bai tap dai so tuyen tinh

Đọc kỹ các bài học trong sách giáo khoa. Để ý những điều quan trọng được in đậm, các công thức và bài tập mẫu. Các thành phần quan trọng của bài học nên được ghi chú, đánh dấu bằng ký hiệu của bạn hoặc tô màu. 4. Hãy làm lại các bài tập mẫu và hoàn thành các bài tập trong sách giáo khoa.

PHAN HUY PHIJ • NGUYEN DOAN TUAN

BAI TAP

NHA XUAT BAN HAI HOC QUOC GIA HA NOI

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Chin trach nhiem xual bcin

Gicim doe: NGUYEN VAN THOA

DOAN 'MAN NGOC QUYEN

Trinh bay Ilia: NGOC ANH

BAI TAP DAI sq TUYEN TINH

Lai NOI DAU

Mon Dai s$ tuygn tinh dude dua vao giang day a hau hat cac trUnng dai hoc va cao dang nhtt 1a mot mon hoc cd se; can thigt d@ tigp thu nhUng mon hoc khan Nham cung cap them mot tai lieu tham khao phut vu cho sinh vien nganh Toan vi cac nganh Ki thuat, chting Col Bien soan cugn "BM tap Dai so tuygn tinh" Cugn each dude chia lam ba chudng bao g6m nhUng van d6 Cd ban cna Dal so tuygn tinh: Dinh thfic va ma trail - Khong gian tuygn tinh, anh xa tuygn tinh, he phticing trinh tuygn tinh - Dang than phttdng

Trong mOi chudng chung toi trinh bay phan torn tat lY thuyat, cac vi du, cac hal tap W giai va cugi mOi chudng c6 phan hudng dan (HD) hoac dap s6 (DS) Cac vi du va bai tap &roc chon be a mac an to trung binh den kh6, c6 nhUng bai tap mang tinh 1± thuygt va nhUng bai tap ran luyen ki nang nham gain sinh vien higu sau them mon lice

Chung toi xin cam on Ban bien tap nha xugt ban Dai hoc Qugc gia Ha Nei da Lao digt, kien de cugn sach som dude ra mat ban doe

Mac du chting tea da sa dung 'Lai lieu nay nhigu narn cho sinh vien Toan Dal hoc Su pham Ha NOi va da co nhieu co gang khi bier, soon, nhUng chat than con có khigm khuygt Cluing toi rat mong nhan dude nhUng y kin clang gap cna dee gia

Ha N0i, thcing 3 !Lam 2001

NhOni bien soan

3

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Chudng 2 KHONG GIAN VECTO - ANH XA TUYEN TINH

§1 'thong gian vec to va anh xa tuyeen tinh 96

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§1 Khong gian vec td va anh xn tuyin tinh 11(

§1 Dang song tuy6n tinh aol xUng va dang town phuong 139

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Chuang 1

DINH THUG - MA TRAN

A - TOM TAT Lt THUYET

§1 PHEP THE

Met song anh o tit tap 11, 2,

met phep the bac n, ki hieu la

'1 2 3

n} len chinh no duet goi la

\ GI a 2 G3

15 del a, = a(1), 02 = a(2), , a„ = a(n)

Tap cac phep the bac n yeti phep nhan anh xa lap thanh met nhom, goi la nh6m del xeing bac n, ki hieu S S6 cac Olen t3 cua nhom S„ bang n! = 1, 2 n

Khi n > 1, cap s6 j} (khong thu tv) dude pi IA met nghich the cem a n6u s6 - j) (a, - a) am Phep the a &foe goi la than ndeM s6 nghich thg cim a chan, a &toe goi la phep the le n6u s6

nghich the ciaa a le

1 neM s la phep the chan

Ki hieu sgna =

-1 net} a la phep th6 le

va sgna goi IA deu am, phep the a Neu a vat la hai phOp the cling bac, thi sgn(a = sgn(a) sgn( )

