LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University

LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0

3x + y + 2z 4 x + y + z x y + 5z 7 x + y + z 3x + y + 2z 4 x y + 5z 7 3 2 4 5 7 3 2 4 5 7 x + y + z 2y z 2y 6z 6 0 2 0 2 6 6 x + y + z 2y z 0 0 0 2 0 0 0 0 z y x z (parameter) - { x + y + z y + 2 z 2 { x + 2 z 3 2 y + 2 z 2 0 2 2 0 0 0 0 0 3 2 2 0 2 2 0 0 0 0

z t 3 2 2 x t + 3 2 2 y t 2 2 z t x y z t 2 + 2 0 n m a x + a 2 x 2 + + a n x n b a 2 x + a 22 x 2 + + a 2n x n b 2 a m x + a m2 x 2 + + a mn x n b m a ij b k a ij (double index) x, y, z x, x 2,, x n a a 2 a n b a 2 a 22 a 2n b 2 a m a m2 a mn b m b, b 2,, b m a a 2 a n a 2 a 22 a 2n a m a m2 a mn 3x + y + 2z 2 x + y + z 2 9x y + 5z 6 x 4, y 2, z 8 { x + 3y + 2z 3x 9y 6z 3 2 3x + y + 2z 4 x + y + z x y + 5z [ 3 2 0 0 0 0 ]

x 3t + 2u, y t, z u I 0 0 0 0 2 0 0 3 0 5 2 0 0 0 0 0 0 5 2 0 0 0 0 3 x, x 2, x 3 x, x 2 2, x 3 3 x 2 5t, x 2 t, x 3 t 5 0 0 5 0 0 0 3 0 0 0 4 2 0 0 0 0 0 0 x x 2 x 3 x 4 x 5 0 2 0 + t 5 0 0 0 + u 0 0 2 0 3 0 2 4 0 3 5 0 3 4

2 3 2 2 2 3 (arbitrary) 0 i 0 i c 2 c i ) i i c j c j i 3 2 0 i 2 2 i 2 c 2 4 2 c 2 2 i 2 ) 3 i 2 i 2 c j 2 c j i 2 2 5 3 0 i 3 3 3 i 3 c 3 4

2 2 a x + a 2 x 2 + + a n x n b a 2 x + a 22 x 2 + + a 2n x n b 2 a m x + a m2 x 2 + + a mn x n b m 3x + y + 2z 4 x + y + z x y + 5z 7 a a 2 a n b a 2 a 22 a 2n b 2 a m a m2 a mn b m 2 2 3 2 4 5 7 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 { x + 2 z 3 2 y + 2 z 2 0 3 2 2 0 2 2 0 0 0 0 0 5 0 0 0 0 0 0 x + 5z 0 y + z 0 0 2

2 (rank) A rank A 2 rank 0 5 0 0 0 0 0 0 rank 3, rank 5 0 0 5 0 0 0 3 0 0 0 4 2 0 0 0 0 0 0 0 5 0 0 0 0 3 2 22 A n A A I 0 0 2 () (2) 2 2 2

22 a x + a 2 x 2 + + a n x n 0 a 2 x + a 22 x 2 + + a 2n x n 0 a m x + a m2 x 2 + + a mn x n 0 3x + y + 2z 0 x + y + z 0 x y + 5z 0 x x 2 x n 0 a a 2 a n 0 a 2 a 22 a 2n 0 a m a m2 a mn 0 3 2 0 0 5 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r l,, r l i l {i, i 2,, i r, j, j 2,, j n r } {, 2,, n} x i + n r k c kx jk 0 x i2 + n r k c 2kx jk 0 x ir + n r k c rkx jk 0 x j,, x jn r 2 3

