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. 學 大 大 ( ) 線性

, 學 , 線性代數

. , 線性代數

, .

, . 學

. , (Question).

, 大

. , 線性代數 . ,

, ,

, . , 性

, . , .

v

(3)

## Determinant

determinant ( ). n× n matrix,

row vectors “ ”. , determinant

row vector , column vector .

8.1. Signed Area in R2 and Properties of Determinant Function

A 2× 2 matrix, det(A) , A

row vectors sign area. n× n

matrix determinant 性 .

A =

[ a b c d

]

, A determinant det(A) = ad− bc. R2

u = (a, b), v = (c, d). u , a = 0, b = 0, det(A) = 0. u̸= O, u π/2 Rπ/2(u) = (−b,a).

Rπ/2(u)· v = (−b,a) · (c,d) = ad − bc = det(A). (8.1)

u v θ 0≤θ ≤ π/2 ( ). u, v

, u , ∥v∥sinθ. ∥u∥∥v∥sinθ.

Rπ/2(u) u π/2 , ∥Rπ/2(u)∥ = ∥u∥ Rπ/2(u) v (π/2) − θ. sinθ = cos((π/2) − θ), u, v

∥u∥∥v∥sinθ = ∥u∥∥v∥cos((π/2) − θ) = ∥Rπ/2(u)∥∥v∥cos((π/2) − θ) = Rπ/2(u)· v = det(A).

..

u

. v

.

θ

.

π2−θ

. Rπ/2(u)

163

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u v θ π/2 < θ ≤ π, u, v , u ,

∥v∥sinθ. Rπ/2(u) v θ −(π/2). sinθ = cos(θ −(π/2))

u, v ∥u∥∥v∥sinθ = ∥u∥∥v∥cos(θ − (π/2)) = Rπ/2(u)· v = det(A).

u v θ −π/2 ≤ θ < 0 ( ). u, v

, u , −∥v∥sinθ. −∥u∥∥v∥sinθ.

Rπ/2(u) v (π/2) − θ. u, v

−∥u∥∥v∥sinθ = −∥u∥∥v∥cos((π/2) − θ) = −Rπ/2(u)· v = −det(A).

..

u

.

v .

θ

.

π 2θ

.

Rπ/2(u)

u v θ −π < θ < −π/2 , u, v −det(A).

, A =

[ a b c d

]

, u = (a, b), v = (c, d). det(A)

u, v . det(A) v u . ,

det(A) v u . det(A) u, v

, u, v 性. det(A) u, v

signed area.

n× n matrix. A n× n matrix, v1, . . . , vn∈ Rn

A row vectors. det(A) v1, . . . , vn Rn

signed volume. , det(A) v1, . . . , vn Rn

“ ” , v1, . . . , vn 性. 大 ? n≥ 4

,Rn n “ ” , ?

, R2 R3 .

Rn n . 言 ,

signed area 性 , determinant ( ) Mn×n R 數

( det ), 性 .

, . R2

(1, 0), (0, 1) ( ) 1,

. (1, 0) (0, 1) π/2, 前 性 . det(I2) = 1

(1, 0), (0, 1) 1 . Rn

, standard basis e1, . . . , en 1,

e1, . . . , en 性 . e1, . . . , en

signed volume 1, det(In) = 1.

(5)

Rn n v1, . . . , vn 性 ? R2 , u, v , v

u , v, u , u v . 2×2 determinant

det [ a b

c d ]

= ad− bc = −(cd − ad) = −det [ c d

a b ]

n v1, . . . , vn, vi, vi+1 性.

A∈ Mn×n, A row A, det(A) =−det(A).

v1, . . . , vn 性 vi −vi. R2 ,

u, v , −u,v . , u, v , −u,v . :

..

u

. v

.

−u

.

v

.

u

.

−u

2× 2 determinant

det

[ −a −b

c d

]

= det

[ a b

−c −d ]

=−ad + bc = −(ad − bc) = −det [ a b

c d ]

det(A) =−det(A).

r , r . 2× 2 determinant

det

[ ra rb c d

]

= det

[ a b rc rd

]

= rad− rbc = r(ad − bc) = r det [ a b

c d ]

rvi r . r > 0, A∈ Mn×n row r

A, det(A) = r det(A). A row −r

A′′, row r A A row −1,

det(A′′) =−det(A) =−r det(A). 言 , r 數 數, A row

r A, det(A) = r det(A).

