3.3. Ha . ng cu ’ a ma trˆa . n 111 Gia ’ i. Ta t`ım ha . ng cu ’ a ma trˆa . nd˜a cho theo phu . o . ng ph´ap I. Hiˆe ’ n nhiˆen ma trˆa . n A c´o d i . nh th´u . c con ∆ 2 = −10 01 = −1 =0. Ta t´ınh c´ac di . nh th ´u . c con ∆ 3 bao ∆ 2 .Tac´o ∆ (1) 3 = −100 011 111 =(−1) 11 11 =0; ∆ (2) 3 = −100 011 423 = −1 =0. Nhu . vˆa . y c´o mˆo . td i . nh th´u . cbao∆ (2) 3 = 0. Ta t´ınh di . nh th´u . c bao cu ’ a ∆ (2) 3 .Tac´o δ (1) 4 = −1001 0 112 1 111 4 231 =0 (ta . i sao ?). T`u . d´o suy ra r( A)=3. V´ı d u . 2. T`ım ha . ng r(A)nˆe ´ u A = 1 −325 −24 31 0 −2711 7 −15 −72 −11 56 Gia ’ i. Ta gia ’ i theo phu . o . ng ph´ap I. Hiˆe ’ n nhiˆen ma trˆa . n A c´o di . nh th ´u . c con ∆ 2 = 1 −3 −24 = −2 =0. 112 Chu . o . ng 3. Ma trˆa . n. D - i . nh th´u . c Tˆa ´ tca ’ c´ac di . nh th´u . c con bao ∆ 2 : 1 −32 −243 0 −27 ; 1 −35 −24 1 0 −211 ; 1 −32 −24 3 7 −15 −7 ; 1 −35 −241 7 −15 2 ; 1 −32 −243 −115 ; 1 −35 −243 −116 d ˆe ` ub˘a ` ng 0. Do d´o r(A)=2. V´ı d u . 3. B˘a ` ng c´ac ph´ep biˆe ´ nd ˆo ’ iso . cˆa ´ p, t´ınh ha . ng cu ’ a c´ac ma trˆa . n 1) A = 12 3 5 3 −14−2 53108 ;2)B = −1001 0 112 1 111 4 231 3 120 . Gia ’ i. 1) Ta thu . . chiˆe . n ph´ep biˆe ´ ndˆo ’ iso . cˆa ´ p theo h`ang v`a thu du . o . . c A = 12 3 5 3 −14−2 53108 h 2 −3h 1 → h 2 h 3 −5h 1 → h 3 ∼ ∼ 12 3 5 0 −7 −5 −7 0 −7 −5 −17 h 3 − h 2 → h 3 ∼ 12 3 5 0 −7 −5 −17 00 0 0 . D ´o l`a ma trˆa . n h`ınh thang v`a hiˆe ’ n nhiˆen n´o c´o ha . ng b˘a ` ng 2. Do d´o r(A)=2. 3.3. Ha . ng cu ’ a ma trˆa . n 113 2) Ta c´o B = −1001 0112 1111 4231 3210 h 3 + h 1 → h 3 h 5 +4h 1 → h 4 h 5 +3h 1 → h 5 ∼ −1001 0 112 0 112 0 235 0 123 h 3 −h 2 → h 3 h 4 − 2h 2 → h 4 h 5 −h 2 → h 5 ∼ −1001 0 112 0 000 0 011 0 011 ∼ −1001 0 112 0 011 0 000 0 011 h 5 − h 3 → h 5 ∼ −1001 0 112 0 011 0 000 0 000 . T`u . d ´o t h u d u . o . . c r(B)=3. V´ı d u . 4. T´ınh ha . ng cu ’ a c´ac ma trˆa . n 1) A = 12 4 5 2 23 1 1 3 01 7 9 1 1311143 ;2)B = 13205 2 6 9712 −2 −524 5 1 4 8420 . Gia ’ i. 1) Ta thu . . chiˆe . n c´ac ph´ep biˆe ´ nd ˆo ’ i sau: A = 12 4 5 2 23 1 1 3 01 7 9 1 1311143 ∼ 12452 −0 −1 −7 −9 −1 01791 01791 ∼ 12452 01791 01791 01791 ∼ 12452 01791 00000 00000 ∼ 12452 01791 T`u . d´o suy r˘a ` ng r(A)=2. 