線性 代數

大學數學

數學 大 學 線性代數 , 大

數 .

大 學 線性代數學 ,

學 線性代數 . 大學線性代數

. 學 大 大 ( ) 線性

代數 , . 大學 線性代數 , 大

學 數學 . 大 大 數學

, 學 , 線性代數

. , 線性代數

數學 . , .

, .

學 , 數學 學 , 線性

代數 . 線性代數 .

, . 學

. , (Question).

, 大

. , 線性代數 . ,

學 線性代數 學 線性代數 , .

, ,

代. , .

, . , 性

, . , .

v

Determinant

determinant ( ). n× n matrix,

row vectors “ ”. , determinant

性 , 性 determinant . , Rⁿ 大

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 R_π/2(u) = (−b,a).

R_π/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.

R_π/2(u) u π/2 , ∥R_π/2(u)∥ = ∥u∥ R_π/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

. R_π/2(u)

163

u v θ π/2 < θ ≤ π, u, v , u ,

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

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

u v θ −π/2 ≤ θ < 0 ( ). u, v

, u , −∥v∥sinθ. −∥u∥∥v∥sinθ.

R_π/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 2−θ

.

R_π/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, . . . , v_n∈ Rⁿ

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

” signed volume. , det(A) v₁, . . . , v_n Rⁿ

“ ” , v₁, . . . , v_n 性. 大 ? n≥ 4

,Rⁿ n “ ” , ?

, R² R³ .

Rⁿ n . 言 ,

性 , .

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, . . . , e_n 1,

e₁, . . . , en 性 . e₁, . . . , en

signed volume 1, det(I_n) = 1.

Rⁿ n v₁, . . . , v_n 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 ]

性 . Rⁿ

n v₁, . . . , v_n, v_i, v_i+1 性.

A∈ Mn×n, A row A^′, det(A^′) =−det(A).

v₁, . . . , v_n 性 v_i −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 ]

性 . Rⁿ n v₁, . . . , vn, v_i −vi

性. A∈ Mn×n, A row −1 A^′,

det(A^′) =−det(A).

性 ? r 數,

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 ]

性 . r > 0, Rⁿ n v₁, . . . , v_n, v_i

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, . . . , v_n v_i , v_i= w_i+ w^′_i, v₁, . . . , v_i, . . . , v_n v₁, . . . , w_i, . . . , v_n

v₁, . . . , w^′_i, . . . , vn . R²

:

..

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) + (a^′d− b^′c) = 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^′

]

性 . 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).

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+ rv^′_i— ...

— vn—











= det











— v₁— ...

— vi— ...

— vn—









 + r det











— v₁— ...

— v^′_i— ...

— vn—









 .

Question 8.1. determinant multi-linear 性 det

[ ra₁+ sa₂ rb₁+ sb₂ tc₁+ uc₂ td₁+ ud₂

]

det

[ ai bi

c_j d_j ]

線性 .

8.2. Uniqueness of the Determinant Function

det 數 性 , 數

. 性 , 性 數 .

性 , 數 性 . ,

性 數 , . , 數

數 性 , .

大 , 數 , 性 ?

數學 . , ,

, 代

, . , 數 代 ,

性 .

, 前 性 性 , 性 數

, 數 ,

性 .

, 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 :











 ...

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

det(I_n) = 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

數 ( ) det(A) = det(A^′). det(A) = det(A^′) =−det(A) det(A) = 0. 

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(I_n) = r. 性 (3)

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

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 v_k, for k = 1, . . . , n. A i-th row r j-th row A^′, A^′ j-th row v_j+ rvi, A^′ k-th row v_k, for k̸= j. B n× n matrix j-th row v_i, k-th row v_k, for k̸= j. det multi-linear 性 ( 性 (3) (4)), det(A^′) = det(A) + r det(B). B i-th row

j-th row v_i, Lemma 8.2.2 det(B) = 0. det(A^′) = det(A),

性 .

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(I_n) = 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).

數學 , E₁, . . . , E_k elementary matrices,

det(E_k···E1A) == det(E_k)···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(A^T) = 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, . . . , E_k A^′= E_k···E1A, (8.2) det(A^′) = det(E_k)···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. . . E_k A = Ek···E1. (8.2) det(A) = det(Ek)···det(E1).

det(E₁), . . . , det(E_k) 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, . . . , E_k A = E_k···E1. (8.2) det(A) = det(E_k)···det(E1)

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

(3) A invertible (1) det(A) = 0 A^T invertible (

Proposition 3.5.5(2)). det(A^T) = 0 = det(A). A invertible.

elementary matrices E1, . . . , E_k A = Ek···E1 A^T= E₁^T···E_k^T ( Proposition

3.2.4). E row elementary matrix row 數 r

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

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

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

n× n matrix determinant 數 數 ,

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

det(A^T) = det(A). 