5 0 0 5 0 0 0 0 3 0 0 0 0 4 2 0 0 0 0 0 0 0 0 x + 5x 2 + 5x 5 + ( )x 6 0 x 2 + 0x 2 + 3x 5 + x 6 0 x 3 + 0x 2 + 4x 5 + 2x 6 0 i, i 2 3, i 3 4, j 2, j 2 5, j 3 6 j, j 2, j 2 x 2, x 5, x 6 22 n r n r x x 2 x n 0 n > r n r n r 0 0 0 0 0 0 0 0 0 x 0 x 2 0 x n 0 23 24 m n r r m r n r m r n 2 4

3 3 3 m n a a 2 a n a 2 a 22 a 2n A a m a m2 a mn m n (m, n) A [a ij ] 2 (m, n) m n 3 n [a, a 2,, a n ] n m m a a 2 a m 4 A j a j a 2j a j a mj A j i a i [a i, a i2,, a in ] A i A A [a, a 2,, a n ] a a 2 a m 3

5 i j (i, j) A (i, j) a ij 32 A B m n c A+B ca c a + b a 2 + b 2 a n + b n ca ca 2 ca n a 2 + b 2 a 22 + b 22 a 2n + b 2n ca 2 ca 22 ca 2n A+B, ca a m + b m a m2 + b m2 a mn + b mn ca m ca m2 ca mn x t + 3 2 2 y t 2 2 z t x y z 2 t + 3 2 2 t 2 t t 2 2 + 33 A (a i,j ) m r B (b k,l ) r n m n C (c s,t ) 3 2 2 0 r c s,t a s,u b u,t a s, b,t + a s,2 b 2,t + + a s,r b r,t u C AB A B ru a,u b ru u, a,u b u,2 ru a 2,u b ru u, a 2,u b u,2 C AB ru a m,u b ru u, a m,u b u,2 ru a,u b u,n ru a 2,u b u,n ru a m,u b u,n 3 A [ 0 2 0 ] AB BA, B [ 0 2 0 2 5 3 6 4 7 2 5 3 6 4 7 ] 2 5 3 6 4 7 [ 0 2 0 ] [ 0 9 7 3 2 5 9 3 6 2 4 7 5 AB BA AB BA 3 2 ]

2 A 3 2 5 b, x Ax 4 7 x x 2 x 3 3 2 5 x x 2 x 3 3x + x 2 + 2x 3 x + x 2 + x 3 x x 2 + 5x 3 Ax b 3 A a a 2 a n a 2 a 22 a 2n a m a m2 a mn, x x x 2 x n, b b b 2 b m Ax b Ax a x + a 2 x 2 + + a n x n a 2 x + a 22 x 2 + + a 2n x n a m x + a m2 x 2 + + a mn x n b b 2 b m A, B n n AB BA n n n 2 m n 0 0 m,n A m n A + 0 m,n 0 m,n + A A, A0 n,l 0 m,l, 0 l,m A 0 l,n 0 n,n 0 n 3 3

3 i i (i, i) n 0 I I n E E n A m n B n m AE A EB B 3 () A + B B + A (2) A + (B + C) (A + B) + C (3) A(BC) (AB)C (4) A(B + C) AB + AC (A + B)C AC + BC (5) ca (ci)a 34 m n A [a i,j ] (i, j) a j,i (j, i) a i,j n m A A t A a a 2 a n a 2 a 22 a 2n a m a m2 a mn A t a a 2 a m a 2 a 22 a m2 a n a 2n a mn A t t A, A T, T A A t 3 4

4 Ax b a a 2 a n x b a 2 a 22 a 2n x 2 b 2 A, x, b a m a m2 a mn ax b a A 4 A AB BA I B A (invertible matrix) [ (nonsingular matrix)] B A B A A B A Ax b B x n Bb BAx Ix x x Bb Ax A(Bb) (AB)b Ib b Bb Bb Ax b b m 4 2 2 [ ] [ a b A A d b c d ad bc c a ] [ a b AA c d ] [ d b ad bc c a ] [ ad bc 0 ad bc 0 ad bc ] I A A ad bc [ d b c a ] [ a b c d ] [ ad bc 0 ad bc 0 ad bc ] I [ 2 5 3 ] [ 3 5 2 ad bc 0 ad bc 0 2 2 [ ] a a 2 A a 2 a 22 4 ]