Rn n v1, . . . , vn vi , vi= wi+ wi, v1, . . . , vi, . . . , vn v1, . . . , wi, . . . , vn

v1, . . . , wi, . . . , vn . R2

:

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..

u .

v1 .

v2

. v1+ v2

u , u, v1+ v2 u, v1 u, v2

. 2× 2 determinant det

[ a + a b + b

c d

]

=

(a + a)d− (b + b)c = (ad− bc) + (ad− bc) = det [ a b

c d ]

+ det

[ a b c d

] ,

det

[ a b

c + c d + d ]

=

a(d + d)− b(c + c) = (ad− bc) + (ad− bc) = det [ a b

c d ]

+ det

[ a b c d

]

i-th row , A, B,C row , det(A) = det(B) + det(C).

determinant 性 :

(1) det(In) = 1.

(2) n× n matrix A row A, det(A) =−det(A).

(3) n×n matrix A row 數 r A, det(A) = r det(A).

(4) A, B,C n× n matrix, A i-th row B C i-th row ,

A, B,C row , det(A) = det(B) + det(C).

(3), (4) 性 , determinant multi-linear 性 . ,

A = B + rC det(A) = det(B) + r det(C), row 線性

row , determinant row 線性 . 大

det







— v1— ...

— vi+ rvi— ...

— vn







= det







— v1— ...

— vi— ...

— vn







+ r det







— v1— ...

— vi— ...

— vn







.

Question 8.1. determinant multi-linear 性 det

[ ra1+ sa2 rb1+ sb2 tc1+ uc2 td1+ ud2

]

det

[ ai bi

cj dj ]

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8.2. Uniqueness of the Determinant Function

det 數 性 , 數

. 性 , 性 數 .

, 代

, . , 數 代 ,

, 前 性 性 , 性 數

, 數 ,

, det , 性 det

. , 前 “ det

” . det ,

. , “ det ” .

det elementary row operation . det 性

(2) A row . A row

. row A i-th row j-th

row . 性, i < j, A i-th row i + 1-th row ,

i + 1-th row i + 2-th row , i-th row j− 1-th

row. i + 1-th j− 1-th row row,

( j− 1) − i row . j-th row , j− 1-th row

( i-th row j-th row), row

j-th row i-th row. j-th row j− 1-th row

i + 1-th row i-th row j− i row .

i-th row j-th row ( j− 1 − i) + ( j − i) = 2( j − i) + 1

row . 3-th row 6-th row , 3-rd

4-th row , 4-th 5-th row , 3-rd row 5-th row .

(6− 1) − 3 = 2 row :

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







 ...

— v3

— v4

— v5

— v6— ...

















 ...

— v4

— v3

— v5

— v6— ...

















 ...

— v4

— v5

— v3

— v6— ...









6-th 5-th row ( 3-rd 6-th row ),

5-th 4-th row , 4-th 3-rd row 6-th row 3-rd row

. 6− 3 = 3 row :









 ...

— v4

— v5

— v3

— v6— ...

















 ...

— v4

— v5

— v6

— v3— ...

















 ...

— v4

— v6

— v5

— v3— ...

















 ...

— v6

— v4

— v5

— v3— ...









row det , 2( j− 1) + 1 數, det

. 性 .

Lemma 8.2.1. A n× n matrix. A row A,

det(A) =−det(A).

, A i-th j-th row elementary row operation

A elementary matrix E. E identity matrix In i-th j-th row . Lemma 8.2.1, det(E) =−det(In). det 性 (1)

det(In) = 1, det(E) =−1. Lemma 8.2.1 A i-th j-th row

EA −det(A), E i-th j-th row elementary row

operation elementary matrix, det(EA) =−det(A) = det(E)det(A).

Lemma 8.2.1, det 數 , 性 .

Lemma 8.2.2. A n× n matrix A row . det(A) = 0.

Proof. A i-th row j-th row . A i-th j-th row A, Lemma 8.2.1 det(A) =−det(A). A= A, det

elementary row operation row 數.

elementary row operation det 性 (3). ,

elementary row operation A i-th row 數 r

A elementary matrix E. E identity matrix In i-th row r. det 性 (1)(3), det(E) = r det(In) = r. 性 (3)

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det(EA) = r det(A), det(EA) = r det(A) = det(E) det(A).

det 數 , 性 .

