ad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign(

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1 I n n A AX = I, YA = I () n XY A () X = IX = (YA)X = Y(AX) = YI = Y X Y () XY A A AB AB BA (AB)(B A ) = A(BB )A = AA = I (BA)(A B ) = B(AA )B = BB = I (AB) = B A (BA) = A B A B A = B = 5 5 A B AB BA A n B () B = I + A + A + A + B I A A ( ) a b A = c d 56

2 ad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign(σ) = i j σ(i) σ(j) i j σ sign(σ) sign(σ) = σ() σ() σ() σ() σ() σ() σ() σ() = = σ() σ() στ ( σ τ ) στ sign(στ) = sign(σ) sign(τ) σ() σ() 57

3 A = (a ij ) det A = sign(σ)a σ(), a σ(n), σ S n S n n ) A λ λ A = λ λ n det A = λ λ λ n I ) A, B n C = AB n det C = det A det B det I = A det(a ) = det A ) n A det A = A det A = A A A A A ( ) A = 5 6 (, )- 58

4 : A ( ) ( ) ( ) 5 6 = 5 6 : A ( ) ( ) = A t ( ) ( ) ( ) = t 5 6 t 5t 6t : A t ( ) = 5 6 t ( t ) 5t 6 A t ( ) ( ) ( ) t + t + 5t + 6t = A t ( ) ( ) + t = t 5 6 t E det E = t E det E = t t E det E = 59

5 5 E t E t E det E = det E = t det E = 6 A A E A E 7 6 E E ) A α B det B = α det A A α C det C = α det A A A = ( ) A B ( ) 6 B = det A = ( ) = 9 det B = 6 ( ) = 7 det B = det A 5) A B det B = det A 5 A A t B = ( t ) + t det B = ( t) ( + t) ( ) = 9 det A 6) 5 A det A = A det A = A n a,, a n A A = ( a a a n ) 6

6 a,, a n c a + + c n a n + c n a n = c, c,, c n c n λ i = c i /c n (i =,, n ) n i λ i E i E, E,, E n A ( AE E E n = a a a n + ) n i= λ na i = ( a a ) B det B = 5 det A = det B 6 A A = ( ) 6 A det A = ( ) ( 6) = A B = ( ) 6 + ( ) = + ( ) det B = = ( ) 7) A n A n + B a a a n a a a n a a a n A = B = a a a n a n a n a nn a n a n a nn det B = det A 7 A 8 A = A / / 8 = B 6

7 B = C 8 8 C = D D = E det E = 8 8 = ( 8) ( ) ( 8) = 6 / E = A det E = det A det A = det A = = = 8 8 = 8 8 = 8 8 = 8 8 = ( 6) = / 6

8 8 i j a ij a ij i j a ij A A = A I A A E = E A = = A A A E =, E = E E A = = A A A E =, E 5 = E5 E A = = A A E 6 =, E 7 = E7 E 6 A = 6

9 E 7 E 6 E 5 E E E E A = I AA = I E 7 E 6 E 5 E E E E = E 7 E 6 E 5 E E E E AA = A = E 7 E 6 E 5 E E E E I = E 7 E 6 E 5 E E E E A A = E 7 E 6 E 5 E E E E = E, E,, E 7? 6

10 B B = E, E, E, E, E 5 B E 5 E E E E B = = B B = B B B det B = n A ~A = r n r () () r ( ) ~ ~A = Ir = () n r B () () B = A () r A rank A A rank A = n () 65

11 9 6 A 6 B rank A = rank B = A A rank A = A = A B B = rank B = + = ( ) ( ) + ( ) = B 8 n A i)iv) i) A ii) det A iii) iv) A rank A = n 66

12 (m, n) A ~A = r m r () }{{}}{{} r n r ~ ~A = }{{}}{{} r n r r m r A rank A = r rank A = A = A AB 5 A = B = (5) 67

13 x y z x + y z = x + y z = x y = 8 x y = z 8 A A = A Ax = b A x = A b (6 ) A x 5 y = = z n n n A Ax = b 68

14 A b x = A b x x + x + x = x x + x = x x = x + x x = A Ax = b x x + y z = x + y z = x = 8 y z = 8 (6) y z = (7) (7) y z = (6)(7) x, y x x + y z = x + y z = x = y z = (8) y z = 5 (9) (9) (8) y z = 5 y, z t (x, y, z ) = (, 5 + t, t) 69

15 B B = B ~ B = B ~ 6 E E = = v = (x, y, z) Bv = b E Bv ~ = Eb b ~ x x Bx ~ = y = y z z b b ~ Eb = b + b b = b b b b + b + b ~ B ~ b b + b + b b = b = b = 8 ~ b = x = ~ b y z = ~ b b ~ i b ~ i x ~b y = ~b + z z Bv = b (,, ) ( ~ b, ~ b, ) 7

16 Ax = b x A =, x = x x, b = 5 x 9 x x x = x ( ) ( ) ( ) ( ) x x = + x s, t x x x x = + s + t x x x x = x 8 + s + t B x x B x x = x

17 n n m n > m x + x x + x = x + x z x = x x x = 8 x C =, x = x x, b = 8 x A A = A Cx = b A x x 5 x = 7 x t x 5 x x = 7 + t x n < m n n x D =, x = x x, b = x Dx = b 7

漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト

漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト https://www.hmg-gen.com/tuusin.html https://www.hmg-gen.com/tuusin1.html 1 2 OK 3 4 {a n } (1) a 1 = 1, a n+1 a n = 2 (2) a 1 = 3, a n+1 a n = 2n a n a n+1 a n = ( ) a n+1 a n = ( ) a n+1 a n {a n } 1,