det A a a 22 a 2 a 2 a, b, c, d ad bc [ ] b b 2 B b 2 b 22 det(ab) [ ] a b + a 2 b 2 a b 2 + a 2 b 22 det a 2 b + a 22 b 2 a 2 b 2 + a 22 b 22 (a b + a 2 b 2 )(a 2 b 2 + a 22 b 22 ) (a b 2 + a 2 b 22 )(a 2 b 2 + a 22 b 22 ) (a a 22 a 2 a 2 )(b b 22 b 2 b 2 ) det A det B AB I det I det A det B det I det A a a 22 a 2 a 2 0 4 2 2 A [ a a 2 a 2 a 22 () A, B 2 2 det AB det A det B ] det A a a 22 a 2 a 2 (2) A det A a a 22 a 2 a 2 0 A [ ] a22 a 2 det A a 2 a 2 2 42 A n I I n n C [A, I] n 2n C D D [I, B] B A D I A A rank A n 4 2

42 A 2 2 2 0 3 2 C 2 2 0 0 2 0 0 0 3 2 0 0 2 2 0 0 2 0 0 0 3 2 0 0 2 2 0 0 0 3 4 2 0 0 4 5 3 0 0 0 2 2 0 0 4 5 3 0 0 0 2 2 0 0 0 4 3 2 2 0 0 0 3 4 2 0 3 2 0 0 2 2 0 0 0 0 4 5 3 0 0 0 2 2 0 0 0 4 3 0 0 2 2 0 0 2 5 4 0 0 4 3 A 2 2 A 2 5 4 4 3 43 A 5 5 2 0 C 5 0 0 5 0 0 2 0 0 0 5 0 0 5 0 0 2 0 0 0 5 0 0 2 0 0 0 0 0 0 0 5 0 0 0 0 0 0 2 0 0 0 4 3

[I, B] rank A 2 rank A 2 rank A 3 A A rank A 42 ( ) 0 0 2 0 3 0 2 4 ( 0 ) 4 4

5 I I m m E i,j (i, j)- i j m P (i; c) I + (c )E i,i, P (i, j) I E i,i E j,j + E i,j + E j,i, P (i, j; c) I + ce i,j c A [a i,j ] (i, j)- a i,j m n 2 2 3 P (i; c) A P (i; c)a a a 2 a n a 2 a 22 a 2n ca i ca i2 ca in a m a m2 a mn P (i; c) A A i c P (i, j) A i < j a a 2 a n a 2 a 22 a 2n P (i, j)a a j a j2 a jn a i a i2 a in a m a m2 a mn P (i, j) A A i j 5

P (i, j; c) A P (i, j; c)a a a 2 a n a i a i 2 a i n a i + ca j a i2 + ca j2 a in + ca jn a i+ a i+2 a i+n a m a m2 a mn P (i, j; c) A A j i i P (i; c)p (i; /c) P (i; /c)p (i; c) I P (i; c) P (i; /c) 2 P (i, j)p (i, j) I P (i, j) P (i, j) 3 P (i, j; c)p (i, j; c) I P (i, j; c) P (i, j; c) 5 P, Q n P (P ) P P Q (P Q) Q P P, P 2,, P n, P n n P n P n P 2 P (P n P n P 2 P ) P P 2 P n P n Q P P Q Q Q I, P QQ P P P I (P Q) Q P 52 A m n A B m P P A B A P, P 2,, P n, P n P P n P n P 2 P B P n P n P 2 P A P A P, P 2,, P n, P n 5 P 42 5 2

