Đang tải... (xem toàn văn)
toan cao cap b2
Chu . o . ng 1. MA TR ˆ A . N-D - I . NH TH ´ U . C (8+4) I. Ma trˆa . n * Cho m, n nguyˆen du . o . ng. Ta go . i ma trˆa . nc˜o . m × n l`a mˆo . tba ’ ng sˆo ´ gˆo ` m m × n sˆo ´ thu . . cd¯u . o . . cviˆe ´ t th`anh m h`ang, n cˆo . t c´o da . ng nhu . sau: (a i,j ) m×n = a 1,1 a 1,2 . a 1,n a 2,1 a 2,2 . a 2,n . . . . a m,1 a m,2 . a m,n trong d¯´o c´ac sˆo ´ thu . . c a i,j ,i= 1,m,j = 1,n d¯ u . o . . cgo . il`ac´ac phˆa ` ntu . ’ cu ’ a ma trˆa . n,chı ’ sˆo ´ i chı ’ h`ang v`a chı ’ sˆo ´ j chı ’ cˆo . tcu ’ a phˆa ` ntu . ’ ma trˆa . n. * Ma trˆa . nc˜o . 1 × n d¯ u . o . . cgo . il`ama trˆa . n h`ang, ma trˆa . nc˜o . m × 1d¯u . o . . cgo . il`ama trˆa . ncˆo . t, ma trˆa . nc˜o . n × n d¯ u . o . . cgo . il`ama trˆa . n vuˆong cˆa ´ p n. *Trˆen ma trˆa . n vuˆong cˆa ´ p n,d¯u . `o . ng ch´eo gˆo ` m c´ac phˆa ` ntu . ’ a i,i ,i= 1,n d¯ u . o . . cgo . il`ad¯ u . `o . ng ch´eo ch´ınh,d¯u . `o . ng ch´eo gˆo ` m c´ac phˆa ` ntu . ’ a i,n+1−i ,i= 1,n d¯ u . o . . cgo . il`ad¯ u . `o . ng ch´eo phu . cu ’ a ma trˆa . n. * Ma trˆa . n vuˆong cˆa ´ p n c´o c´ac phˆa ` ntu . ’ n˘a ` m ngo`ai d¯u . `o . ng ch´eo ch´ınh d¯ˆe ` ub˘a ` ng 0, ngh˜ıa l`a: a i,j =0,∀i = j d¯ u . o . . cgo . il`ama trˆa . n ch´eo. * Ma trˆa . n ch´eo c´o a i,i =1,i = 1,n d¯ u . o . . cgo . il`ama trˆa . nd¯o . nvi . cˆa ´ p n,k´yhiˆe . u I n . * Ma trˆa . nc˜o . m× n c´o a i,j =0,∀i, j : i>j d¯ u . o . . cgo . il`ama trˆa . nbˆa . c thang. * Ma trˆa . nc˜o . m × n c´o c´ac phˆa ` ntu . ’ d¯ ˆe ` ub˘a ` ng 0 d¯u . o . . cgo . il`ama trˆa . n khˆong,k´y hiˆe . u0 m,n . *Tago . i ma trˆa . n chuyˆe ’ nvi . A T =(a j,i ) n×m = a 1,1 a 2,1 . a m,1 a 1,2 a 2,2 . a m,2 . . . . a 1,n a 2,n . a m,n Typeset by A M S-T E X 2 cu ’ a ma trˆa . n A =(a i,j ) m×n = a 1,1 a 1,2 . a 1,n a 2,1 a 2,2 . a 2,n . . . . a m,1 a m,2 . a m,n l`a ma trˆa . n c´o d¯u . o . . ct`u . A b˘a ` ng c´ach chuyˆe ’ n h`ang th`anh cˆo . t, cˆo . t th`anh h`ang. * Hai ma trˆa . nc`ung c˜o . (a i,j ) m×n v`a (b i,j ) m×n d¯ u . o . . cgo . il`ab˘a ` ng nhau nˆe ´ u c´ac phˆa ` n tu . ’ o . ’ t`u . ng vi . tr´ıd¯ˆe ` ub˘a ` ng nhau: a i,j = b i,j ,∀i = 1,m,∀j = 1,n. +Tˆo ’ ng (hiˆe . u) cu ’ a hai ma trˆa . n c`ung c˜o . m × n l`a mˆo . t ma trˆa . nc˜o . m × n, trong d¯´o phˆa ` ntu . ’ cu ’ a ma trˆa . ntˆo ’ ng (hiˆe . u) l`a tˆo ’ ng (hiˆe . u) c´ac phˆa ` ntu . ’ o . ’ vi . tr´ıtu . o . ng ´u . ng: (c i,j ) m×n =(a i,j ) m×n ± (b i,j ) m×n v´o . i c i,j = a i,j ± b i,j ,∀i = 1,m,∀j = 1,n. +T´ıch vˆo hu . ´o . ng cu ’ asˆo ´ thu . . c α v´o . i ma trˆa . nc˜o . m× n l`a ma trˆa . nc˜o . m× n, trong d¯´o mˆo ˜ i phˆa ` ntu . ’ l`a t´ıch cu ’ a α v´o . i phˆa ` ntu . ’ o . ’ vi . tr´ıtu . o . ng ´u . ng cu ’ a ma trˆa . n ban d¯ˆa ` u: (c i,j ) m×n = α.(a i,j ) m×n v´o . i c i,j = α.b i,j ,∀i = 1,m,∀j = 1,n. +T´ıch vˆo hu . ´o . ng c´o t´ınh phˆan bˆo ´ v´o . i ph´ep cˆo . ng c´ac ma trˆa . n: α.(A+B)=α.A+α.B, v´o . i ph´ep cˆo . ng c´ac hˆe . sˆo ´ :(α + β).A = α.A + β.B, c´o t´ınh kˆe ´ tho . . p: α.(β · A)=(α.β) · A. +T´ıch cu ’ a hai ma trˆa . n A =(a i,j ) m×n v`a B =(b j,k ) n×q l`a ma trˆa . n C = A × B =(c i,k ) m×q , v´o . i c i,k = n j=1 a i,j b j,k ,∀i = 1,m,∀k = 1,q. V´ıdu . . 132 247 356 × 13 1 −1 32 = 1.1+3.1+2.31.3− 3.1+2.2 2.1+4.1+7.32.3− 4.1+7.2 3.1+5.1+6.33.3− 5.1+6.2 = 10 4 27 16 26 16 3 +Ph´ep nhˆan hai ma trˆa . n c´o t´ınh kˆe ´ tho . . p: A × (B × C)=(A × B) × C, t´ınh phˆan phˆo ´ id¯ˆo ´ iv´o . i ph´ep cˆo . ng: A × (B + C)=A × B + A × C;(A + B) × C = A × C + B × C. Ngo`ai ra, nˆe ´ u A c´o c˜o . m × n,th`ı A × I n = I m × A = A. II. D - i . nh th´u . c * Cho E = {1, 2, 3, .,n}.Tago . i ho´an vi . cu ’ atˆa . p E l`a mˆo . t song ´anh f : E → E, k´yhiˆe . u f : 12 . n f(1) f(2) . f(n) hay (f(1),f(2), .,f(n)) (c´o tˆa ´ tca ’ n! ho´an vi . kh´ac nhau). V´ıdu . . Cho E = {1, 2, 3}. ´ Anh xa . f : E → E x´ac d¯i . nh bo . ’ i: f(1) = 1,f(2) = 3,f(3) = 2 l`a mˆo . t ho´an vi . cu ’ a E,k´yhiˆe . ul`a 123 132 ho˘a . c (1, 3, 2). * Cho mˆo . t ho´an vi . f : 12 . n f(1) f(2) . f(n) ta th`anh lˆa . p c´ac c˘a . pth´u . tu . . (f(i),f(j)),∀i = j, s˜ec´oC 2 n c˘a . pth´u . tu . . nhu . thˆe ´ ;mˆo . tc˘a . p(f(i),f(j)) d¯u . o . . cgo . il`anghi . ch thˆe ´ nˆe ´ u (i − j)(f(i) − f(j)) < 0. Go . i N(f) l`a sˆo ´ c´ac nghi . ch thˆe ´ cu ’ a ho´an vi . f (c´o trong C 2 n c˘a . pth´u . tu . . trˆen). V´ıdu . . T`ım sˆo ´ nghi . ch thˆe ´ cu ’ a ho´an vi . f : 12345 32154 . 