x- a1 a, an
1.19. a) Hay tinh dinh thing ca pn san:
a+13 (A.13 0 0 0 1 a+11 a.{3 0 ... 0
14„ = 0 1 a+^ 0
0 0 0 0 1 a+ 0
1)) Chung to rang dinh tithe sal' khong plat thuhe vao y,, ) ••• Ynt 1 g1+ Yl (x1 1111 1)(x1 11 379) 1 x2 +y1 (x9 + N. 1)(x9 + y2) (x n +y 1 ) (x„ +y 1 )(x 1 +y2) D„= n-1 11 (X 1 + Yi ) 1.20. Cho ma 14.'0
Hay tinh A u1° .
n-I Fl(x.)+yi) n(xn +)'i) 1=1 i=1 ( 1 -2 1 \ A= -1 1 0 -2 0 1
1.21. (ha si:t ma trgn A e Mat(m, n, va rang A = 1. Jhisng minh rang cac ma trgn B e M(m, 1, K) va C e Mat(1, 1g A= B. C.
1.22. Chung minh rang ma trap A = a b ) ( d thOa man pIntong trinh:
X2 - (a + d)X + (ad - bc) = 0, 0 do 1 2 la ma Iran don vi cap hai.
1.23. Cho ma trgn
9. 1 0 v 4 1 A=
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:
(-1)H- CV yin
vOi i = j 0 vai i>j 6 do Chi.,'
kHrn-k)! - . Chung minh rang A2 =1,1 . (9, la ma Iran chin vi).
1.26. Gth sa X = ())) e Mat(n, R), 1-1
a do x ii 4- (-1)")! Chitng minh X' = I. 01- 4,
Chit Se voi aeR k e N, ki hiqu
)a, a(a - 1)... (a - k +1) - a ())) )0 )- 1 0 k! ;
1.27. Gia = yXx„ x.„„ x„) vdi (i = 1, 2, ..., n) la ham
[ cua cac bi8n doe lap x 1 , x2, .. x„. Ma tran J = J(Y, X) = aYi
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).
1.30. 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.
D - HDONG DAN HOAC DAP S6
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).
b) (1, 6, 3) (2 5 4) -= (1, 3) (1, 6) (2, 4) (2, 5).
1.3. DS. S6 Dat ca cac phep the a e a(i) # i vai moi
f n
(- 0
n +1 . NMI yky A, (i =1, 2,..., n-F1) i =1,n la n! E
k0 = kl
1.4. Xet tap A gam at ce eke Minn vj (Ma 1, 2, ..., 14, n+1 co thing k nghich the. Ki hieu A i la b° phkn eaa A gdm cac imam NO
an, an+1) ma ai =
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(
U Ch2
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
Thep the f, thi khi nhan viii chuyen tri (a, (i), hai yang xich se Thep lai lam met (136 de nay a thing chfing minh). Tr/ do neu g a phop the. va T la met chuyen tri tin dp Mem cua g oT khong vire); qua do giam cfia g ceng them 1. Vi the nen f phAn tich luec thanh h chuyen tri thi do giam cUa f khong \wet qua h.
1.6. H.D. Do a va n nguyen t6 vfii nhau, nen a k khong chic het cho n vdi mm k = 1, 2.... n-1. do de r(ak, n) la nhfing so phan hied.
1.7. Vi moi phep the phan tich due() thanh tich cat ghee chuyOn tri, nen chi can chfing minh bai town cho phep chuyen tri(1,j)e Sk vei 1, j # 1. Ta ce (i. j) = (1, i) . (1, ) . (1, i).
1.8. Xem vi du 1.1 - 1-1(n+D a) DS: (-1) 2 .
r(1121) h) DS: (-1) 2
1.10. Iasi trien cot dau, to ca. D ykk =(a2 _ " " --12n-2 • Do D2 = -13.2 nen D 2n = (a 2
1.11. A = (aid , detA = E sgna ai,(1) ana(N
E S..
Trong moi tich a ko(k) a ngfrik , tong E(k +cr(k) )= n(n +1) k=1
la sir than; nen re met se cliSn the thfia so ak.,0.;) ma tong k+G(k) le. VI Vey khi thay ha; nen i + j 1e, thi cfic tich
1.12. a) COng dOng thu nhat vdi (long the./ ba, ta dude don tY le vdi (long thu hai.
b) Nhan dong thd nhat vdi (-1), rdi cUng vao die (long th hai va Ulf( ba, ta dupe hai dOng CY 15.
1.13. Ma tran A, B e Mat(2, K). Ta có hai bat bian detAB detBA va tr(AB) = tr(BA). Vi vay:
fx+y=30 {x = 20 {x =10
hoar
x.y= 200 {y=10 {y=20
20 14 \ a) x = 20 va y = 10 o BA=