Phep the a chicly goi IA met yang xich do dai k n6u c6 k s6 i„

•-• , i k doi mot khac nhau dr coo = 1 2, coo = i3, a(ic) = i1

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va a(i) = i vdi moi i x i„ i k Vong )(felt do dttoc ki hieu IA

ik) M9i phep th6 dau &tan tfch the thanh tfch nhung yang xfch doe lap

Met vOng xfch do dal 2 dude goi IA met chuygn trf Vong xfch ••• , i k ) phan tfch chive thanh tfch 0 1 ,

§ 2 DINH THUG

I Gia sit K IA met trueng (trong cuan sich nay to din yau xet K la &Ong s6thvc K hoac truang s6 phitc C) Ma tran kidu (m, n) vdi cox phan tit troll twang IC la met bang chit nhat gfim

m hang, n cet cac phan tit K, i = 1,m, j = 1,n Tap cac ma tran kidu (m, n) chive kf hieu M(m, n, R) Ma trail vuong cap n

IA ma tran co n dong, n cot Tap cac ma trail vu8ng cap n vdi cac phan tit thuoc truong K ki hiOu IA Mat(n, K)

2 Cho ma tr4n A vuong cap n, A = (ad, i, j = 1, 2, , n Dinh thitc ciia ma tran A, kf hieu det A la met flan tit dm K dude xac dinh nhu sau:

detA = zsgn(a)a mo)

E S n

3 Tinh eh& ceta Binh that

a) Neu dgi cho hai dong (hoac hai cot) nao do cim ma tram

A, thi dinh auk cim no ddi da:u

b) N6u them veo met dong (hoac met cot) cim ma tran A met to hdp tuygn tinh cim nhUng thing (hoac nhung khac, thi dinh auk khong thay ddi

8

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4 Cdch tinh dinh that

a) Cho ma tran A E Mat(n, K) Kf hi'911 Mi; la dinh that cua

ma trail alp (n-1) nhan dine bAng cach gach be clOng thU i, cot

thu j cut ma tram A vb Aij = (-1)H M u clucic g9i la pha'n phu dai s6cUa phgn to a ii cna ma trait A Ta có CAC tong thtic:

det A ngu i = k

O ngu i x k det A ngu i = k

Nhu fly detA = EamAki (k = 1, 2, n)

1=1 heat detA = Z

•

a ikAik /=1

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detA = ZED' Si k+31+ A ik

do j1. jk la k cot cgdinh Tgng &toe lgy theo tat ca cac bQ ik) sao cho < i2 < < i k Cong thdc troll dude goi la c8ng thdc khai trim dinh thae theo k cot ji, 'along tV, to có gong thdc khai trim theo k clang Khi k = 1, to &roc gong tilde da not trong muc a

§ 3 MA TRANI

1 Ma trgn kigu (m, n) vgi cac phan tU tr8n trang K da dude gidi thigu trong §2 Tap the ma tran kigu (m, n) vdi cac phan ti tren tragng K dude ki hiQu la Mat(m, n, K) A E Mat(m, n K) (Woe vigt A = (aii) i = 1, m ; j = 1, 2 n hay ro rang hon:

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amt amt arnn,

2 Cac phep todn tren Mat(m, n,

Cho A = (a y ), B= (b,j ) thuOc Mat(m n, K)

Ta có:

a) Ma tran C = (cg) a do cy = a ii +

&toe goi la tong cua hai ma tran A va B va ki hien la A +B

Ma tran D= (d,,) a do di; = a ij -

dude goi la hiOu cila ma trail A va B va Id hi'eu la A - B

b) Vdi k E At, ma trail kA c8 cac phAn to la (ka ii ) duoc goi

la tick cua ma tran A vdi ph&n td k cua trudng K

c) Neu A = (aii ) c Mat(m, n, K) va

&toe goi litich caa hai ma tram B ye A

Vol A, B e Mat(n, K), to có det(AB) = detA detB

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d) Tap Mat(n, K) con ma tran yang cap n vdi phep toan cOng lap thanh mot nhom giao hoan, con vdi phep Wan rang ma tran va phep nhan ma trail lap thanh mat vanh khong giao hodn, co don vi