A n I I n n C [A, I] n 2n C D D [I, B] B A D I A A rank A n X, Y n [X, Y ] P X Y P [X, Y ] [P X, P Y ] C [A, I] D [I, B] C P, P 2,, P n, P n P P n P n P 2 P [I, B] D P n P n P 2 P C P [A, I] [P A, P ] P A I B P P P P A I P P I P P A A B P A D L P A L I L m 0 rank L r < n 22 n r n r x x 2 x n 0 n > r n r n y 0 Ly 0 A L P A 0 L 0 L Ly Iy y 0 A 53 A, B n AB I A B BA I B n y 0 By 0 0 A0 ABy Iy y 0 5 3

B B IB ABB A BA BB I A n n y 0 Ay 0 Ax b [A, b] P P [A, b] [P A, P b] P Ax P b P Ax b P Ax P b x Ax b P Ax P b x P Ax P b Ax b 5 4

6 2 2 det A n n n 6 N {, 2,, n} N (j, j 2,, j n ) N P n 2 ρ (j, j 2,, j n ) P n l(ρ) {(r, s) r < s n, j r > j s } ρ 3 ρ sgn(ρ) ( ) l(ρ) ρ (sign, signature) 6 n P {()} 2 n 2 P 2 {(, 2), (2, )} {, 2} 2 3 n 3 P 3 {(, 2, 3), (, 3, 2), (2,, 3), (2, 3, ), (3,, 2), (3, 2, )} {, 2, 3} 6 4 n 4 P 4 {(, 2, 3, 4), (, 2, 4, 3), (, 3, 2, 4), (, 3, 4, 2), (, 4, 2, 3), (, 4, 3, 2), (2,, 3, 4), } 24 4 3 2 4! 5 n 2 n 3 n 2 S S {, 2,, n} P n n (n ) (n 2) 2 n! S S 6

62 (3, 4,, 5, 2) 3 (3, ), (3, 2) 4 (4, ), (4, 2) (5, 2) l(3, 4,, 5, 2) 2 + 2 + 0 + 5 2 sgn(3, 4,, 5, 2) ( ) 5 3 (, 2, 3,, n) 0 i < j i j ρ (, 2,, i, j, i +,, j, i, j +,, n) l(ρ) (j i) + (j i) 2(j i) + sgn(ρ) 4 P {()} l() 0, sgn() P 2 P 3 P 2 (, 2) (2, ) l 0 sgn P 3 (, 2, 3) (, 3, 2) (2,, 3) (2, 3, ) (3,, 2) (3, 2, ) l 0 2 2 3 sgn n 62 A [a ij ] n det A ρ(j,j 2,,j n) P n sgn(ρ)a,j a 2,j2 a n,jn A A ρ P n n! n, 2, 3, 4, 2, 6, 24 2 sgn(ρ)a,j a 2,j2 a njn sgn(ρ) ± A n n 6 2

n det A A [a ij ] n det A a a 2 n 2 det A a a 2 a 2 a 22 sgn(, 2)a a 22 + sgn(2, )a 2 a 2 a a 22 a 2 a 2 3 n 3 det A a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 sgn(, 2, 3)a a 22 a 33 + sgn(, 3, 2)a a 23 a 32 + sgn(2,, 3)a 2 a 2 a 33 + sgn(2, 3, )a 2 a 23 a 3 + sgn(3,, 2)a 3 a 2 a 32 + sgn(3, 2, )a 3 a 22 a 3 a a 22 a 33 a a 23 a 32 a 2 a 2 a 33 + a 2 a 23 a 3 + a 3 a 2 a 32 a 3 a 22 a 3 4 n 4 24 n 4 63 n 2 det [ 3 2 5 ] 3 2 3 5 ( 2) 7 5 a b 0 d ad 2 n 3 det 2 0 3 2 2 0 3 2 ( ) ( ) + 2 3 + 2 0 2 3 2 ( ) ( ) 0 2 6 3