Lemma 8.2.3. A n× n matrix A row 0. det(A) = 0.

Proof. A i-th row 0. A i-th 2 A,

det 性 (3) det(A) = 2 det(A). A = A, det 數 ( )

det(A) = det(A). det(A) = det(A) = 2 det(A) det(A) = 0. 

elementary row operation row 數 r row.

A n×n matrix A k-th row vk, for k = 1, . . . , n. A i-th row r j-th row A, A j-th row vj+ rvi, A k-th row vk, for k̸= j. B n× n matrix j-th row vi, k-th row vk, for k̸= j. det multi-linear 性 ( 性 (3) (4)), det(A) = det(A) + r det(B). B i-th row

j-th row vi, Lemma 8.2.2 det(B) = 0. det(A) = det(A),

Lemma 8.2.4. A n× n matrix. A i-th row r j-th row A, det(A) = det(A).

, A i-th row 數 r j-th row elementary row operation A elementary matrix E. E identity matrix In

i-th row 數 r j-th row. Lemma 8.2.4, det(E) = det(In) = 1

det(EA) = det(A). E i-th row 數 r j-th row

elementary row operation elementary matrix, det(EA) = det(A) = det(E) det(A).

elementary row operations det , .

Theorem 8.2.5. A n× n matrix. E elementary matrix, det(EA) = det(E) det(A).

Theorem 8.2.5 determinant 性 ,

determinant 性 . elementary row operations elementary

matrices determinant 0, determinant :

(1) row elementary matrix E, det(E) =−1.

(2) row 數 r elementary matrix E, det(E) = r.

(3) row 數 r row elementary matrix E,

det(E) = 1.

, E1, E2 elementary matrices, Theorem 8.2.5

det(E2E1A) = det(E2(E1A)) = det(E2) det(E1A) = det(E2) det(E1) det(A).

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det(Ek···E1A) == det(Ek)···det(E1) det(A). (8.2)

elementary matrices determinant, determinant 性 .

Theorem 8.2.6. A, B n× n matrices.

(1) A invertible det(A)̸= 0.

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

(3) det(AT) = det(A).

Proof. (1) A invertible A elementary row operations echelon form A pivot 數 n, rank(A) < n ( Theorem 3.5.2). A

pivot 數 n, A row ( n-th row) 0. Lemma 8.2.3

det(A) = 0. elementary matrices E1, . . . , Ek A= Ek···E1A, (8.2) det(A) = det(Ek)···det(E1) det(A). elementary matrices determinants det(E1), . . . , det(Ek) 0, det(A) = 0 det(A) = 0. A invertible,

A elementary matrices ( Proposition 3.5.7). elementary

matrices E1. . . Ek A = Ek···E1. (8.2) det(A) = det(Ek)···det(E1).

det(E1), . . . , det(Ek) 0 det(A)̸= 0.

(2) A invertible (1) det(A) = 0. B n× n matrix,

AB invertible ( Proposition 3.5.5(3)). (1) det(AB) = 0, det(AB) = 0 = det(A) det(B). A invertible. elementary matrices E1, . . . , Ek A = Ek···E1. (8.2) det(A) = det(Ek)···det(E1)

det(AB) = det(Ek···E1B) = det(Ek)···det(E1) det(B) = det(A) det(B).

(3) A invertible (1) det(A) = 0 AT invertible (

Proposition 3.5.5(2)). det(AT) = 0 = det(A). A invertible.

elementary matrices E1, . . . , Ek A = Ek···E1 AT= E1T···EkT ( Proposition

3.2.4). E row elementary matrix row 數 r

elementary matrix E = ET, E i-th row 數 r j-th row

elementary matrix , ET j-th row 數 r i-th row elementary matrix, elementary matrix E det(E) = det(ET).

det(AT) = det(E1T···EkT) = det(E1T)···det(EkT) = det(E1)···det(Ek).

n× n matrix determinant 數 數 ,

det(E1)···det(Ek) = det(Ek)···det(E1) = det(A),

det(AT) = det(A). 