3 Hang ctia ma tran; Ma trim nghich ddo

Gig A E Mat(m, n, K), ta dinh nghia hang ciat ma tran A

la cap cao nhgt cua dinh thric con khgc khong rut ra W ma tran

A KM A E Mat(n, K) va hang A = n (ta cling dung ki hi3u hang

A la rang A) thi ma tran A goi la khong suy bign, khi do detA * 0 va ton tai duy nhgt ma tran B thuOc M(n, K) A.B = B.A = I„; d do I lit ma trail don vi Ma tran B &roc goi 11

ma tran nghich dgo cna ma tran A va ki hi3u la A'

Gig su A= (A u )la ma trail plw hpp cim ma tran A = (ad,

Ab la Olga Ow dal see mitt phgn ht aii ; A t la ma tran chuya'n

vi cua A Khi do:

At detA

Vay a có 5 nghich the nen sgna = -1

b) Ta hay tinh sS nghich the cua boa)) vi (1, 4, 7 3n-2, 2,

1 khong tham gia vao nghich the

4 tham gia yen 2 nghich the voi the s6 thing sau no

7 tham gia van 4 nghich the

3n - 2 tham gia vao 2(n - 1) nghich the voi the se dung sau no

2 khong tham gia vao nghich the nao vdi the se dung sau no

5 tham gia yen 1 nghich the voi the se dung sau n6

S tham gia vao 2 nghich the vdi cae s6 dUng sau n6

3n - 1 tham gia vim (n - 1) nghich the voi the s6 thing sau no Cae s6 3, 6, 9 , 3n khong tham gia vao nghich the nao voi the s6 (hang sau

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Vay co tat ca 2 + 4 + 2(n-1) + 1 + 2 + (n 1) - 3(n -1)n

2

(n-1 )n

nghich the trong hoot vi da neu va do d6 sgn S = (-1) 2

Khi n = 4k hoac n = 4k + 1 thi sgn 5 = 1

con neu n = 4k + 2 ho4 n = 4k + 3 thi sgn = -1

Vi du 1.2

_ 1 2 3 Cho phep th'e' f - en dgu la (-1) 1

Lift( gidi:

a) Vi sgn f sgn = sgn (f

= sgn(Id) = 1 non sgn (e 1)= sgn (f) = (-1) k

b) X4t phep the"a = 1 2 nj

n -1 1 thi g = f a

Do gay sgn g = sgnf sgn a

n(n-1) Nhung sgn a = (-1) 2 nen sgn g = (_])k+C;',

14

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Vida 1.3

ChUng minh rang vier nhan mat phep th6 vdi ehuy6n trf

j) v6 ben trai Wring throng v6i viac dai cha car s6 i, j a clang drah cna phep the Cling nhu vay, nhan mat phop th6 Nth ehuynn trf (i, j) v6 been phai tunng during voi del eh?, I, j a dong tit 66a phep th6

chuy6n tri Xet

Trunng hop nhan ben phai dude ant Wring fib

Vi dy 1.4 Cho f va g la hal phop th6cua n strtn nhien clAu tien a) Chung minh rang có the' cilia f va g bang khong qua (n-1) phop chuyan trf (nghia la ton tai k phop chuyan trf a l , cr2 , ak ,

± m2 + =

rang mOt vong xfch (a l , a2, am) la mOt plop the

a cac s6 tv nhien Ui 1 den n sao cho a(a) = a 1+1 (i = m-1) va a(a„,) = a l , con a(l) = 1 nen 1 yen moi i = 1, ,., m Vong xfch (a l , a2, u„) goi la ce do, dai m