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 , .

大 , 性 前 det 性

. 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, . . . , E_k A = E_k···E1. Theorem 8.2.6 (2) det(A) = det(E_k)···det(E1) det^′(A) = det^′(E_k)···det^′(E₁). elementary matrix E det(E) = det^′(E),

det(A) = det(E_k)···det(E1) = det^′(E_k)···det^′(E₁) = det^′(A).

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

學 , n× n matrix determinant,

性 ? , invertible matrix elementary matrix

. 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

( 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 a_{i 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/a_{i 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^′ I_n.

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

det(A) = a_{1 1}···an ndet(A^′) = a_{1 1}···an n 

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

0 ( a_{i j}= 0, for i < j). det(A) = a_{1 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

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(A^T) = det(A). determinant row

性 , column 性 . determinant (3)(4)

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

大 ( column vector):

det



  

v₁ ··· vi+ rv^′_i ··· vn



 = det



  

v₁ ··· vi ··· vn



 + rdet



  

v₁ ··· v^′i ··· vn



.

3×3 matrix A =



 a_{1 1} a_{1 2} a_{1 3} a_{2 1} a_{2 2} a_{2 3} a3 1 a3 2 a3 3



. determinant column multi-linear

性 , A 1-st column a_{1 1}



1 0 0



 + a_{2 1}



0 1 0



 + a_{3 1}



0 0 1



,

det



 a_{1 1} a_{1 2} a_{1 3} a_{2 1} a_{2 2} a_{2 3} a_{3 1} a_{3 2} a_{3 3}



 =

a_{1 1}det



 1 a_{1 2} a_{1 3} 0 a_{2 2} a_{2 3} 0 a3 2 a3 3



 + a_{2 1}det



 0 a_{1 2} a_{1 3} 1 a_{2 2} a_{2 3} 0 a3 2 a3 3



 + a_{3 1}det



 0 a_{1 2} a_{1 3} 0 a_{2 2} a_{2 3} 1 a3 2 a3 3



.

det



 1 a_{1 2} a_{1 3} 0 a_{2 2} a_{2 3} 0 a_{3 2} a_{3 3}



 , , elementary

row operation echelon form. 1-st row,

[ a_{2 2} a_{2 3} a_{3 2} a_{3 3}

]

echelon form. det



 1 a_{1 2} a_{1 3} 0 a_{2 2} a_{2 3} 0 a3 2 a3 3



 = det[

a_{2 2} a_{2 3} a_{3 2} a_{3 3}

]

. ,

det 性 ,

det



 0 a1 2 a1 3

1 a2 2 a2 3

0 a_{3 2} a_{3 3}



 = −det



 1 a2 2 a2 3

0 a1 2 a1 3

0 a_{3 2} a_{3 3}



 = −det[

a1 2 a1 3

a_{3 2} a_{3 3} ]

,

det



 0 a_{1 2} a_{1 3} 0 a2 2 a2 3

1 a3 2 a3 3



 = −det



 0 a_{1 2} a_{1 3} 1 a3 2 a3 3

0 a2 2 a2 3



 = det



 1 a_{3 2} a_{3 3} 0 a1 2 a1 3

0 a2 2 a2 3



 = det[

a_{1 2} a_{1 3} a2 2 a2 3

] . determinant 性 det(A) “ ”

det



 a_{1 1} a_{1 2} a_{1 3} a_{2 1} a_{2 2} a_{2 3} a3 1 a3 2 a3 3



 = a_{1 1}det

[ a_{2 2} a_{2 3} a3 2 a3 3

]

− a2 1det

[ a_{1 2} a_{1 3} a3 2 a3 3

]

+ a_{3 1}det

[ a_{1 2} a_{1 3} a2 2 a2 3

] .

, det 性 , M3×3 R 數

( ). 數 初

性 . 初 性 ,

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



 = a_{1 1}det

[ 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

a_{1 1} a_{1 2} a_{1 3} a_{3 1} a_{3 2} a_{3 3}



 = a_{2 1}det

[ a1 2 a1 3

a_{3 2} a_{3 3} ]

− a1 1det

[ a2 2 a2 3

a_{3 2} a_{3 3} ]

+ a_{3 1}det

[ a2 2 a2 3

a_{1 2} a_{1 3} ]

,

det



 a1 1 a1 2 a1 3

a3 1 a3 2 a3 3

a2 1 a2 2 a2 3



 = a_{1 1}det

[ 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

[ a_{3 2} a_{3 3} a_{2 2} a_{2 3}

]

=−det

[ a_{2 2} a_{2 3} a_{3 2} a_{3 3}

] , det

[ a_{2 2} a_{2 3} a_{1 2} a_{1 3}

]

=−det

[ a_{1 2} a_{1 3} a_{2 2} a_{2 3}

] .

det



 a2 1 a2 2 a2 3

a1 1 a1 2 a1 3

a_{3 1} a_{3 2} a_{3 3}



 = −det



 a1 1 a1 2 a1 3

a2 1 a2 2 a2 3

a_{3 1} a_{3 2} a_{3 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



.

