(୲౰: ୩‫ޫށ‬ত)
3.10
‫ྻߦٯ‬
(5) ͋Δྻʹଞͷྻͷఆ਺ഒΛՃ͑Δ.
࣮਺ a ʹର͠, ax = xa = 1 ͱͳΔ࣮਺ x ͕ଘࡏ͢Δ
ͱ͖, x Λ a ͷ‫͏͍ͱ਺ٯ‬. a ͷ‫਺ٯ‬͸ a 6= 0 ͷͱ͖ʹ‫ݶ‬
Γͨͩͻͱͭଘࡏͯ͠, x = 1/a = a−1 Ͱ͋Δ.
ಉ༷ͷ͜ͱΛߦྻͰߟ͑ͯΈΔ.
n ࣍ߦྻ A ʹରͯ͠, AX = XA = En ͱͳΔ n ࣍ߦ
ྻ X ͕ଘࡏ͢Δͱ͖, A Λਖ਼ଇߦྻͱ‫ͼݺ‬, X Λ A ͷ‫ٯ‬
ߦྻͱ͍͏. ͜ͷΑ͏ͳ X ͸ଘࡏͨ͠ͱ͢Ε͹ͨͩͻͱ
ͭʹఆ·Δ. ͦΕΛ A−1 ͱॻ͘.
ྫ. 1 ࣍ߦྻ (a) ͸࣮਺ a ͱಉҰࢹͰ͖Δ. ·ͨ, E1 =
(1) = 1 Ͱ͋Δ. Αͬͯ, 1 ࣍ߦྻʹ͓͚Δ‫ྻߦٯ‬͸‫਺ٯ‬
ͱಉٛͰ͋Δ.
!
a b
ྫ. 2 ࣍ߦྻ A =
͸ ∆ = ad − bc 6= 0 ͷͱ͖ʹ
c d
‫ݶ‬Γਖ਼ଇͰ, ‫ྻߦٯ‬͸
!
d −b
1
−1
A =
∆ −c a
ͱͳΔ. ྫ͑͹ A =
1 2
!
ͳΒ, ∆ = 1 · 4 − 2 · 3 =
3 4
−2 6= 0 ΑΓ A ͸ਖ਼ଇͰ, ‫ྻߦٯ‬͸
!
!
−2
1
4 −2
1
−1
=
A =
−2 −3 1
3/2 −1/2
!
1 2
ͱͳΔ. Ұํ, B =
͸ ∆ = 1 · 6 − 2 · 3 = 0 ΑΓ
3 6
ඇਖ਼ଇͰ͋Δ (B −1 ͸ଘࡏ͠ͳ͍).
3.11
9
ߦྻͷ‫ج‬ຊม‫ܗ‬
ߦྻ A ʹରͯ࣍͠ͷม‫ܗ‬Λ͢Δ͜ͱΛ, ߦྻ A ʹର͠
ͯߦ‫ج‬ຊม‫͢ܗ‬Δͱ͍͏.
(6) ;ͨͭͷྻΛަ‫͢׵‬Δ.
Ҏ্ͷ (1)ʙ(6) Λ͢΂ͯ߹Θͤͯ, ୯ʹ‫ج‬ຊม‫͏͍ͱܗ‬
!.
0 2
ྫ. ߦ‫ج‬ຊม‫ܗ‬Λ‫܁‬Γฦ͠ߦ͏͜ͱͰ, 2 ࣍ߦྻ
1 3
͸୯Ґߦྻ E2 ʹม‫͢ܗ‬Δ͜ͱ͕Ͱ͖Δ:
!
!
0 2 1 3
1 ↔
2
−−−−→
1 3
0 2
!
1 3
2 × 12
−−−→
0 1
!
1 0
1 +
2 ×(−3)
= E2
−−−−−−−−→
0 1
஫ҙ. ߦ‫ج‬ຊม‫ܗ‬ͷաఔ͸্ͷྫͷΑ͏ʹ໼ҹΛ࢖ͬͯ
ද͢. ౳߸ʹΑΔࣜม‫ͱܗ‬͸ҟͳΔͷͰ, ஫ҙͤΑ.
3.12
ߦྻͷ֊਺
[४උ] ߦྻ Cm,n (r) (r ͸ 0 ≤ r ≤ min{m, n} ͳΔ੔਺)
Λ࣍ͷΑ͏ʹఆΊΔ: (1, 1) ੒෼, (2, 2) ੒෼,...,(r, r) ੒෼
͸ 1 Ͱͦͷଞͷ੒෼͸͢΂ͯ 0 ͷ (m, n) ߦྻ. ͨͩ͠,
r = 0 ͷͱ͖͸ྵߦྻ.
ߦྻ Cm,n (r) ͸࣍ͷΑ͏ʹද͢͜ͱ΋Ͱ͖Δ:
!
Er O
Cm,n (r) =
O O
ྫ. 2 ࣍ߦྻͷ৔߹͸࣍ͷ 3 छྨ.
!
1 0
= E2 , C2,2 (1) =
C2,2 (2) =
0 1
!
0 0
= O2
C2,2 (0) =
0 0
1
0
!
0
,
0
(1) ͋ΔߦΛ 0 Ͱͳ͍਺Ͱఆ਺ഒ͢Δ.
ͦͷଞͷྫ.
(2) ͋Δߦʹଞͷߦͷఆ਺ഒΛՃ͑Δ.
(3) ;ͨͭͷߦΛަ‫͢׵‬Δ.
ҎԼͷ‫Ͱྫࢉܭ‬͸, ͜ΕΒͷม‫ܗ‬Λ࣍ͷΑ͏ͳ‫߸ه‬Λ༻
͍ͯද͢͜ͱʹ͢Δ.
ɾ
ɾୈ 3 ߦΛ (−2) ഒ͢Δ.
• 3 × (−2)ɾ
• ɾ
ɾୈ 2 ߦʹୈ 1 ߦͷ 4 ഒΛՃ͑Δ.
2 +
1 × 4ɾ
ɾ
ɾୈ 1 ߦͱୈ 2 ߦΛަ‫͢׵‬Δ.
• 1 ↔
2ɾ
·ͨ, ্ͷม‫ܗ‬ͷʮߦʯΛʮྻʯʹஔ͖‫ͨ͑׵‬΋ͷΛྻ
C2,3 (2) =
1
0
0
1