Ta da hiet rad yang xfch do, dai m deu phan tich duo thanh m - 1 chuyen trf Vi vay g o e' phan tfch duo thanh 'Lich

caa i(m i -1)= n -p = k phep chuyen trf

f ve g duo Wang it hdn n - 1 phep chuyen trf

on f = Ta se chUng to rang killing dua

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Do f chi co met phAn tei chinh quy ye g co n phan tV chinh quy, vi vgy khong thg clua f vg g bring it hon n - 1 phep chuygn tri

Vi du 1.5

Chung minh rang vdi mei so k (0 <k < C;) t6n tai met phep thg a e S„ co dung k nghich thg

1231 giai

Ditch 1: Ta hay cluing minh met It& gug manh

Nigu a = (a,, la met hogn vi cda 1, 2, n va ace 1: nghich thg, 0 < k < , thi có thg ct6i che hai phfin tri nao

do de thu ridge hog') vi c -3 co k + 1 nghich thg That Nay, int& hgt ta nhan thy rang negi > vdi moi i = 1, 2, , n-1 thi

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2cosa, = 2coi2 a - 1 = cos2a

Gia sa D1 = cosia \TM moi = 1, k

Ta có

Dk,I = 2cosa Dk - Dk_ i

= 2.cosa coska - cos(k -Da

= (cos(k+Da + cos(k-1)x) - cos(k-1)a = cos(k+1)a Nhn vay D„ = cosna

Khai trin theo cot tht nhEt, to c6:

A n = (e P +e -P)A n _i:

e 21' - e -2(P Nlinn xet rang 4 1 = 6 9 +Cc =

A 2 = ((i e ro e

636 - -39

e (P -

e (1+1)6 - e -(lrv1),p Girt sit AR -

P(x +1) P(x +1)

P(x + n) P(x +n)

Ong vao clang Hirt nhgt vgi tat ca k=2, n+2)

Khi do, phAn tii dung dau có clang:

poc + 0 + P(k) (x +0.(x +11.) k o k = n)

k!

k=1

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Pkx+1.) P(x+n)

PTUx +1) P(n)(x+n) PT+I kx+1) PT+I kx+n)

Ta ki hieu dinh thge a v6 phai bai C va ma tr5n Wong ring

hai (6-1 Vi da thge P(x) = n(x + i) Ken P'" 0(x) (n+1) !, vi vay

i=0 the s6 hang a dOng cu6i &au bang (n+1) ! dgn gian ki higu

va each viek ta dirt xk= x+ k, k = 0, 1, n d dOng thg hai tit

&leg len ciaa ( ta co:

(Pw(x0), P("ax,), PT)(x.,)) ((n+1) hco + a l , , (n+1) tx„+

a do a l la hang s6 &do do Khi nhan (long cue"' ding voi

r6i Ong vao clang trail no, ta dtta ma trgn ( ) va dang

P(x 0 ) P(x ] ) P (x i) )

P (11-1) (x 0 ) Pth-e (x l ) PT-1) (x n ) (n+1)!x 0 (n+1)!x 1 (n+1)!x n (n+1)! (n+1)! (n+1)!

a l

(n+1)!

(*)

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.D n

Jk!

k=1

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6 do D„ la dinh thiic Vandermonde cua clic so" x„, x„

n-1 D6' thay D„= n(x k - x i )= n (c_i) = 11(n-o!= 1

Lbi gidi:

Nhan xet rang neh to them vim mit phan to a jj nao do cha

ma tran vuemg A mot s6 than, thi dinh thfic cha ma tran nhan

dticic se sai khac vat dinh thiic cha ma tran A mot s6 than Vi

the ne'u to hat di 2000 don vi a nhUng phan t,i bang 2001 cha A, thi tinh than le cha dinh thfic cila A khong thay d6i, nghia la:

detA = detB (mod 2) a do B = (14 hi; = 0 n6u i= j va = 1 ngu it j

Ta có:

0 1

1 0 det B =

1 1 0

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nhan Bong ddu veli -1 fee cling vao cac dOng con lei ta dttuc:

detB =

1 0 0 -1 Deng vao cot thu nhat ca car cot con lai ta cO:

cosmp - sinmp sinmp cosny

ma 2000 chia het cho 4

\ray A 200° =(A) 500 =I, (I, la ma tran thin vi cep hai)

Vi du 1.17

Ma Iran vuong A e Mat(n, K), A = (ad throe goi la Ina tran phan del xang netu aii + = 0 vdi moi i, j = 1, n

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MATHEDUCARE.COMHay chang minh: 'rich caa hai ma tran phan doi xang A va

la mat ma tran phan doi 'ding khi va chi khi AB = -BA

Nha vay AB phan del xang <=> c, = V i, k

tt=> = vet mot i, k c> AB = -BA

Khai tri6n theo n clang dal) (thee dinh 12) Laplace), ta co

Mat khac, bi6n &it ma Iran C bai phep bi6n ct6i sd ca) sau: Nhan cot thu nhal vat b 1, cot thil hat Null cot thin n vet btl; r6i Ong vac) Ot thu n + j (j = 1, 2, n), ta dude ma tran D Bang sau ma dinh thiic cua D va cua C bang nhau:

Khai tri6n theo n cot cuoi (theo dinh 157 Laplace) ta có

detD = det(dij) = det(A B) (2) Tit (1) va (2) va do detC = detD nen ta co:

det(A0 B) = detA detB

Lari gidi:

Ni111u X = 1 thi re rang X giao ho an vdi moi ma Han

\along cling cap

Ngnoc lai, gia sit X = (X„) giao haul vdi moi ma tran yang cdp n

VOi i„ # j 0, to chiing minh x inio = 0 Mudn stay chon A = (ad trong do a joio =1 con one phan tit khac ddu bAng Thong Phan to ding i o cot j„ cim ma tran XA Wing x •' can phan tit a thing io

cot j, cim AX la 0 Tir di6u kien AX = XA suy ra =O Nhu

y X co dang:

k r 0

„

0 k„

Cho ma tran A = (a) a do a ] , = 1 vii moi i, j Khi do phan tit

o dung i cat j cim ma Han XA la X , con phan to 0 ding i cot j dia ma tran AX la 2 nen k, =

b a) Chung minh detA = (a-b)° (a + (n-1)b)

b) Trong trudng hOp detA s 0 Hay Huh ma trdn nghich dao A+ oda ma tran A

X=

33

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Lai giai:

a) Gang the clang vao dOng this nhat ro'i nit

a + (n-1)b a clang dau, ta (Woe

1 1 1

b a b detA = (a + (n-1)b) x

b b a Lay dOng chin aria (Anil th.fie tit nhan voi -b r6i Ong vao the dong sau, ta eó:

1 0 0

0 a - b 0 detA = (a + (n-1)b) x

0 0 a - b detA = (a + (n-1)b) (a-b)" -i

h) A kha nghieh <=> detA 0 (=> a b va a + (n-Gb O Goi B = (b ii) la ma tran nghich (lac) cila A = (a„)

b b a Theo phan a) thi A i = (a-b)" -2 (a + (n-2)b)

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Vay b„ -

det A (a 1- (n -1)1).(a - b) Vol i # j = (-1)H , d do kl„ la chub thue e6p n-1,

co dupe ba' lig each x6a &Ong thu i va cat tha j eila ma tran A Do

A dovi ming nen = Gia sU rang i < j, khi do cot thfi i va dung thu j-1 cua M o gain town nhang phan t& b N6u d6i eh 6 Bong j - 1 len tren Ming dau (Mu nguyen cac dung khac), rei lai MS( nit i len 6;4 thu nhai (va van gib nguyen the cat khac), thi