3 n 4 a a 2 a 3 0 a 22 a 23 0 0 a 33 2 3 4 2 0 5 4 2 7 2 4 9 3 a a 22 a 33? 0 4 24 n 2 n 3 n 4 63 n A [a ij ] i > j i, j a ij 0 i < j i, j a ij 0 i j i, j a ij 0 6 A [a ij ] n A det A a a 22 a nn det A ρ(j,j 2,,j n ) P n sgn(ρ)a,j a 2,j2 a n,jn sgn(ρ)a,j a 2,j2 a n,jn 0 a,j, a 2,j2,, a n,jn n 0 a n,n j n n n 0 a n,n a n,n j n n j n n j n n (j, j 2,, j n ) (, 2,, n) 0 sgn(, 2,, n) det A a a 22 a nn 6 4

7 det A ρ(j,j 2,,j n) P n sgn(ρ)a,j a 2,j2 a n,jn A [a ij ] n A det A a a 22 a nn 2 (a) c c (b) c c a a 2 a n a 2 a 22 a 2n ca i ca i2 ca in a n a n2 a nn c a a 2 a n a 2 a 22 a 2n a i a i2 a in a n a n2 a nn a a 2 ca j a n a 2 a 22 ca 2j a 2n a n a n2 a nj a nn sgn(ρ)a,j a 2,j2 a n,jn c c c c (a) 0 0 (b) 0 0 0 0 0 7

3 a a 2 a n a j a j2 a jn a i a i2 a in a n a n2 a nn a a 2 a n a i a i2 a in a j a j2 a jn a n a n2 a nn sgn(ρ)a,j a 2,j2 a n,jn i k sgn(ρ)a,j a i,ji a k,ji a i+,ji+ a k,jk a i,jk a k+,jk+ a n,jn sgn(ρ) sgn(j, j 2,, j n ) sgn(j, j i, j k, j i+,, j k, j i, j k+,, j n ) i k i, k n S {j 2,, j n j > j n {l S j > l} + {l S l > j n } + S {l S l > j } + S {l S j n > l} + 2 S + ( {l S j n > l} + {l S l > j } ) (a) (b) - d d d d 0 4 (a) (b) 7 2

a a 2 a n a i a i 2 a i n a i + ca j a i2 + ca j2 a in + ca jn a i+ a i+2 a i+n a n a n2 a nn a a 2 a n a i a i 2 a i n a i a i2 a in a i+ a i+2 a i+n a n a n2 a nn a a 2 a n a i a i 2 a i n a i + b i a i2 + b j2 a in + b jn a i+ a i+2 a i+n a n a n2 a nn a a 2 a n a i a i 2 a i n a i a i2 a in a i+ a i+2 a i+n a n a n2 a nn + a a 2 a n a i a i 2 a i n b i b i2 b in a i+ a i+2 a i+n a n a n2 a nn ρ(j,j 2,,j n ) P n sgn(ρ)a,j (a i,ji + b i,ji ) a n,jn ρ(j,j 2,,j n ) P n sgn(ρ)a,j a i,ji a n,jn + ρ(j,j 2,,j n) P n sgn(ρ)a,j b i,ji a n,jn a a 2 a n a i + ca j a i2 + ca j2 a in + ca jn a j a j2 a jn a n a n2 a nn 7 3

a a 2 a n a i a i2 a in a j a j2 a jn a n a n2 a nn a a 2 a n a i a i2 a in a j a j2 a jn a n a n2 a nn + c a a 2 a n a j a j2 a jn a j a j2 a jn a n a n2 a nn c c 2 2 3 c c 2 2 3 7 2 2 3 2 0 2 0 0 0 0 0 2 2 2 0 0 0 0 7 4

0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 0 4 0 2 0 2 0 0 0 0 0 0 0 0 4 0 2 0 2 0 0 0 0 0 0 0 0 3 0 0 2 0 3 0 0 0 0 0 0 0 0 3 0 0 0 2 3 0 0 0 2 2 0 0 0 0 0 0 3 0 0 0 2 3 0 0 0 0 6 3 a b b b b a b b b b a b b b b a, x x 2 x n x 2 x 2 2 x n 2 x 3 x 2 3 x n 3 x n x 2 n x n n 7 5