1 0


C3,2 (1) = 0 0
0 0

!
0
,
0
࿅श. ࣍ͷߦྻΛ্ͷΑ͏ʹ۩ମతʹॻ͖දͤ.
C3,3 (3), C4,2 (1), C3,4 (2).
ఆཧ-ఆٛ. (m, n) ߦྻ A ͸, ߦ‫ج‬ຊม‫( ܗ‬1)ʙ(3) ͱྻͷ
ަ‫( ׵‬6) Λ༗‫ݶ‬ճ࢖ͬͯ,
Er
O
∗
O
!
‫ج‬ຊม‫͏͍ͱܗ‬.
ͷ‫ʹܗ‬ม‫͢ܗ‬Δ͜ͱ͕Ͱ͖Δ. (஫ҙ. ͜Ε͸ Cm,n (r) ͷ
(4) ͋ΔྻΛ 0 Ͱͳ͍਺Ͱఆ਺ഒ͢Δ.
ӈ্ϒϩοΫ͕ O Ͱͳͯ͘΋ྑ͍ͱ͍͏ҙຯͰ͋Δ.)
1
͞Βʹ, ྻ‫ج‬ຊม‫( ܗ‬4)(5) Λ‫ͤڐ‬͹,
!
Er O
Cm,n (r) =
O O
0
ͷ‫Ͱ·ʹܗ‬ม‫͢ܗ‬Δ͜ͱ͕Ͱ͖Δ.
ߦྻ Cm,n (r) ͸‫ج‬ຊม‫ܗ‬ͷ࢓ํʹΑΒͣʹ A ʹΑͬͯ
ͷΈܾ·Δ. ͦ͜Ͱ, Cm,n (r) Λ A ͷ‫ج‬ຊม‫ʹܗ‬ΑΔඪ
४‫͏͍ͱܗ‬. ·ͨ, ੔਺ r Λ A ͷ֊਺ (ϥϯΫ) ͱ͍͍,
rank A = r ͱද͢.
!
0 2
ྫ. લઅͰ‫ͨݟ‬Α͏ʹ, 2 ࣍ߦྻ A =
͸ߦ‫ج‬ຊ
1 3
ม‫Ͱܗ‬୯Ґߦྻ E2 ʹม‫͞ܗ‬ΕͨͷͰ, ‫ج‬ຊม‫ʹܗ‬ΑΔඪ
४‫ܗ‬͸ E2 = C2,2 (2) Ͱ͋Δ. ͦͯ͠, A ͷ֊਺͸ 2 Ͱ͋
Δ. ͢ͳΘͪ, rank A = 2.
1
0