114 Chu . o . ng 3. Ma trˆa . n. D - i . nh th´u . c 2) Ta thu . . chiˆe . n c´ac ph´ep biˆe ´ nd ˆo ’ i B = 13205 2 6 9712 −2 −524 5 1 4 8420 ∼ 1320 5 0057 2 016415 016415 ∼ 1320 5 016415 0057 2 016415 ∼ 1320 5 016415 0057 2 0000 0 T`u . d´o suy r˘a ` ng r(B)=3. B ` AI T ˆ A . P T`ım ha . ng cu ’ a c´ac ma trˆa . n: 1. A = 12 3 −1 .(D S. r(A)=2) 2. A = −13 2 −6 .(D S. r(A)=1) 3. A = 12 36 .(DS. r(A)=1) 4. A = 12 34 56 .(D S. r(A)=2) 5. A = 1 −21 −143 .(D S. r(A)=2) 3.3. Ha . ng cu ’ a ma trˆa . n 115 6. A = 01 3 03−1 02 0 .(D S. r(A)=2) 7. A = 1 −23 2 −46 514 .(D S. r(A)=2) 8. A = 132 264 −13−2 .(DS. r(A)=1) 9. A = 1 −240 −1351 2 −140 .(D S. r(A)=3) Su . ’ du . ng c´ac ph´ep biˆe ´ ndˆo ’ iso . cˆa ´ pdˆe ’ t`ım ha . ng cu ’ a ma trˆa . n: 10. A = −10 3 −2 23−1 −3 361−8 .(DS. r(A)=2) 11. A = 2 −21 −31−1 541 100 .(D S. r(A)=3) 12. A = 49072 −11 603 0 −121−3 4 −3 −19 6 (D S. r(A)=4) 13. A = −1 −3 −21−3 4124−1 −69−1 −26 46112−4 .(D S. r(A)=3) 116 Chu . o . ng 3. Ma trˆa . n. D - i . nh th´u . c 14. A = 2 −9 −5 −29−5 4437−44 −2 −3 −1 −33−3 22−12−62 −11 3−11−1 .(DS. r(A)=4) T`ım ha . ng cu ’ a ma trˆa . nb˘a ` ng phu . o . ng ph´ap di . nh th ´u . c bao: 15. A = 11000 23000 00500 00060 00008 .(DS. r(A)=5) 16. A = 1 234 −1301 2 418 1 769 0 10 1 10 .(D S. r(A)=3) 17. A = 11 3 3 2 22−1 −14 11 3 3 2 (D S. r(A)=2) 18. A = 24−2331 −1 −21172 12−14103 .(DS. r(A)=2) 19. A = 112 3 −1 021 2 2 003 3 −3 000 4 0 13612−2 133 5 1 .(D S. r(A)=4) 3.3. Ha . ng cu ’ a ma trˆa . n 117 20. V´o . i gi´a tri . n`ao cu ’ a λ th`ı ma trˆa . n A = λ −1 12 c´o ha . ng b˘a ` ng 1 ? (D S. λ = − 1 2 ) 21. V´o . i gi´a tri . n`ao cu ’ a λ th`ı ha . ng r(A)=2,nˆe ´ u A = λ 01 341 1 −12 ?(D S. λ = 7 9 ) 22. V´o . i gi´a tri . n`ao cu ’ a λ th`ı ha . ng r(A)=3nˆe ´ u A = 10−1 2 λ −23 10 4 ?(D S. λ =2) 23. V´o . i gi´a tri . n`ao cu ’ a λ th`ı ha . ng r(A)=3nˆe ´ u A = 10λ 234 008 ?(D S. ∀λ ∈ R) 24. V´o . i gi´a tri . n`ao cu ’ a λ th`ı ha . ng: 1) r(A)=1;2)r(A)=2; 3) r(A)=3nˆe ´ u: A = 1 λ 2 214 428 ? (D S. 1) λ = 1 2 ;2)λ = 1 2 ; 3) Khˆong tˆo ` nta . i) 118 Chu . o . ng 3. Ma trˆa . n. D - i . nh th´u . c 3.4 Ma trˆa . n nghi . ch da ’ o 3.4.1 D - i . nh ngh˜ıa Nˆe ´ u A l`a ma trˆa . n vuˆong cˆa ´ p n th`ı ma trˆa . n vuˆong B cˆa ´ p n tho ’ a m˜an d iˆe ` ukiˆe . n AB = BA = E n trong d´o E n l`a ma trˆa . ndo . nvi . cˆa ´ p n du . o . . cgo . il`ama trˆa . n nghi . ch da ’ o dˆo ´ iv´o . i ma trˆa . n A v`a du . o . . ck´yhiˆe . ul`aB = A −1 . Nhu . vˆa . y theo d i . nh ngh˜ıa AA −1 = A −1 A = E n . D - i . nh l´y. Ma trˆa . n vuˆong A c´o ma trˆa . n nghi . ch da ’ o khi v`a chı ’ khi ma trˆa . n A khˆong suy biˆe ´ n(t´u . c l`a khi detA =0) v`a khi d ´o A −1 = 1 detA P A , (3.12) P A = A 11 A 21 A n1 A 12 A 22 A n2 . . . . . . . . . . . . A 1n A 2n A nn trong d ´o A ij l`a phˆa ` nb`uda . isˆo ´ cu ’ a phˆa ` ntu . ’ a ij (i, j = 1,n) cu ’ ama trˆa . n A. Ma trˆa . n P A du . o . . cgo . i l`a ma trˆa . n phu . ho . . pcu ’ a ma trˆa . n A. T´ınh chˆa ´ t 1 + Nˆe ´ u ma trˆa . n A c´o ma trˆa . n nghi . ch d a ’ ov`am = 0 th`ı ma trˆa . n mA c˜ung c´o ma trˆa . n nghi . ch d a ’ ov`a (mA) −1 = 1 m A −1 . 3.4. Ma trˆa . n nghi . ch da ’ o 119 2 + Nˆe ´ u A v`a B l`a hai ma trˆa . n vuˆong c`ung cˆa ´ pv`adˆe ` u c´o ma trˆa . n nghi . ch da ’ oth`ı (AB) −1 = B −1 A −1 . 3 + Nˆe ´ u A c´o ma trˆa . n nghi . ch da ’ o A −1 th`ı A −1 c˜ung c´o ma trˆa . n nghi . ch d a ’ ov`a A −1 −1 = A. 3.4.2 Phu . o . ng ph´ap t`ım ma trˆa . n nghi . ch d a ’ o Phu . o . ng ph´ap I gˆo ` m c´ac bu . ´o . c sau Bu . ´o . c1.T´ınh detA +Nˆe ´ u detA =0th`ıA khˆong c´o ma trˆa . n nghi . ch da ’ o. +Nˆe ´ u detA = 0 th`ı chuyˆe ’ n sang bu . ´o . c2. Bu . ´o . c2. T`ım ma trˆa . n phu . ho . . p P A .T`u . d ´o ´ap du . ng cˆong th´u . c (3.12) ta thu d u . o . . c ma trˆa . n A −1 . Phu . o . ng ph´ap II (phu . o . ng ph´ap Gauss-Jordan) Dˆa ` u tiˆen ta viˆe ´ t ma trˆa . ndo . nvi . c`ung cˆa ´ pv´o . i ma trˆa . n A v`ao bˆen pha ’ i ma trˆa . n A v`a thu d u . o . . c ma trˆa . n M = A|E n . (3.13) Tiˆe ´ p theo thu . . chiˆe . n c´ac ph´ep biˆe ´ ndˆo ’ iso . cˆa ´ p trˆen c´ac h`ang cu ’ a ma trˆa . n M d ˆe ’ du . a khˆo ´ i ma trˆa . n A vˆe ` ma trˆa . nd o . nvi . E n c`on khˆo ´ i E n trong (3.13) th`anh ma trˆa . n B: A|E n −→ E n |B . Khi d´o B = A −1 . C ´ AC V ´ IDU . 120 Chu . o . ng 3. Ma trˆa . n. D - i . nh th´u . c V´ı d u . 