8 8 A n det A det A t det A sgn(ρ)a,j a 2,j2 a n,jn ρ(j,j 2,,j n ) P n A t [b ij ] b ij a ji det A t sgn(ρ)b,j b 2,j2 b n,jn ρ(j,j 2,,j n ) P n sgn(ρ)a j,a j2,2 a jn,n ρ(j,j 2,,j n) P n sgn(ρ )a,k a 2,k2 a n,kn ρ (k,k 2,,k n) P n det A ρ (j, j 2,, j n ) a j,a j2,2 a jn,n a,k a 2,k2 a n,kn ρ (k, k 2,, k n ) l(ρ) l(ρ ) sgn(ρ) sgn(ρ ) ρ (j, j 2, j 3, j 4, j 5 ) (2, 5,, 4, 3) a j,a j2,2 a j5,5 a 2 a 52 a 3 a 44 a 35 a 3 a 2 a 35 a 44 a 52 ρ (k, k 2, k 3, k 4 k 5 ) (3,, 5, 4, 2) l(ρ) l(ρ ) 5 sgn(ρ) sgn(ρ ) rho (, 2, 3, 4, 5) (2, 5,, 4, 3) rho (, 2, 3, 4, 5) (3,, 5, 4, 2) A B A t 8

a a,i + ca,j a,j a,n a 2 a 2,i + ca 2,j a 2,j a 2,n a n a n,i + ca n,j a n,j a n,n a a 2 a n a i + ca ij a 2i + ca 2j a ni + ca nj a j a 2j a nj a n a 2n a nn a a 2 a n a i a 2i a ni a a,i a,j a,n a 2 a 2,i a 2,j a 2,n a j a 2j a nj a n a n,i a n,j a n,n a n a 2n a nn A B 2 2 det AB det A det B 4 (2) n 82 A B n det AB det A det B c c P (i; c) 2 2 P (i, j) 3 P (i, j; c) det P (i; c) c, det P (i, j), det P (i, j; c) 8 2

P (i; c)a c A P (i; c) A P (i, j)a A P (i, j) A P (i, j; c)a A P (i, j; c) A P, P,, P m A B P P m P 2 P B P A P m P 2 P A P m P 2 P A P A A B B P A P A B I 0 I I B P A A P A A 53 B I 83 n A det A 0 53 A P m P 2 P 0 A P m P 2 P P m P 2 P 0 0 B P A B I B B 0 B 0 B P A P A P 0 A 0 82 A 53 AB P B P B A B A A P P A 0 P AB 0 AB P P AB P P AB 0 A B 8 3

84 A det A det A AA I det I det AA det A det A 82 0 A n B AB BA I det A 0 53 det A det B det AB det I A, B n AB I A B BA I AB I det I det AB det A det B det A det B 0 A B BA BABB BIB BB I B A A B 8 Vandermonde x x 2 x n x 2 x 2 2 x n 2 x 3 x 2 3 x3 n i x j ) i>j(x x n x 2 n xn n n x 2 x 2 2 x n 2 2 x 3 x 2 3 x n 2 2 x 4 x 2 4 x4 n 2 i x j ) i>j(x x n x 2 n xn n 2 8 4 n n j ij+ n n j2 ij+ (x i x j ) (x i x j )

x x 2 x n x 2 x 2 2 x n 2 x n x 2 n xn n x x 2 x n 0 x 2 x x 2 2 x 2 x2 n x n 0 x n x x 2 n x 2 xn n x n x 2 x x 2 2 x 2 x2 n x n x 3 x x 2 3 x 2 x3 n x n x n x x 2 n x 2 xn n x n x 2 x x 2 2 x 2 x2 n x x n 2 2 x 3 x x 2 3 x 2 x3 n x x n 2 3 x n x x 2 n x 2 xn n x xn n 2 x 2 x x 2 2 x x 2 x2 n x x n 2 2 x 3 x x 2 3 x x 3 x3 n x x n 2 3 x n x x 2 n x x n xn n x xn n 2 (x 2 x )(x 3 x ) (x n x ) n n j ij+ (x i x j ) x 2 x 2 2 x n 2 2 x 3 x 2 3 x2 n 2 x 4 x 2 4 x4 n 2 x n x 2 n xn n 2 8 5