4 T`u . ho´an vi . n`ay, ta c´o c´ac c˘a . pth´u . tu . . (3, 2), (3, 1), (3, 5), (3, 4), (2, 1), (2, 5), (2, 4), (1, 5), (1, 4), (5, 4), trong d¯´o ta c´o c´ac nghi . ch thˆe ´ : (3, 2), (3, 1), (2, 1), (5, 4), suy ra N(f)=4 * Cho ma trˆa . n(A) n,n . D - i . nh th´u . ccu ’ a A l`a mˆo . tsˆo ´ thu . . c, k´yhiˆe . u v`a x´ac d¯i . nh nhu . sau: det(A)= f∈S n (−1) N(f ) a 1,f(1) a 2,f(2) .a n,f(n) trong d¯´o S n l`a tˆa . ptˆa ´ tca ’ n! ho`an vi . cu ’ a n phˆa ` ntu . ’ {1, 2, .,n}.Nhu . vˆa . y, d¯i . nh th´u . ccu ’ a ma trˆa . n A l`a mˆo . tsˆo ´ : +b˘a ` ng tˆo ’ ng d¯a . isˆo ´ cu ’ a n!ha . ng tu . ’ da . ng a 1,f(1) a 2,f(2) .a n,f(n) +mˆo ˜ iha . ng tu . ’ l`a t´ıch cu ’ a n phˆa ` ntu . ’ a i,j m`a mˆo ˜ i h`ang, mˆo ˜ icˆo . t pha ’ ic´omˆo . t v`a chı ’ mˆo . t phˆa ` ntu . ’ tham gia v`ao t´ıch d¯´o. +dˆa ´ ucu ’ amˆo ˜ iha . ng tu . ’ phu . thuˆo . c v`ao sˆo ´ nghi . ch thˆe ´ cu ’ a ho´an vi . tu . o . ng ´u . ng. *Tago . i d¯ i . nh th´u . ccˆa ´ p2l`a gi´a tri . t´ınh d¯u . o . . ct`u . ba ’ ng 2 h`ang, 2 cˆo . tnhu . sau: a 1,1 a 1,2 a 2,1 a 2,2 = a 1,1 a 2,2 − a 2,1 a 1,2 *Tago . i d¯ i . nh th´u . ccˆa ´ p3l`a gi´a tri . t´ınh d¯u . o . . ct`u . ba ’ ng 3 h`ang, 3 cˆo . tnhu . sau: 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 a 2,2 a 3,3 + a 2,1 a 3,2 a 1,3 + a 3,1 a 1,2 a 2,3 − a 3,1 a 2,2 a 1,3 − a 2,1 a 1,2 a 3,3 − a 1,1 a 3,2 a 2,3 +D - ˆe ’ t´ınh nhanh d¯i . nh th´u . ccˆa ´ p 3, ta viˆe ´ tcˆo . tth´u . nhˆa ´ t v`a th´u . hai tiˆe ´ p theo v`ao bˆen pha ’ iba ’ ng n´oi trˆen: a 1,1 a 1,2 a 1,3 a 1,1 a 1,2 a 2,1 a 2,2 a 2,3 a 2,1 a 2,2 a 3,1 a 3,2 a 3,3 a 3,1 a 3,2 th`ı 3 phˆa ` ntu . ’ lˆa ´ ydˆa ´ ucˆo . ng l`a t´ıch c´ac phˆa ` ntu . ’ n˘a ` m trˆen c´ac d¯u . `o . ng ch´eo song song v´o . id¯u . `o . ng ch´eo ch´ınh, ba phˆa ` ntu . ’ lˆa ´ ydˆa ´ utr`u . l`a t´ıch c´ac phˆa ` ntu . ’ n˘a ` m trˆen c´ac d¯ u . `o . ng ch´eo song song v´o . id¯u . `o . ng ch´eo phu . (quy t˘a ´ c Serrhus) 5 *Tago . i d¯ i . nh th´u . ccˆa ´ p n l`a gi´a tri . t´ınh d¯u . o . . ct`u . ba ’ ng: a 1,1 a 1,2 . a 1,n a 2,1 a 2,2 . a 2,n . . . . a n,1 a n,2 . a n,n = a 1,1 D 1 − a 2,1 D 2 +···+(−1) n+1 a n,1 D n trong d¯´o D k l`a d¯i . nh th´u . ccˆa ´ p n − 1thud¯u . o . . ct`u . ba ’ ng d¯˜a cho b˘a ` ng c´ach bo ’ cˆo . t th´u . nhˆa ´ t v`a h`ang th´u . k, k = 1,n. V´ıdu . . 