性 (3), (4) , multi-linear 性 .

det



 a1 1+ rb1 1 a1 2+ rb1 2 a1 3+ rb1 3

a_{2 1} a_{2 2} a_{2 3} a_{3 1} a_{3 2} a_{3 3}



 = (a_{1 1}+ rb_{1 1}) det

[ a2 2 a2 3

a_{3 2} a_{3 3} ]

−

a_{2 1}det

[ a1 2+ rb1 2 a1 3+ rb1 3

a_{3 2} a_{3 3} ]

+ a_{3 1}det

[ a1 2+ rb1 2 a1 3+ rb1 3

a_{2 2} a_{2 3} ]

. (8.3)

2× 2 matrix determinant multi-linear 性 , det

[ a1 2+ rb_{1 2} a1 3+ rb_{1 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+ rb_{1 2} a1 3+ rb_{1 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)

( a_{1 1}det

[ a_{2 2} a_{2 3} a_{3 2} a_{3 3}

]

− a2 1det

[ a_{1 2} a_{1 3} a_{3 2} a_{3 3}

]

+ a_{3 1}det

[ a_{1 2} a_{1 3} a_{2 2} a_{2 3}

]) + r

( b_{1 1}det

[ a_{2 2} a_{2 3} a_{3 2} a_{3 3}

]

− a2 1det

[ b_{1 2} b_{1 3} a_{3 2} a_{3 3}

]

+ a_{3 1}det

[ b_{1 2} b_{1 3} a_{2 2} a_{2 3}

]) . 3× 3 matrix determinant

det



 a1 1+ rb_{1 1} a1 2+ rb_{1 2} a1 3+ rb_{1 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

a_{3 1} a_{3 2} a_{3 3}



 = det



 a1 1 a1 2 a1 3

a2 1 a2 2 a2 3

a_{3 1} a_{3 2} a_{3 3}



+rdet



 a1 1 a1 2 a1 3

b2 1 b2 2 b2 3

a_{3 1} a_{3 2} a_{3 3}





det



 a1 1 a1 2 a1 3

a_{2 1} a_{2 2} a_{2 3} a_{3 1}+ rb_{3 1} a_{3 2}+ rb_{3 2} a_{3 3}+ rb_{3 3}



 = det



 a1 1 a1 2 a1 3

a_{2 1} a_{2 2} a_{2 3} a_{3 1} a_{3 2} a_{3 3}



+rdet



 a1 1 a1 2 a1 3

a_{2 1} a_{2 2} a_{2 3} b_{3 1} b_{3 2} b_{3 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, A_{i j} . a^′_{i j}= (−1)^{i+ j}det(A_{i j}), A (i, j) cofactor.

, 初 det(A)

det(A) = a_{1 1}a^′_{1 1}+ a_{2 1}a^′_{2 1}+ a_{3 1}a^′_{3 1}.

det(A) det(A) = a_{1 2}a^′_{1 2}+ a_{2 2}a^′_{2 2}+ a_{3 2}a^′_{3 2}, 2-nd column ,

det



 a1 1 a1 2 a1 3

a2 1 a2 2 a2 3

a_{3 1} a_{3 2} a_{3 3}



 = −a1 2det

[ a2 1 a2 3

a_{3 1} a_{3 3} ]

+ a_{2 2}det

[ a1 1 a1 3

a_{3 1} a_{3 3} ]

− a3 2det

[ a1 1 a1 3

a_{2 1} a_{2 3} ]

.

前 det ( cofactor

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

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



 = a_{1 3}det

[ 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(A^T), A A^T 1-st column det



 a_{1 1} a_{2 1} a_{3 1} a_{1 2} a_{2 2} a_{3 2} a_{1 3} a_{2 3} a_{3 3}



 = a_{1 1}det

[ a_{2 2} a_{3 2} a_{2 3} a_{3 3}

]

− a1 2det

[ a_{2 1} a_{3 1} a_{2 3} a_{3 3}

]

+ a_{1 3}det

[ a_{2 1} a_{3 1} a_{2 2} a_{3 2}

] .