9 Cramer A a 0 B a B 9 n A i j n M i,j C i,j ( ) i+j M i,j A (i, j) (k, l) C l,k C t à A à [ã i,j] ã i,j ( ) i+j M j,i M j,i A j i n 9 A 2 4 3 9 4 6 A à à ( ) + M ( ) +2 M 2 ( ) +3 M 3 ( ) 2+ M 2 ( ) 2+2 M 22 ( ) 2+3 M 32 ( ) 3+ M 3 ( ) 3+2 M 23 ( ) 3+3 M 33 3 9 + 4 6 9 6 3 + 4 2 4 4 6 4 + 6 2 4 2 4 + 3 9 4 9 2 + 3 ( ) i+j + ( ) i+i + à + M M 2 + M 3 M 4, M 2 + M 22 M 32 + M 4,2 + M 3 M 23 + M 33 M 4,3 M 4 + M 24 M 34 + M 4,4 9

[ ] a b A c d [ d b à c a ] A ad bc 0 à det A A Aà ÃA det A I n det A 0 0 δ i,j (Kronecker) i j i j 0 i, j I [δ i,j ] (i, j) δ i,j 9 A [a i,j ] n à [ã i,j] A n () i, k a i,j ã j,k δ i,k det A j n (2) i, k ã i,j a j,k δ i,k det A j (3) Aà ÃA det A I (4) det A 0 A à det A (3) (4) A 2 4 3 9 4 6 Vandermonde (4 2)(4 3)(3 2) 2 0 A A à det A 9 2

3 9 + 4 6 9 2 6 3 + 4 2 +2 6 +6 7 +2 5 + 2 + 2 4 4 6 4 + 6 2 4 2 4 + 3 9 4 9 2 + 3 (3) AÃ i, k () ÃA i, k (2) () (2) (3) () i k ã j,k ã j,i ( ) i+j M i,j (i, j) n n det A a i,j ã j,i ( ) i+j a i,j M i,j j j i a i,j (i, j) i (2) i k n n det A ã i,j a j,i ( ) i+j a j,i M j,i j j i a i a 2i a ni a i 0 0 0 a 2i + + a a i a n a 2 a 2i a 2n det A a n a ni a nn 0 0 0 a ni 9 3

a 0 a n n a j a ji a jn j a n 0 a nn a ji a j a jn 0 a a n n j 0 a n 0 a nn a a n n ( ) i+j a ji omit row i and column j j a n a nn n ( ) i+j a j,i M j,i j i k i (i, i) a j,k a j,i n ã i,j a j,k j i k 0 n ã i,j a j,k δ i,k det A j () 92 2 3 4 2 0 5 0 4 2 7 2 4 9 3 9 4

0 9 Cramer 92 (Cramer ) A [a i,j ] n Ax b A i b A i x i A i, i, 2,, n A x A b 9 x i A A A A A i A x Ãb A n ã ij b j j n ã ij b j j a 0 a n n 0 b i 0 j a n 0 a nn a b a n a i b i a in a n b n a nn 93 2 4 3 9 4 6 x y z Cramer 2 x 5 2 4 8 3 9 4, y 5 4 8 9 2 2 2, z 2 5 3 8 2 2 0 4 6 0 6 4 0 5 8 0 9 5