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row operation transpose transpose column operation. Theorem 8.2.6 (3) elementary column operation

determinant elementary row operation determinant . 言 ,

Theorem 8.2.5 .

Corollary 8.2.7. A n× n matrix. E elementary matrix, det(AE) = det(E) det(A).

, det 數 , .

. 數, 性 前 .

Theorem 8.2.8. 數 det :Mn×n→ R (1) det(In) = 1.

(2) n× n matrix A row A, det(A) =−det(A).

(3) n×n matrix A row 數 r A, det(A) = r det(A).

(4) A, B,C n× n matrix, A i-th row B C i-th row ,

A, B,C row , det(A) = det(B) + det(C).

Proof. det :Mn×n→ R det:Mn×n→ R . n× n matrix A, A invertible, Theorem 8.2.6 (1) det(A) = det(A) = 0. A invertible,

elementary matrices E1, . . . , Ek A = Ek···E1. Theorem 8.2.6 (2) det(A) = det(Ek)···det(E1) det(A) = det(Ek)···det(E1). elementary matrix E det(E) = det(E),

det(A) = det(Ek)···det(E1) = det(Ek)···det(E1) = det(A).

A∈ Mn×n, det(A) = det(A), det det 數. 

. det 數 ,

elementary matrix determinant. 數

. det ,

invertible matrix elementary matrix , determinant.

elementary row operation n×n matrix determinant.

elementary row operations echelon form. echelon form

(1) row ( determinant ) (3) row 數 r

row ( determinant ), row operations. pivot 數

n, determinant 0. pivot 數 n,

row row operation, echelon form determinant determinant

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( 數 , 數 ). echelon form determinant ? n× n matrix echelon form upper triangular matrix ( ),

diagonal ( 線) ( (i, i)-th entry) 0 ( ai j= 0, for

i > j), determinant diagonal entries .

Proposition 8.2.9. A = [ai j] n× n upper triangular matrix det(A) = a1 1···an n, det(A) A diagonal entries .

Proof. A diagonal entry ai i 0, A upper triangular, A

echelon form pivot 數 n. A invertible, det(A) = 0.

A diagonal entries a1 1···ai i···an n 0, det(A) = 0 = a1 1···an n.

A diagonal entry 0, A row elementary row

operation: i∈ {1,...,n}, A i-st row 1/ai i. A,

det(A) = a1 1···an ndet(A). A diagonal entry 1, echelon form reduced echelon form ( Section 2.2), row ( n-th row)

row 數 row A In.

elementary row operations determinant, det(A) = det(In) = 1.

det(A) = a1 1···an ndet(A) = a1 1···an n 

Question 8.2. A = [ai j] n× n lower triangular matrix, A diagonal

0 ( ai j= 0, for i < j). det(A) = a1 1···an n, det(A) A diagonal

entries .

Example 8.2.10. elementary row operation



0 2 −1 1

1 2 0 2

1 4 −2 6 2 6 −1 8



 determi-

nant. 1-st, 2-nd row



1 2 0 2

0 2 −1 1 1 4 −2 6 2 6 −1 8



 ( determinant ).

1-st row −1,−2 3-rd, 4-th row



1 2 0 2

0 2 −1 1 0 2 −2 4 0 2 −1 4



 ( deter-

minant ). 2-nd row −1,−1 3-rd, 4-th row echelon form



1 2 0 2

0 2 −1 1 0 0 −1 3

0 0 0 3



 ( determinant ). Proposition 8.2.9

echelon form determinant −6, echelon form

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row row operation, determinant ,

det



0 2 −1 1

1 2 0 2

1 4 −2 6 2 6 −1 8



 = −det



1 2 0 2

0 2 −1 1 0 0 −1 3

0 0 0 3



 = 6.

8.3. Determinant of 3× 3 Matrix

determinant 性 , 3× 3 matrix determinant

, 3× 3 matrix determinant . determinant

R3 sign volume.

Theorem 8.2.6 (3) , det(AT) = det(A). determinant row

multi-linear 性 , row , column multi-linear 性 .

det

v1 ··· vi+ rvi ··· vn

 = det

v1 ··· vi ··· vn

 + rdet

v1 ··· vi ··· vn

.