1452 0331 2040 0021 =1. 331 040 021 − 0. 452 040 021 +2. 452 331 021 − 0. 452 331 040 =14 +D - i . nh th´u . c khˆong thay d¯ˆo ’ inˆe ´ u ta d¯ˆo ’ i h`ang th`anh cˆo . t +D - i . nh th´u . cd¯ˆo ’ idˆa ´ unˆe ´ u ta d¯ˆo ’ ichˆo ˜ hai h`ang (ho˘a . c hai cˆo . t) v´o . i nhau +D - i . nh th´u . c c´o hai h`ang (ho˘a . c hai cˆo . t) ty ’ lˆe . v´o . i nhau nhau th`ı b˘a ` ng 0 +Th`u . asˆo ´ chung cu ’ amˆo . t h`ang hay cˆo . tc´othˆe ’ d¯ u . a ra ngo`ai dˆa ´ ucu ’ ad¯i . nh th´u . c +D - i . nh th´u . c khˆong thay d¯ˆo ’ inˆe ´ u ta d¯ˆo ` ng th`o . icˆo . ng v`ao c´ac phˆa ` ntu . ’ cu ’ amˆo . t h`ang (hay mˆo . tcˆo . t) n`ao d¯´o c´ac phˆa ` ntu . ’ cu ’ amˆo . t h`ang (hay mˆo . tcˆo . t) kh´ac nhˆan v´o . ic`ung mˆo . tsˆo ´ . V´ıdu . . Gia ’ iphu . o . ng tr`ınh: 11 1 . 1 11− x 1 . 1 112− x . 1 . . . . . 11 1 . n− x =0. D - i . nh th´u . co . ’ vˆe ´ tr´ai cu ’ aphu . o . ng tr`ınh l`a d¯a th´u . cbˆa . c n nˆen c´o khˆong qu´a n nghiˆe . m kh´ac nhau. Thay x =0,x =1,x =2, .,x = n − 1 v`ao d¯i . nh th´u . c, ta luˆon c´o hai h`ang v´o . i c´ac phˆa ` ntu . ’ b˘a ` ng 1, nˆen d¯i . nh th´u . cb˘a ` ng 0. Vˆa . yphu . o . ng tr`ınh c´o n nghiˆe . m x =0,x=1,x=2, .,x = n − 1. *D - i . nh th´u . ccu ’ a ma trˆa . n vuˆong A =(a i,j ) n×n ,k´yhiˆe . u det(A) l`a d¯i . nh th´u . ccˆa ´ p n cu ’ aba ’ ng a 1,1 a 1,2 . a 1,n a 2,1 a 2,2 . a 2,n . . . . a n,1 a n,2 . a n,n v`a c´o t´ınh chˆa ´ t: + det(αA)=α n . det(A) + det(A × B) = det(A). det(B) III. Ma trˆa . n nghi . ch d¯a ’ o 6 * Ma trˆa . n A =(a i,j ) n×n d¯ u . o . . cgo . il`ama trˆa . n kha ’ nghi . ch nˆe ´ utˆo ` nta . i ma trˆa . n A −1 sao cho: A × A −1 = A −1 × A = I n . Khi d¯´o, ma trˆa . n A −1 d¯ u . o . . cgo . il`ama trˆa . n nghi . ch d¯a ’ o cu ’ a A. + Ma trˆa . n A kha ’ nghi . ch khi v`a chı ’ khi det A =0. * Cho A =(a i,j ) m×n .Mˆo . t d¯ i . nh th´u . c con cˆa ´ p k (1 ≤ k ≤ n) cu ’ a A l`a mˆo . td¯i . nh