1.3 Tim s6 tat ca the phep the a e S„ sao oho a(i) x i vol

mm i = 1, 2, n Cluing t8 rang khi n chan, so" one phep th6 clang tren la m54 3616

1.4 Ki hi8u (n, k) la secac hoan vj rim 1, 2 n ce dung k nghich the Chiing minh cong thile truy 146i sau:

(n+1, k) = (n, k) + (n k-1) + + (n, k-n)

vdi guy vac (n, j) = 0 nevu j < 0 hoac j > 9

1.5 Ta goi 45 giam ena phdp the f Ia hi5u cna s6 the phan tic khting bat dog (nghia la s6 the pilau tit i ma f(i) i) va s6 clic Yong xich <IQ dai ion him 1 trong phan tich cua f thanh tich car yang xich 458 lap

a) Chung minh f cotang tinh chat chan la vOi do giam cua no h) Chling minh rang s6 161 thidu one nhan tic trong phan tich cna f thanh tich the chuydri tri bgng 45 giAm cua f

1.6 DM \h hai s6 x va n nguyen, n x 0, to ki hi5u r(x, n) IA s6 chi khi chia x cho n : 0 r(x, n) < n Chiang minh rang nal

n > 2 va a la so? nguyen, nguytin to' d61 vdi n, thi tthing Ung ki—>r(ak, n) la mot phan tit cua S„,, k e 11, , n-1}

1.7 Cluing minh rang moi phep th6 cap k > 1, dou phan tich dupe thanh tich nhiing chuydn tri dang (1, i) vdi i = 1, k

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MATHEDUCARE.COM1.8 Xac dinh davu cua cac phop the' sau:

b) Dinh attic ay Ion nhig la 1

c) Dinh thew c)41) ba ma cac pga'n tit la 1 hoac 0 dot gia tri IOn nheit biing 2

1.10 Tinh dinh thew cap 2n D = det(cM),

1.12 Chung to cac dinh thew sau day bang khong:

1 cosa costa cos3a

cosa costa cos3a cos4a

a)

costa cos3a cos4a cos5a

costa cos4a cos5a cos6a

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-1 -1 -1

0 -1 -1

1

0 -1

MATHEDUCARE.COM1.16 Hay tinh dinh thdc sau:

1.17 Cho da thite P x) = (x - al)(x - (x - a„)

a d6 a, la cac so thvc dot mot phan biet Hay tinh climb thde sau:

P(x) P(x) P(x) x-a 1 x-a, x-a n

Way chang minh det J # 0 Hay tinh ma tr'nn AJ tit do suv

ra gin tri detA Hay nen hal Loan Wong to kin A ya J cite ma 1.rn cap n

1.19 a) Hay tinh dinh thing cap n san:

(x 1 1111 1)(x1 11 379)

1

x 2 + y1 (x9 + N 1)(x9 + y2)

(x n +y 1 )

(x„ +y 1 )(x 1 +y2) D„=

( 1 -2 1 \ A= -1 1 0 -2 0 1

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90

vOi 4 x 0 Hay tinh A- '

1.24 Cho ma tran vuong c5p 4

cosa sincx cosa sina

cns2a :;in2a 2cos2a 2sin2a cos3o sin3a 3cos3a 3sin3a cos4a sin1a 4cos4a 4sin4a Chung minh rang A khg nghich khi va chi khi a ♦ kg (Ice Z) 1.25 Cho ma trgn A = (a) e Mat (n, 1H) ma the phan tit doge cho bai tong that:

dx• 1 i

dliciC goi ]a ma tran Jacobi cua phdp bi6n din, con Binh thud dm

no duck goi la Jacobien dm phep biers den do

Bay gin /cot m6i quan hq gilla n 2 ham vii va n 2 bien xii dune cho Isdi ding thiic:

Y= A.X.B, a do Y= (y i) , X = (xi),

A B e Mat(n, R) la hai ma tran the trn6c

Chung minh rAng det(J(Y, X)) = (detA)" (detB)"