3×3 matrix A =

a1 1 a1 2 a1 3 a2 1 a2 2 a2 3 a3 1 a3 2 a3 3

. determinant column multi-linear

1 0 0

 + a2 1

0 1 0

 + a3 1

0 0 1

,

det

a1 1 a1 2 a1 3 a2 1 a2 2 a2 3 a3 1 a3 2 a3 3

 =

a1 1det

 1 a1 2 a1 3 0 a2 2 a2 3 0 a3 2 a3 3

 + a2 1det

 0 a1 2 a1 3 1 a2 2 a2 3 0 a3 2 a3 3

 + a3 1det

 0 a1 2 a1 3 0 a2 2 a2 3 1 a3 2 a3 3

.

det

 1 a1 2 a1 3 0 a2 2 a2 3 0 a3 2 a3 3

 , , elementary

row operation echelon form. 1-st row,

[ a2 2 a2 3 a3 2 a3 3

]

echelon form. det

 1 a1 2 a1 3 0 a2 2 a2 3 0 a3 2 a3 3

 = det[

a2 2 a2 3 a3 2 a3 3

]

. ,

det 性 ,

det

 0 a1 2 a1 3

1 a2 2 a2 3

0 a3 2 a3 3

 = −det

 1 a2 2 a2 3

0 a1 2 a1 3

0 a3 2 a3 3

 = −det[

a1 2 a1 3

a3 2 a3 3 ]

,

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det

 0 a1 2 a1 3 0 a2 2 a2 3

1 a3 2 a3 3

 = −det

 0 a1 2 a1 3 1 a3 2 a3 3

0 a2 2 a2 3

 = det

 1 a3 2 a3 3 0 a1 2 a1 3

0 a2 2 a2 3

 = det[

a1 2 a1 3 a2 2 a2 3

] . determinant 性 det(A) “

det

a1 1 a1 2 a1 3 a2 1 a2 2 a2 3 a3 1 a3 2 a3 3

 = a1 1det

[ a2 2 a2 3 a3 2 a3 3

]

− a2 1det

[ a1 2 a1 3 a3 2 a3 3

]

+ a3 1det

[ a1 2 a1 3 a2 2 a2 3

] .

, det 性 , M3×3 R 數

( ). 數 初

3× 3 matrix determinant ( ).

det(I3) = 1.

det

 1 0 0 0 1 0 0 0 1

 = 1det[ 1 0 0 1

]

− 0det [ 0 0

0 1 ]

+ 0 det [ 0 0

1 0 ]

= 1.

det(I3) = 1 .

row determinant .

det

a1 1 a1 2 a1 3

a2 1 a2 2 a2 3

a3 1 a3 2 a3 3

 = a1 1det

[ a2 2 a2 3

a3 2 a3 3

]

− a2 1det

[ a1 2 a1 3

a3 2 a3 3

]

+ a3 1det

[ a1 2 a1 3

a2 2 a2 3

] ,

det

a2 1 a2 2 a2 3

a1 1 a1 2 a1 3 a3 1 a3 2 a3 3

 = a2 1det

[ a1 2 a1 3

a3 2 a3 3 ]

− a1 1det

[ a2 2 a2 3

a3 2 a3 3 ]

+ a3 1det

[ a2 2 a2 3

a1 2 a1 3 ]

,

det

a1 1 a1 2 a1 3

a3 1 a3 2 a3 3

a2 1 a2 2 a2 3

 = a1 1det

[ a3 2 a3 3

a2 2 a2 3

]

− a3 1det

[ a1 2 a1 3

a2 2 a2 3

]

+ a2 1det

[ a1 2 a1 3

a3 2 a3 3

] .

2× 2 matrix row determinant ,

det

[ a3 2 a3 3 a2 2 a2 3

]

=−det

[ a2 2 a2 3 a3 2 a3 3

] , det

[ a2 2 a2 3 a1 2 a1 3

]

=−det

[ a1 2 a1 3 a2 2 a2 3

] .

det

a2 1 a2 2 a2 3

a1 1 a1 2 a1 3

a3 1 a3 2 a3 3

 = −det

a1 1 a1 2 a1 3

a2 1 a2 2 a2 3

a3 1 a3 2 a3 3

,

det

a1 1 a1 2 a1 3

a3 1 a3 2 a3 3

a2 1 a2 2 a2 3

 = −det

a1 1 a1 2 a1 3

a2 1 a2 2 a2 3

a3 1 a3 2 a3 3

.