th´u . cta . o th`anh t`u . ma trˆa . n A b˘a ` ng c´ach bo ’ d¯ i m − k h`ang v`a n − k cˆo . t. * Cho ma trˆa . n vuˆong cˆa ´ p n kha ’ nghi . ch A = a 1,1 a 1,2 . a 1,n a 2,1 a 2,2 . a 2,n . . . . a n,1 a n,2 . a n,n Phˆa ` nb`ud¯a . isˆo ´ cu ’ a phˆa ` ntu . ’ a i,j ,l`asˆo ´ A i,j =(−1) i+j D i,j trong d¯´o D i,j l`a d¯i . nh th´u . ccˆa ´ p n− 1cu ’ aba ’ ng thu d¯u . o . . ct`u . ma trˆa . n A b˘a ` ng c´ach ga . ch bo ’ h`ang th´u . i v`a cˆo . tth´u . j. + Cho A l`a ma trˆa . n vuˆong kha ’ nghi . ch cˆa ´ p n v`a ∆ = det A = 0. Khi d¯´o ma trˆa . n nghi . ch d¯a ’ ocu ’ a A d¯ u . o . . c x´ac d¯i . nh mˆo . t c´ach duy nhˆa ´ tbo . ’ i: A −1 = 1 ∆ A i,j T = 1 ∆ A 1,1 A 2,1 . D n,1 A 1,2 A 2,2 . D n,2 . . . . A 1,n A 2,n . D n,n V´ıdu . . Ma trˆa . n nghi . ch d¯a ’ ocu ’ a A = 1 −11 211 112 l`a: A −1 = 1 5 13−2 −31 1 1 −23 v`ı: ∆ = det A = (1)(1)(2)+(2)(1)(1)+(1)(−1)(1)−(1)(1)(1)−(2)(−1)(2)−(1)(1)(1) = 5 =0 7 v`a: A 1,1 =(−1) 1+1 11 12 =1; A 1,2 =(−1) 1+2 21 12 = −3; A 1,3 =(−1) 1+3 21 11 =1; A 2,1 =(−1) 2+1 −11 12 =3; A 2,2 =(−1) 2+2 11 12 =1; A 2,3 =(−1) 2+3 1 −1 11 = −2 A 3,1 =(−1) 3+1 −11 11 = −2; A 3,2 =(−1) 3+2 11 21 =1; A 3,3 =(−1) 3+3 1 −1 21 =3 +T´ınh chˆa ´ t: − Cho A kha ’ d¯ a ’ ov`ak = 0, th`ı: (kA) −1 = 1 k A −1 − Cho A, B c`ung cˆa ´ p v`a kha ’ d¯ a ’ o, th`ı: (A × B) −1 = B −1 × A −1 − Cho A kha ’ d¯ a ’ oth`ıA −1 c˜ung kha ’ d¯ a ’ ov`a A −1 −1 = A Phˆa ` n I.4: Ha . ng cu ’ a ma trˆa . n *Tago . i ha . ng cu ’ a ma trˆa . n A =(a i,j ) m×n ,k´yhiˆe . u r(A) l`a cˆa ´ p cao nhˆa ´ tcu ’ a c´ac d¯ i . nh th´u . c con kh´ac 0 cu ’ a A. +Ha . ng cu ’ a ma trˆa . n0 m×n l`a 0, ha . ng cu ’ a ma trˆa . n A =(a)v´o . i a = 0 l`a 1. +Ha . ng cu ’ a ma trˆa . n khˆong thay d¯ˆo ’ i qua c´ac ph´ep biˆe ´ nd¯ˆo ’ iso . cˆa ´ p sau d¯ˆay: a. D - ˆo ’ ichˆo ˜ hai h`ang ho˘a . c hai cˆo . t cho nhau; b. Nhˆan mˆo . t h`ang (hay mˆo . tcˆo . t) v´o . imˆo . tsˆo ´ kh´ac 0; c. Cˆo . ng v`ao mˆo . t h`ang (hay mˆo . tcˆo . t) v´o . imˆo . t h`ang (hay mˆo . tcˆo . t) kh´ac nhˆan v´o . imˆo . tsˆo ´ . D - ˆe ’ t`ım ha . ng cu ’ a ma trˆa . n A mtimesn , c´o thˆe ’ d`ung c´ac phu . o . ng ph´ap sau: + Phu . o . ng ph´ap theo d¯i . nh