1.28 Cho X = (x„) e Mat(n, R) la ma tran tam gide dudi;

va Y = X X

Chung minh rAng det(J(Y, X)) =

1.29 Cho Z lA tap cac sqinguyen; A, S hai ma tran vu8ng cap n, cac phAn t> la nhilng s6nguyen (ta vie) A, S e Mat(n, Z)) Hun nua detA = 1, det S x 0

Dal B = A S; Chung minh rAng cOs6m nguyen dding

de Bm e Mat(n, Z)

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MATHEDUCARE.COM1.30 GM sit trong ma tran A = (a u) c Mat(n, R) da cho

rude tat ea the phan ti a d (i # j) Chang minh rang có thk dien

Mo &rang char) chinh the s60 hoc 1 de ma tran A kheng suy Bien 1.31 Tim tat ca cac ma tran A e Mat(n, K), A= (dj), 0

na tan toi ma tran nghich dao A-' cling có the phiin t5 khOng am 1.32 Cho ma tran vuong A co the phan to la s6 nguyen rim diet' kien can va du de ma train nghich dao cung c6 the

-Men to la s6 nguyen

Xet T e S„ (n 2 2) gie sii i < j va T = j, T (j) =

T (k) = k vdi moi k s i, j Khi do the nglach the eaa

ji, k} Nob < k < j j/, j} Nob / i + 1, j - 1 With vey co tat ek la U - i) + (j - i - 1) = 2U - I) - 1 nghich the Vi so nghich the le nen t la phop the le

1.2 a) Phop th6 da cho phan Deb thanh hai yang xich chic lap (1 8 2) (4 6 5 7) = (1, 2) (1, 8) (4, 7) (4, 5) (4, 6)

a n , a n+1 ) ma ai =

la nhang tap con rot nhau caa A va A = A, U A2 U U U AnA

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Xet Anki = {a = (a p a 2 , , an+1)1 = n +1}

NMI 0y so cac nghich the cna a bAng se cac nghich the cue

1 9

nghia la bling 1+ Dieu do cheng to so ea(

Jhfin to cria A„,, bang (n, k)

Xet tap Ai (i = 1, , n) cis su a e Ai, a = (a, a j , a„ 4 ,) Theo dinh nghia a ; = n+1 Nhti vay (aj, khong la nghich th( vdi j < i va (x„ ai) la nghich the vdi j > i Do do a, tham gia vac n+1-i nghich the Xet hoan vi a' = (a l — aa,_„ (>61, cCia S2 S5 nghich the maa a' bang sernghich the cna a trii NM; vay ta có met song anh tit A ; len tap cac hofin vi cua S„ co thing k-n-l+i nghich the., do do so' phan t& cera A ; IA (n, k-n-1+i) Td

do, sephan to cUa A la:

(n + 1, k) = (n, k) + (n, k-1) + (n, k-2)+ + (n, k-n)

1.5 a) Xet phan tich f = a l o a, o a,„ thanh tich cac veng xich dec lap di) dai > 1 Gia sa do dai a l, la d k; ta they f(i) # i khi

va chi khi i thuoc met trong cac yang xich do Vay do giam coal

IA d,+€1.2 M6i vOng xich dai d k phan tich dude thanh

dk -1 chuydn trf Do vay f phan tick deck thhnh d 1 + + (.16-1n chuyen trf Do do kha'ng dinh a) driec chting mink

b) Goi / a di) giam can f Theo a) f phan tich (kw thanh / chuydn trf N5u có phfin tich f thanh h chuyen tri nao do, ta phai chring minh h 2 1 Ta có bd de sau: Neu a, 13 tham gia van met yang ;rich cas phep the f, thi khi nhAn f vdi chuydn trf (a, f3) (ve ben trai hay ben phai) yang xich dji not phan thimh hai

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