det

a1 1+ rb1 1 a1 2+ rb1 2 a1 3+ rb1 3

a2 1 a2 2 a2 3 a3 1 a3 2 a3 3

 = (a1 1+ rb1 1) det

[ a2 2 a2 3

a3 2 a3 3 ]

a2 1det

[ a1 2+ rb1 2 a1 3+ rb1 3

a3 2 a3 3 ]

+ a3 1det

[ a1 2+ rb1 2 a1 3+ rb1 3

a2 2 a2 3 ]

. (8.3)

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2× 2 matrix determinant multi-linear 性 , det

[ a1 2+ rb1 2 a1 3+ rb1 3 a3 2 a3 3

]

= det

[ a1 2 a1 3

a3 2 a3 3

] + r det

[ b1 2 b1 3

a3 2 a3 3

] ,

det

[ a1 2+ rb1 2 a1 3+ rb1 3 a2 2 a2 3

]

= det

[ a1 2 a1 3

a2 2 a2 3

] + r det

[ b1 2 b1 3

a2 2 a2 3

] . (8.3)

( a1 1det

[ a2 2 a2 3 a3 2 a3 3

]

− a2 1det

[ a1 2 a1 3 a3 2 a3 3

]

+ a3 1det

[ a1 2 a1 3 a2 2 a2 3

]) + r

( b1 1det

[ a2 2 a2 3 a3 2 a3 3

]

− a2 1det

[ b1 2 b1 3 a3 2 a3 3

]

+ a3 1det

[ b1 2 b1 3 a2 2 a2 3

]) . 3× 3 matrix determinant

det

a1 1+ rb1 1 a1 2+ rb1 2 a1 3+ rb1 3 a2 1 a2 2 a2 3

a3 1 a3 2 a3 3

 = det

a1 1 a1 2 a1 3

a2 1 a2 2 a2 3

a3 1 a3 2 a3 3

+rdet

b1 1 b1 2 b1 3

a2 1 a2 2 a2 3

a3 1 a3 2 a3 3

2-nd row 3-rd row det

a1 1 a1 2 a1 3

a2 1+ rb2 1 a2 2+ rb2 2 a2 3+ rb2 3

a3 1 a3 2 a3 3

 = det

a1 1 a1 2 a1 3

a2 1 a2 2 a2 3

a3 1 a3 2 a3 3

+rdet

a1 1 a1 2 a1 3

b2 1 b2 2 b2 3

a3 1 a3 2 a3 3

det

a1 1 a1 2 a1 3

a2 1 a2 2 a2 3 a3 1+ rb3 1 a3 2+ rb3 2 a3 3+ rb3 3

 = det

a1 1 a1 2 a1 3

a2 1 a2 2 a2 3 a3 1 a3 2 a3 3

+rdet

a1 1 a1 2 a1 3

a2 1 a2 2 a2 3 b3 1 b3 2 b3 3

2× 2 matrix determinant 3× 3 matrix determinant ,

“ ” . , 數學

n× n matrix determinant . determinant 1-st

column , 2-nd column 3-rd column . ,

.

Deﬁnition 8.3.1. A = [ai j] 3× 3 matrix. A i-th row j-th column 2× 2 matrix, A (i, j) minor matrix, Ai j . ai j= (−1)i+ jdet(Ai j), A (i, j) cofactor.

, 初 det(A)

det(A) = a1 1a1 1+ a2 1a2 1+ a3 1a3 1.

det(A) det(A) = a1 2a1 2+ a2 2a2 2+ a3 2a3 2, 2-nd column ,

det

a1 1 a1 2 a1 3

a2 1 a2 2 a2 3

a3 1 a3 2 a3 3

 = −a1 2det

[ a2 1 a2 3

a3 1 a3 3 ]

+ a2 2det

[ a1 1 a1 3

a3 1 a3 3 ]

− a3 2det

[ a1 1 a1 3

a2 1 a2 3 ]

.

