1 )(

3 ( 0

0 3

3 4

2

4 5

3

6 0

3

1 4

2

2 5

3 )

det( 3 1

) 2 (

13 = =

=

=

= C +

A

3 5)

)( 3 5 6 (

) 3 1 )(

5 ( 6 0 3

0 2 5 3

6 0

3

1 4

2

2 5

3 )

det( 5

3 5

2 2 1 5

3 5

2

5) (4

12 = =

=

=

= C +

A

(34)



### 解：

=

0 2

3 1

1

3 4

2 1

3

3 2

1 0

1

1 2

3 1

2

2 3

1 0

2

A

1 0

0 3

4 6

5 1

3 2

1 1

2 3

1 2 (1)(-1)

1 0

0 0

3

4 6

5 0

1

3 2

1 0

1

1 2

3 1

2

2 3

1 0

2

0 2

3 1

1

3 4

2 1

3

3 2

1 0

1

1 2

3 1

2

2 3

1 0

2

)

det( 2 2

) 1 ( 25

) 1 (

24

=

=

= +

r

r

A

(35)

6 5

13

2 1 8

5 0

0

6 5

13

2 1 8

3 1

8 )

1 (1)(

1 0

0 0

4 6

5 13

3 2

1 8

2 3

1 8

) 11 ( 21 )

3 (

41 = − 4 4 − − = − −

= +

r C

1 8 −

135 ) 27 )(

5 (

5 13

1 1) 8

5( 1 3

=

=

− −

= +

(36)





### 定理 3.5：矩陣相乘的行列式

(1) det (EA) = det (E) det (A) det (AB) = det (A) det (B)

) det(

) det(

)

det(A+ BA + B (2)

(3)

33 32

31

23 22

21

13 12

11

33 32

31

23 22

21

13 12

11

a a

a

b b

b

a a

a

a a

a

a a

a

a a

a

+

=

33 32

31

23 23

22 22

22 22

13 12

11

a a

a

b a

b a

b a

a a

a

+ +

+

(37)







 −

=

1 0

1

2 3

0

2 2

1 A





=

2 1

3

2 1

0

1 0

2 B

7 1

0 1

2 3

0

2 2

1

|

| = −

=

A 11

2 1

3

2 1

0

1 0

2

|

| =

= B

(38)





=









 −

=

1 1

5

10 1

6

1 4

8

2 1

3

2 1

0

1 0

2

1 0

1

2 3

0

2 2

1 AB

1 4

8

77 1

1 5

10 1

6

1 4

8

|

| = −

=

|AB| = |A| |B|



(39)



4 2

1 40

20

10



### 若A是一個n × n 矩陣並且c是一個純量，則

det (cA) = cn det (A)

5 1

3 2

5 0

3

4 2

1 ,

10 30

20

50 0

30

40 20

10

=

= A

=

1 3

2

5 0

3

4 2

1 10

A (1000)(5) 5000

1 3

2

5 0

3

4 2

1

103 = =

=

(40)





0 2 −1





=

1 2

0



≠ 0

= 0 ⇒ A



 



 −

=

1 2

3

1 2

3 A



 



=

1 2

3

1 2

3 B

A是不可逆(奇異)

= 12 0

B B是可逆(非奇異)

(41)



) A det(

) 1 A

det( 1 =



) det(

)

det(AT A

A是一方陣，則 =



1 = ? A

=

0 1

2

2 1 0

3 0

1

A (a) (b) AT = ?

4 0

1 2

2 1 0

3 0

1

|

| A = − =

4

1

1 = 1 =

A A

= 4

= A AT

(42)

### 若A是一個n × n矩陣，下列敘述是等價的

(1) A是可逆

(2) 對每一個n × 1矩陣b，Ax = b 具有唯一解



### 非奇異矩陣的等價條件

(3) Ax = 0 只有顯然解 (4) A列等價於In

(5) A可以寫為一些基本矩陣的相乘 (6) det (A) ≠ 0

(43)



### 範例 5：下列系統何者有唯一解?

(a)

4 2

3

4 2

3

1 2

3 2

1

3 2

1

3 2

=

− +

= +

=

x x

x

x x

x

x x

4 2

3x1 + x2x3 = − (b)

4 2

3

4 2

3

1 2

3 2

1

3 2

1

3 2

= +

+

= +

=

x x

x

x x

x

x x

(44)

b x = (a) A

= 0 Q A

(b) Bx = b

0 12 ≠

=

B

(45)



×n矩陣，在Rn



A：n×n 矩陣

A：n×n 矩陣

x： Rn

x Ax =

(46)



=

3 2

4

A 1

=

1 1 x1

1

1 5

1 5 1 5

5 1

1 3 2

4

1 x

Ax

=

=

=

=



 

= −

1 2 x2

1

1 5

5 1 5

1 3

2 x

Ax

=

=

=

=

2

2 ( 1)

1 1 2

1 2 1

2 3

2

4

1 x

Ax

= −

− −

=

=

=

(47)



×n 矩陣A，

0 )

I (

⇒ − =

= x A x

Ax





n×n

0 )

I ( )

I

det(

A =

A =

n + cn1

n1 +

+ c1

+ c0 = 0

) I

(

⇒ − =

= x A x

Ax

(

I − A)x = 0

det(

I A) = 0

(48)



=

3 2

4 A 1

4 1 −

= −

− λ

λ

0 )

1 )(

5 (

5 4

3 2

4 ) 1

I (

2 − − = − + =

=

= −

λ λ

λ λ

λ λ A λ

1 = 5,

2 = −1 1

, 5 −

=

(49)

5 )

1

(

1 =

0

1 , 1

0 0 2

2

4 ) 4

I (

2 1

2 1 1

=

=

=

= −

t t t

t x

x

x x x

A

1 )

2

( λ2 = −

0

1 , 2 2

0 0 4

2

4 ) 2

I (

2 1

2 1 2

 ≠

 

= −



 

= −



 



 

= 



 



 

= −

t t t

t x

x

x x x

A

(50)



=

0 1

1

1 2

1

2 2

1 A

### 解：特徵方程式：

0 )

3 )(

1 )(

1 (

1 1

1 2

1

2 2

1

I − − − = − + − =

=

A

3 ,

1 ,

1 =1

2 = −

3 =

### 特徵值

Waddill, Using matrix techniques to establish properties of a generalized Tri- bonacci sequence, In: Application of Fibonnaci Numbers, Vol 4, G.E.. Bergun etal, Eds.,

Larsen

 lower triangular matrix: 下三角矩陣.  upper triangular matrix:

After lots of tests, we record two players’ winning probabilities and average number of rounds to figure out mathematical principles behind the problems and derive general formulas

Rodman, Canonical forms for symmetric/skew-symmetric real matrix pairs under strict equivalence and congruence, Linear Algebra and its Applications 406 (2005),

Chebyshev 多項式由 Chebyshev 於 1854 年提出, 它在數值分析上有重要的地位 [11], 本文的目的是介紹 Chebyshev 多項式及線性二階遞迴序列之行列式。 在第二節中, 我們先介

sort 函式可將一組資料排序成遞增 (ascending order) 或 遞減順序 (descending order)。. 如果這組資料是一個行或列向量，整組資料會進行排序。

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ðWinner winner, chicken

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for some constant  1 and all sufficiently  large  , then  Θ.

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FPGA（Field Programmable Gate Array）為「場式可程式閘陣列」的簡稱，是一 個可供使用者程式化編輯邏輯閘元件的半導體晶片

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