0 0 x, x 2,, x n y, y 2,, y n n f(x ) y, f(x 2 ) y 2,, f(x n ) y n n n f(t) c 0 + c t + + c n x n c 0 + c x + + c n x n y c 0 + c x 2 + + c n x2 n y 2 c 0 + c x n + + c n xn n y n c 0, c,, c n x x 2 x n x 2 x 2 2 x n c 0 y 2 A x 3 x 2 3 x3 n c y 2, x, b x n x 2 n xn n c n y n Ax b A Vandermonde x, x 2,, x n A n n j ij+ (x i x j ) 0 A b Ax b x n f(t) 0 f(2) 5, f(3) 8, f(4) 0 f(t) c 0 + c t + c 2 t 2 2 4 3 9 4 6 c 0 c c 2 5 8 0 c 0 4, c /2, c 2 /2 f(t) 4 + 2 t 2 t2 0

02 A i,j l i m j B j,k m j n l A A A 2 A q A 2 A 22 A 2q A p A p2 A pq AB C, B C C 2 C r C 2 C 22 C 2r C p C p2 C pr C il l i n l p C ik A ij B jk j B B 2 B r B 2 B 22 B 2r B q B q2 B qr a x + a 2 x 2 + + a n x n b a 2 x + a 22 x 2 + + a 2n x n b 2 a m x + a m2 x 2 + + a mn x n b m A a a 2 a n a 2 a 22 a 2n a m a m2 a mn, x x x 2 x n, b b b 2 b m Ax b 2 x 0 Ax 0 b n x Ax b A(x x 0 ) Ax Ax 0 b b 0 0 2

y x x 0 x x 0 + y y Ay 0 n A Ay 0 y x x 0 + y Ax b Ax A(x 0 + y) Ax 0 + Ay b + 0 b Ax b Ax 0 Ax b x 0 x x 0 + y Ax 3 2 5 x x 2 x 3 4 7 x x 2 x 3 t 2 2 + 3 2 2 0 t 2 2 Ax 0 3 2 2 0 3 (a) P (i; c) P (i, j) P (i, j; c) (b) (c) 0 3

4 A n (a) A (b) A I (c) A n (d) Ax 0 (e) Ax b b (f) A (g) det A 0 A [a i,j ] B [b i,j ] n det A sgn(ρ)a,j a 2,j2 a n,jn ρ(j,j 2,,j n ) P n sgn(ρ)a j,a j2,2 a jn,n ρ(j,j 2,,j n) P n P n {, 2,, n} sgn(ρ) ( ) l(ρ) l(ρ) ρ A A 2 A det A a a 22 a nn 3 (a) c c (b) 2 (c) (a) c c (b) 2 0 4

(c) 4 0 0 0 5 A [a, a 2,, a i,, a n ] a i a i + a i A a, a 2,, a i + a i,, a n a, a 2,, a i,, a n + a, a 2,, a i,, a n a a 2 A a n a i a i + a i a a a A a i + a i a i + a i a n a n a n 6 det A t det A 7 det AB det A det B 8 A [a i,j ] n à [ã i,j] A n (a) i, k a i,j ã j,k δ i,k det A j n (b) i, k ã i,j a j,k δ i,k det A j (c) Aà ÃA det A I (d) det A 0 A à det A ã i,j ( ) i+j M j,i M j,i A j i n 0 5

02 a b b b b a b b b b a b b b b a (a + (n )b)(a b) n a + (n )b n a + (n )b a b b b b a b b b b a b b b b a a + (n )b b b b a + (n )b a b b a + (n )b b a b a + (n )b b b a a + (n )b b b b 0 a b 0 0 0 0 a b 0 0 0 0 a b (a + (n )b)(a b) n 0 6

m k n b < k < n a 2 n m 3 n m 4 n m 43 k 7 b a k a b a (k ) (n ) b a k, n, b a a b H [h ij ] n m h ij i j 0 HH t n a b J n HH t (a b)i + bj (a + (n )b)(a b) n a b HH t H n n m n > m H P H 0 P HH t n m HH t H JH kj kj H t J HJ aj aj JH t aj(h t ) J H t H H t HH t (H t ) H t ((a b)i + bj)(h t ) (a b)h t (H t ) + bh t J (a b)i + bh t J(H t ) (a b)i + bkj(h t ) (a b)i + bk a J H nk na a k b