, det(I3) = 1). det 性 ( Theorem 8.2.8),

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determinant 1-st column . 3-rd column

det

a1 1 a1 2 a1 3

a2 1 a2 2 a2 3

a3 1 a3 2 a3 3

 = a1 3det

[ a2 1 a2 2

a3 1 a3 2

]

− a2 3det

[ a1 1 a1 2

a3 1 a3 2

]

+ a3 3det

[ a1 1 a1 2

a2 1 a2 2

] .

det(A) = det(AT), A AT 1-st column det

a1 1 a2 1 a3 1 a1 2 a2 2 a3 2 a1 3 a2 3 a3 3

 = a1 1det

[ a2 2 a3 2 a2 3 a3 3

]

− a1 2det

[ a2 1 a3 1 a2 3 a3 3

]

+ a1 3det

[ a2 1 a3 1 a2 2 a3 2

] .

2× 2 matrix determinant ,

a1 1det

[ a2 2 a2 3 a3 2 a3 3

]

− a1 2det

[ a2 1 a2 3 a3 1 a3 3

]

+ a1 3det

[ a2 1 a2 2 a3 1 a3 2

] .

det(A) = det(AT) = a1 1a1 1+ a1 2a1 2+ a1 3a1 3, determinant 1-st row

. 2-nd row 3-rd row determinant. .

Theorem 8.3.2. A = [ai j] 3× 3 matrix. ai j A (i, j) cofactor,

det(A) = a1 1a1 1+ a2 1a2 1+ a3 1a3 1= a1 2a1 2+ a2 2a2 2+ a3 2a3 2= a1 3a1 3+ a2 3a2 3+ a3 3a3 3

= a1 1a1 1+ a1 2a1 2+ a1 3a1 3= a2 1a2 1+ a2 2a2 2+ a2 3a2 3= a3 1a3 1+ a3 2a3 2+ a3 3a3 3

3× 3 matrix determinant, R3

sign volume. R3 u, v, w row vectors,

A 1-st, 2-nd, 3-rd row u, v, w 3× 3 matrix, det(A) u, v, w

sign volume. det(A) , .

det(A) u, v, w 性. u, v, w

right hand rule ( ) , 大 u ,

v , w u, v, w , .

i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) ( det(I3) = 1 > 0). Section 8.1

u = (a1, a2, a3), v = (b1, b2, b3)∈ R3, u, v cross product ( ) u× v u× v =

( det

[ a2 a3

b2 b3

] , det

[ a3 a1

b3 b1

] , det

[ a1 a2

b1 b2

]) .

, , 數,

. u× v v× v , u× v = O. row

determinant , u× v = −v × u. u× v O ?

u× v = O det

[ a2 a3

b2 b3 ]

= det

[ a3 a1

b3 b1 ]

= det

[ a1 a2

b1 b2 ]

= 0, u = (a1, a2, a3), v = (b1, b2, b3) linearly dependent.

u = (a1, a2, a3), v = (b1, b2, b3), w = (c1, c2, c3) w· (u × v) = c1det

[ a2 a3

b2 b3 ]

+ c2det

[ a3 a1

b3 b1 ]

+ c3det

[ a1 a2

b1 b2 ]

(8.4)

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det

[ a3 a1 b3 b1

]

=−det

[ a1 a3 b1 b3

]

(8.4)

a1 a2 a3 b1 b2 b3

c1 c2 c3

 3-rd row determinant,

w· (u × v) = (u × v) · w = det

a1 a2 a3 b1 b2 b3

c1 c2 c3

 = det

 — u —

— v —

— w —

. (8.5)

(u× v) · w u, v, w sign volume.

, w = u w = v , u, v, w row vector row ,

determinant 0 (Lemma 8.2.2). (8.5) u·(u×v) = v·(u×v) = 0.

u, v linearly independent , u× v u v . w = u× v, (u× v) · (u × v) = ∥u × v∥2. u, v, u× v sign volume

∥u × v∥2. u, v, u× v u, v ,

u× v u v , ∥u × v∥ . u, v, u× v

∥u × v∥2 u, v ∥u × v∥, u, v

∥u × v∥. u, v, u× v sign volume

∥u × v∥2> 0, u, v, u× v . 性 .

Theorem 8.3.3. u, v∈ R3. u×w ̸= O u, v linearly independent.

u× v u, v , u, v u× v , u, v, u× v

.

w∈ R, (u×v)·w ̸= 0 u, v, w linearly independent. (u×v)·w

u, v, w sign volume.