2× 2 matrix determinant ,

a_{1 1}det

[ a_{2 2} a_{2 3} a_{3 2} a_{3 3}

]

− a1 2det

[ a_{2 1} a_{2 3} a_{3 1} a_{3 3}

]

+ a_{1 3}det

[ a_{2 1} a_{2 2} a_{3 1} a_{3 2}

] .

det(A) = det(A^T) = a_{1 1}a^′_{1 1}+ a_{1 2}a^′_{1 2}+ a_{1 3}a^′_{1 3}, determinant 1-st row

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

Theorem 8.3.2. A = [a_{i j}] 3× 3 matrix. a^′_{i j} A (i, j) cofactor,

det(A) = a1 1a^′_{1 1}+ a2 1a^′_{2 1}+ a3 1a^′_{3 1}= a1 2a^′_{1 2}+ a2 2a^′_{2 2}+ a3 2a^′_{3 2}= a1 3a^′_{1 3}+ a2 3a^′_{2 3}+ a3 3a^′_{3 3}

= a_{1 1}a^′_{1 1}+ a_{1 2}a^′_{1 2}+ a_{1 3}a^′_{1 3}= a_{2 1}a^′_{2 1}+ a_{2 2}a^′_{2 2}+ a_{2 3}a^′_{2 3}= a_{3 1}a^′_{3 1}+ a_{3 2}a^′_{3 2}+ a_{3 3}a^′_{3 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

性 , det(A) > 0 u, v, w , det(A) < 0 .

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)

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, . ,

數學 n× n matrix determinant .

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, A_{i j} . (n−1)×(n−1) matrix determinant , a^′_{i j}= (−1)^{i+ j}det(Ai j), A (i, j) cofactor.

數學 (n− 1) × (n − 1) matrix determinant , n× n matrix A = [ai j], k∈ {1,...,n}, A k-th column ,

det(A) = a_{1 k}a^′_{1 k}+ a_{2 k}a^′_{2 k}+··· + an ka^′_{n k}.

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

n× n matrix determinant 性 .

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

det(In) = a_{k k}a^′_{k k}= a^′_{k k}. A = In (k, k) minor matrix In−1, A = In (k, k) cofactor a^′_{k k}= (−1)^k+kdet(I_n−1) = det(I_n−1). induction , det(I_n−1) = 1, a^′_{k k}= 1, det(In) = 1.

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

A l-th row l + 1-th row B = [b_{i j}]. i̸= l,l + 1

, bi j= a_{i j} bl j = a_{l+1 j}, bl+1 j= a_{l j}. i < l , B (i, k) minor matrix B_{i k}

A (i, k) minor matrix A_{i k} l− 1-th, l-th row (

det(B_{i k}) =−det(Ai k)). i > l + 1 , B_{i k} A_{i 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 b^′_{i k}

(−1)^i+kdet(Bi k) =





(−1)^i+k(−det(Ai k)) =−a^′_{i k}, if i̸= l and i ̸= l + 1;

(−1)^l+kdet(Al+1 k) =−a^′_{l+1 k}, if i = l;

(−1)^i+1+kdet(Al k) =−a^′_{l k}, if i = l + 1;

det(B) = b1 kb^′_{1 k} +··· + bl kb^′_{l k} + bl+1 kb^′_{l+1 k} + ··· + bn kb^′_{n k}

= a1 k(−a^′_{1 k}) +··· + al+1 k(−a^′_{l+1 k}) + al k(−a^′_{l k}) + ··· + an k(−a^′_{n k})

= −det(A)

性 (3), (4) , multi-linear 性 . l∈ {1,...,n − 1}

r∈ R. A = [a_{i j}], B = [b_{i j}], C = [c_{i j}] n× n matrices i̸= l a_{i j}= b_{i j}= c_{i j} al j= b_{l j}+ rc_{l j}. i < l , A (i, k) minor matrix A_{i k} l− 1-th row Bi k l− 1-th row r C_{i k} l− 1-th row ( det(A_{i k}) = det(B_{i k}) + r det(C_{i k})).

i > l + 1 , A_{i k} l-th row B_{i k} l-th row r C_{i k} l-th row (

det(Ai k) = det(Bi k) + r det(Ci k)). Al k Bl k Cl k. A (i, k) cofactor a^′_{i k}

(−1)^i+kdet(Ai k) =

{ (−1)^i+k(det(B_{i k}) + r det(B_{i k}) = b^′_{i k}+ rc^′_{i k}, if i̸= l;

(−1)^l+kdet(Bl k) = (−1)^l+kdet(Cl k) = b^′_{l k}= c^′_{l k}, if i = l;

det(A) = a_{1 k}a^′_{1 k} +··· + a_{l k}a^′_{l k} + ··· + a_{n k}a^′_{n k}

= b_{1 k}(b^′_{1 k}+ rc^′_{1 k}) +··· + (bl k+ rc_{l k})b^′_{l k} + ··· + bn k(b^′_{n k}+ rc^′_{n k})

= b1 kb^′_{1 k}+ rc_{1 k}c^′_{1 k} +··· + bl kb^′_{l k}+ rc_{l k}c^′_{l k} + ··· + bn kb^′_{n k}+ rc_{n k}c^′_{n 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).