A = −1 3
0 0

2

1 0
2 +
1 ×1 
 −2 −−−−−−→ 0
0
1

1
2 × 13 
−−−→ 0
2
3
0


0
1

0 2

1 0
0 0 1

1 0
1 +
3 ×(−2) 
−−−−−−−−→ 0 1
0 0
Αͬͯ, rank A = 3.
3.13
0
0
0
0
0
ߦྻͷ֊਺ʹΑΔਖ਼ଇੑͷ൑ఆ
‫ྻߦٯ‬Λ࣋ͭߦྻΛਖ਼ଇߦྻ, ࣋ͨͳ͍ߦྻΛඇਖ਼ଇߦ
ྻͱ‫Ϳݺ‬ͷͰ͋ͬͨ. ࣍ʹड़΂Δͷ͸, ߦྻͷਖ਼ଇੑͱߦ
ྻͷ֊਺ͱͷؔ܎Λࣔ͢ඇৗʹॏཁͳఆཧͰ͋Δ.
ఆཧ. A Λ n ࣍ߦྻͱ͢Δ. ͜ͷͱ͖, ҎԼ͕੒Γཱͭ.
(1) A ͕ਖ਼ଇߦྻͰ͋Δ͜ͱͱ, rank A = n Ͱ͋Δ͜ͱ
͸ಉ஋Ͱ͋Δ.
(2) A ͕ඇਖ਼ଇߦྻͰ͋Δ͜ͱͱ, rank A < n Ͱ͋Δ͜
ͱ͸ಉ஋Ͱ͋Δ.
ྫ.

஫ҙ. ඪ४‫ܗ‬Λ‫ٻ‬ΊΔͷʹྻͷަ‫͕׵‬ඞཁʹͳΔྫ:




1 0 0
1 0 0

 2 ྻ↔3 ྻ 

A = 0 0 1 −−−−−−→ 0 1 0 = C3,3 (2)

0

0 = C3,3 (3)
1
(1) ͷূ໌͸লུ͢Δ. (2) ͸ (1) ͷର‫ۮ‬Λऔͬͨ΋ͷ
Ͱ͋Δ.
஫ҙ. (1) ͸ਖ਼ଇߦྻͷ‫ج‬ຊม‫ʹܗ‬ΑΔඪ४‫͕ܗ‬୯Ґߦྻ
ʹͳΔ͜ͱΛද͍ͯ͠Δ. ਖ਼ଇߦྻ͸, ྻ‫ج‬ຊม‫ܗ‬Λ࢖Θ
ͣʹ, ߦ‫ج‬ຊม‫ܗ‬ͷΈͰ୯Ґߦྻʹม‫͖Ͱܗ‬Δ͜ͱ͕஌Β
Ε͍ͯΔ.
3.14
ߦ‫ج‬ຊม‫ʹܗ‬ΑΔ‫ྻߦٯ‬ͷ‫ࢉܭ‬
2 ࣍ߦྻͷ‫ྻߦٯ‬͸ެࣜΛ༻͍ͯ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖
Δ͕, ͜͜Ͱ͸Ұൠʹ n ࣍ߦྻʹର͢Δ‫ྻߦٯ‬ͷ‫ํࢉܭ‬
ྫ.
๏Λ༩͑Δ.

1

A = −1
1
2
−1
3


1
2 +
1 ×1 
 4  −−−−−−→ 0
1
10

1
3 +
1 ×(−1) 
−−−−−−−−→ 0
3
2
3


1 7
3 10

2 3

1 7
0 1
7

−11

7 
7

0 −11

1
7 
0 0
0


1 0 0
3 ྻ+1 ྻ×11 

−−−−−−−−→ 0 1 7
0 0 0


1 0 0
3 ྻ+2 ྻ×(−7) 

−−−−−−−−−−→ 0 1 0 = C3,3 (2)

1
1 +
2 ×(−2) 
−−−−−−−−→ 0
0

1
3 +
2 ×(−1) 
−−−−−−−−→ 0
0
1
1
0 0 0
Αͬͯ, rank A = 2.
A Λ n ࣍ߦྻͱ͢Δ. A ͱ୯Ґߦྻ En Λԣʹฒ΂ͨ
(n, 2n) ‫( ྻߦܕ‬A | En ) Λߟ͑Δ. ֊਺Λ‫ٻ‬ΊΔཁྖͰߦ
ྻ (A | En ) Λߦ‫ج‬ຊม‫͠ܗ‬, ࠷ऴతʹ (En | B) ͱ͍͏‫ܗ‬
ʹͳͬͨͱ͢Δ. ͜ͷͱ͖, A−1 = B Ͱ͋Δ.
஫ҙ.
΋͠ࠨଆ͕୯ҐߦྻʹͳΒͳ͔ͬͨΒ, ͦΕ͸
rank A < n Λҙຯ͢ΔͷͰ, લઅͷఆཧΑΓ A ͸‫ߦٯ‬
ྻΛ࣋ͨͳ͍.
஫ҙ. ྻ‫ج‬ຊม‫( ܗ‬4)ʙ(6) Λ࢖ͬͯ͸͍͚ͳ͍.
!
1 2
ྫ. 2 ࣍ߦྻ A =
ͷ‫ྻߦٯ‬Λ‫ٻ‬ΊΔ.
3 4
(A | E2 ) =
1
2
1 0
3
4
0 1
1 2
−−−−−−−−→
0 −2
2 +
1 ×(−3)
1 2
−−−−−−→
0 1
2 ×(− 21 )
1 +
2 ×(−2)
−−−−−−−−→
!
!
1 0
−3 1
1
3/2
1
0
−2
0
1
3/2
!
0
−1/2
!
1
−1/2

参考文献