8.4. Existence of the Determinant Function

, 2× 2 matrix determinant 性

3× 3 matrix determinant, 性. 3× 3 matrix

determinant 性 4× 4 matrix determinant 性, . ,

Deﬁnition 8.3.1 .

Deﬁnition 8.4.1. A = [ai j] n× n matrix. A i-th row j-th column

(n−1)×(n −1) matrix, A (i, j) minor matrix, Ai j . (n−1)×(n−1) matrix determinant , ai j= (−1)i+ jdet(Ai j), A (i, j) cofactor.

det(A) = a1 ka1 k+ a2 ka2 k+··· + an kan k.

(n− 1) × (n − 1) matrix determinant determinant 性

n× n matrix determinant 性 .

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det(In) = 1. In k-th column ek, k-th entry 1, 0. A = [ai j] = In, ai k= 0 for i̸= k ak k= 1.

det(In) = ak kak k= ak k. A = In (k, k) minor matrix In−1, A = In (k, k) cofactor ak k= (−1)k+kdet(In−1) = det(In−1). induction , det(In−1) = 1, ak k= 1, det(In) = 1.

row determinant . A = [ai j], l∈ {1,...,n−1},

A l-th row l + 1-th row B = [bi j]. i̸= l,l + 1

, bi j= ai j bl j = al+1 j, bl+1 j= al j. i < l , B (i, k) minor matrix Bi k

A (i, k) minor matrix Ai k l− 1-th, l-th row (

det(Bi k) =−det(Ai k)). i > l + 1 , Bi k Ai k l-th, l + 1-th row ( det(Bi k) =−det(Ai k)). Bl k Al+1 k Bl+1 k Al k. B

(i, k) cofactor bi k

(−1)i+kdet(Bi k) =



(−1)i+k(−det(Ai k)) =−ai k, if i̸= l and i ̸= l + 1;

(−1)l+kdet(Al+1 k) =−al+1 k, if i = l;

(−1)i+1+kdet(Al k) =−al k, if i = l + 1;

det(B) = b1 kb1 k +··· + bl kbl k + bl+1 kbl+1 k + ··· + bn kbn k

= a1 k(−a1 k) +··· + al+1 k(−al+1 k) + al k(−al k) + ··· + an k(−an k)

= −det(A)

r∈ R. A = [ai j], B = [bi j], C = [ci j] n× n matrices i̸= l ai j= bi j= ci j al j= bl j+ rcl j. i < l , A (i, k) minor matrix Ai k l− 1-th row Bi k l− 1-th row r Ci k l− 1-th row ( det(Ai k) = det(Bi k) + r det(Ci k)).

i > l + 1 , Ai k l-th row Bi k l-th row r Ci k l-th row (

det(Ai k) = det(Bi k) + r det(Ci k)). Al k Bl k Cl k. A (i, k) cofactor ai k

(−1)i+kdet(Ai k) =

{ (−1)i+k(det(Bi k) + r det(Bi k) = bi k+ rci k, if i̸= l;

(−1)l+kdet(Bl k) = (−1)l+kdet(Cl k) = bl k= cl k, if i = l;

det(A) = a1 ka1 k +··· + al kal k + ··· + an kan k

= b1 k(b1 k+ rc1 k) +··· + (bl k+ rcl k)bl k + ··· + bn k(bn k+ rcn k)

= b1 kb1 k+ rc1 kc1 k +··· + bl kbl k+ rcl kcl k + ··· + bn kbn k+ rcn kcn k

= det(B) + r det(C).

det 性, Theorem 8.2.8 性, .

Theorem 8.4.2. 數 det :Mn×n→ R (1) det(In) = 1.

(2) n× n matrix A row A, det(A) =−det(A).

(3) n×n matrix A row 數 r A, det(A) = r det(A).

augmented matrix [A |I 4 ], elementary row operation A

, A echelon form ( reduced echelon form) pivot column vectors.. elementary row operations column

elementary row operations reduced echelon form,.. echelon form Gauss

Theorem (M.Kalkowski, M.Karonski, and F.Pfender, 2010) ([8]) Every connected graph G 6= K 2 is 5-edge weight colorable1. Theorem (T.Bartnicki, J.Grytczuk,

[r]

Q.10 Does your GRSC have any concerns or difficulties in performing the function of assisting the SMC/IMC to review school‐based policies and