ngh˜ıa: t´ınh c´ac d¯i . nh th´u . c con t`u . cˆa ´ p 2 tro . ’ lˆen. Gia ’ su . ’ ma trˆa . nc´o1d¯i . nh th´u . c con cˆa ´ p r kh´ac 0, t´ınh tiˆe ´ p c´ac d¯i . nh th´u . ccˆa ´ p r +1, nˆe ´ u tˆa ´ tca ’ d¯ ˆe ` ub˘a ` ng 0 th`ı kˆe ´ t luˆa . nha . ng ma trˆa . nl`ar,nˆe ´ uc´od¯i . nh th´u . ccˆa ´ p r + 1 kh´ac 0 th`ı t´ınh tiˆe ´ p c´ac d¯i . nh th´u . ccˆa ´ p r +2,c´u . nhu . thˆe ´ d¯ ˆe ´ nd¯i . nh th´u . ccˆa ´ pl´o . n nhˆa ´ t V´ıdu . . T`ım ha . ng cu ’ a ma trˆa . n A = 1235 3249 1014 Ta c´o d¯i . nh th´u . c con cˆa ´ p2: 12 32 = −4 = 0, v`a c´ac d¯i . nh th´u . ccˆa ´ p3: 123 324 101 =0; 125 329 104 =0; 135 349 114 =0; 235 249 014 =0 suy ra r(A)=2 8 + Phu . o . ng ph´ap d`ung ph´ep biˆe ´ nd¯ˆo ’ iso . cˆa ´ p: biˆe ´ nd¯ˆo ’ i ma trˆa . nvˆe ` da . ng bˆa . c thang B = b 1,1 b 1,2 . b 1,r . b 1,n 0 b 2,2 . b 2,r . b 2,n . . . . . . 00 . b r,r . b r,n 00 . 0 . 0 00 . 0 . 0 v´o . i b i,j =0,∀i>jhay i>rv`a b ii =0,i = 1,r th`ı r(A)=r(B)=r. V´ıdu . . T`ım ha . ng ma trˆa . n A = 13205 269712 −2 −524 5 148420 A h2−2h1;h3+2h1;h4−h1 −→ 1320 5 0057 2 016415 016415 h4−h3;h2↔h3 −→ 1320 5 016415 0057 2 0000 0 suy ra r(A)=3 + Ngo`ai ra, c´o thˆe ’ t`ım ma trˆa . n nghi . ch d¯a ’ o qua c´ac ph´ep biˆe ´ nd¯ˆo ’ iso . cˆa ´ p: lˆa . p ma trˆa . n khˆo ´ i A|E (E c`ung c˜o . v´o . i A, thu . . chiˆe . n c´ac ph´ep biˆe ´ nd¯ˆo ’ iso . cˆa ´ p CHI ’ TR ˆ EN H ` ANG, nˆe ´ ud¯u . ad¯u . o . . cvˆe ` da . ng E|B th`ı B l`a nghi . ch d¯a ’ ocu ’ a A. V´ıdu . . A|E = 1 −11| 100 211| 010 112| 001 h2−2h1,h3−h1 −→ 1 −11| 100 03−1 |−210 02 1|−101 h1−h3,h2+h3 −→ 1 −30| 20−1 050|−31 1 021|−10 1 h2( 1 5 ) −→ 1 −30| 20−1 010|−3/51/51/5 021|−101 h1+3h2,h3−2h2 −→ 100| 1/53/5 −2/5 010|−3/51/51/5 011| 1/5 −2/53/5 thu d¯u . o . . ckˆe ´ t qua ’ nhu . c˜u. B ` AI T ˆ A . P 1.1. Khˆong t´ınh, ch´u . ng minh c´ac d¯i . nh th´u . c sau chia hˆe ´ t cho 17: 204 527 255 ; 323 20 9 1 55 2 5 1.2. Ch´u . ng minh c´ac d¯˘a ’ ng th´u . c sau d¯ˆay (khˆong t´ınh d¯i . nh th´u . cb˘a ` ng d¯i . nh ngh˜ıa): 9 a. 0 xyz x 0 zy yz0 x xyz0 = 01 1 1 10z 2 y 2 1 z 2 0 x 2 1 y 2 x 2 0 v´o . i xyz =0 b. 1 xyz 1 yzx 1 zxy =(x − y)(y − z)(z − x) c. 111 xyz x 3 y 3 z 3 =(x + y + z)(x − y)(y − z)(z − x) 1.3. T`ım x sao cho: a. 33− x −x 273 x +1 3x − 7 x =0 b. xx+1 x +2 x +3 x +4 x +5 x +6 x +7 x +8 =0 c. 1 xx 2 31 x 45 1 < 0 d. x 12 1 x 3 x +1 2 −4 > 0 1.4. T´ınh c´ac d¯i . nh th´u . c sau: 0110 0011 1001 1100 ; 1 xxx 1 a 00 10b 0 100c ; x 1111 1 x 111 11x 11 111x 1 1111x ; a + xx x ab+ xx xxc+ x ; x 2 +1 xy xz xy y 2 +1 yz xz yz z 2 +1 ; 1 xx 2 x 3 x 3 x 2 x 1 12x 3x 2 4x 3 4x 3 3x 2 2x 1 ; 0 xyz x 0 zy yz0 x xyz0 ; 2 x 1 x 1 x 2 x 21xx xx21 ; x 0 y 0 0 z 0 t y 0 z 0 0 t 0 x ; axx−x −x x 2aa 00 xa2a 00 −x 002aa −x 00 a 2a ; 12 3 . n 21 2 . n− 1 32 1 . n− 2 . . . . . nn− 1 n− 2 . 1 ; xaa .a axa .a aax .a . . . . . aaaax ; cos(x 1 − y 1 ) cos(x 1 − y 2 ) . cos(x 1 − y n ) cos(x 2 − y 1 ) cos(x 2 − y 2 ) . cos(x 2 − y n ) . . . . cos(x n − y 1 ) cos(x n − y 2 ) . cos(x n − y n ) ; 011 . 11 10x . x x 1 x 0 . x x . . . . . . 1 xx .0 x 1 xx .x0 ; 10 1+x 1 y 1 1+x 1 y 2 . 1+x 1 y n 1+x 2 y 1 1+x 2 y 2 . 1+x 2 y n . . . . 1+x n y 1 1+x n y 2 . 1+x n y n ; a 1 −a 2 0 . 00 0 a 2 −a 3 . 00 00a 3 . 00 . . . . . . 00 0 . a n−1 −a n 11 1 . 11+a n 1.5. Cho A = 212 301 012 v`a B = 1 −2 46 5 −3 .T`ım A 2 ,AB,A −1 . 1.6. T`ım c´ac ma trˆa . n 2 −1 3 −2 n ; a 1 0 a n ; cos x − sin x sin x cos x n 1.7. Cho A = 12 21 .T`ım f(A)v´o . i f(x)=x 2 − 4x +3,f(x)=x 2 − 2x +1. 1.8. a. Cho A = 211 312 1 −10 v`a B = 12−2 23 1 12 2 . 1. T`ım A −1 ,B −1 . 2. T`ım f(A),f(B)v´o . i f(x)=x 2 − x − 1 b. T`ım ma trˆa . n nghi . ch d¯a ’ ocu ’ a A = 2100 3200 1134 2 −123 ; B = 13−57 01 2 −3 00 1 2 00 0 1 . 1.9. a. T`ım ma trˆa . n vuˆong cˆa ´ p hai c´o b`ınh phu . o . ng b˘a ` ng ma trˆa . n khˆong. b. T`ım ma trˆa . n vuˆong cˆa ´ p hai c´o b`ınh phu . o . ng b˘a ` ng ma trˆa . nd¯o . nvi . . 1.10. T`ım ma trˆa . n X sao cho: 12 34 × X = 35 59 ; X × 3 −2 5 −4 = −12 56 ; 12−3 32−4 2 −10 ×X = 1 −30 10 2 7 10 7 8 ; X× 11−1 21 0 1 −11 = 1 −13 432 1 −25 ; 21 32 × X × −32 5 −3 = −24 3 −1 ; 41 3 −1 × X × 21 53 = 50 61 ; X × 111 011 001 − 2 21−1 30 6 = 105 −1 −21 ; 122 254 245 × X + 35 76 21 =3 15 −12 −20 ; . i ha . ng cu ’ a ma trˆa . n A =(a i,j ) m×n ,k´yhiˆe . u r(A) l`a cˆa ´ p cao nhˆa ´ tcu ’ a c´ac d¯ i . nh th´u . c con kh´ac 0 cu ’ a A. +Ha . ng cu