1. T`ım ma trˆa . n nghi . ch da ’ odˆo ´ iv´o . i c´ac ma trˆa . n sau: 1) A = 35−2 1 −32 67−3 ;2)A = 10−21−7 −54 203 1 6 789 2 6 592 3 −2101 Gia ’ i. 1) Ta c´o detA =10= 0. Do d ´o ma trˆa . n A trong 1) c´o ma trˆa . n nghi . ch d a ’ o. Phˆa ` nb`uda . isˆo ´ cu ’ a c´ac phˆa ` ntu . ’ cu ’ a n´o b˘a ` ng: A 11 = −5; A 12 = 15; A 13 = 25; A 21 =1;A 22 =3;A 23 =9;A 31 =4; A 32 = −8; A 33 = −14. T`u . d ´o theo cˆong th´u . c (3.12) ta c´o A −1 = 1 10 −51 4 15 3 −8 25 9 −14 = − 1 2 1 10 2 5 3 2 3 10 − 4 5 5 2 9 10 − 7 5 . 2) Ta t´ınh detA.Lˆa ´ y h`ang th´u . ba cˆo . ng v`ao h`ang th´u . nhˆa ´ t ta c´o detA = 2 6 592 −5 4 203 1 6 789 2 6 592 3 −2101 =0 v`ı trong ma trˆa . nthud u . o . . c c´o h`ang th´u . nhˆa ´ t v`a th´u . tu . giˆo ´ ng nhau. Nhu . vˆa . y ma trˆa . n A trong 2) l`a ma trˆa . n suy biˆe ´ n, do d´o n´o khˆong c´o ma trˆa . n nghi . ch d a ’ o. V´ı d u . 2. D`ung c´ac ph´ep biˆe ´ ndˆo ’ iso . cˆa ´ p t`ım ma trˆa . n nghi . ch da ’ odˆo ´ i [...]... (DS 1 1 0 ) 4 1 9 −2 13 12 1 2 3 4 1 0 0 ` o o (DS Khˆng tˆn tai) 5 1 −3 1 0 3 1 1 1 1 0 1 1 1 (DS 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 1 (DS 1 0 1 1 1 1 1 0 1 2 5 −2 −3 5 (DS 0 5 4 6 5 0 0 0 1 0 ) 1 1 0 1 1 1 0 1 1 0 ) 0 0 1 0 1 1 −2 5 4 1 11 −9 ) 0 5 −4 0 −6 5 ’ 3.4 Ma trˆn nghich dao a 1 0 19 ... 2 1 1 2 2 5 Chu.o.ng 3 Ma trˆn D nh th´.c a -i u 12 8 1 12 1 1 1 13 1 1 14 15 16 17 18 1 1 1 1 1 3 2 1 1 0 1 0 0 0 0 1 1 0 1 1 2 2 5 0 0 0 0 1 1 1 − 3 6 2 5 2 1 1 (DS 0 4 1 − ) 2 2 1 2 1 7 3 − 3 6 2 2 5 2 − 9 9 9 4 1 1 1 2 (DS 1 4 ) − 9 9 9 1 2 2 2 1 − 9 9 9 −3 1 17 15 1 ... 1 2 1 3 5 (DS 1 5 1 ) 13 −3 2 0 1 1 2 1 (DS ) 3 3 0 3 −2 1 1 3 −9 11 5 1 3 5 1 (DS 2 7 −4 13 ) 41 1 4 1 19 5 6 2 ’ 3.4 Ma trˆn nghich dao a 12 7 1 2 −3 4 3 1 5 5 3 1 2 −3 0 5 1 1 4 3 −2 5 1 0 1 6 0 0 2 1 3 1 1 3 4 7 0 1 2 0 1 5 ` o o (DS Khˆng tˆn tai) 13 1 (DS − 17 25 1 6 −3 1 (DS −2 −2 6 0 3 15 12 10 ... ) 5 1 0 2 ) 0 3 11 2 1 (DS − 0 5 +2) 3 0 1 1 1 12 5 (DS 1 17 −7) 0 −2 1 1 0 0 0 0 1 1 1 1 √ −√ √ √ (DS 0 2 2 2 2 ) 1 1 1 1 √ √ −√ 0 √ 2 2 2 2 1 2 2 1 2 2 1 10 2 1 −2 (DS 2 1 −2) 9 2 −2 1 2 −2 1 1 1 1 2 2 2 1 1 1 3 11 3 1 −2 (DS 3 − − ) 2 2 3 1 0 1 1 0 − 2 2 3 8 1 2 1 ... 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 2 1 0 1 1 4 2 1 6 3 3 1 5 4 − 6 3 4 2 1 1 − − 6 3 3 ’ 3.4 Ma trˆn nghich dao a 12 3 h1 − h2 → h1 2 0 0 − − − − 0 3 0 −−−→ 0 0 4 h1 ( 1 ) → h1 1 0 0 2 −− − − −−−→ 0 1 0 h2 ( 1 ) → h2 3 1 h3 ( 4 ) → h4 0 0 1 2 1 0 1 2 1 0 1 1 1 0 1 − 2 1 2 1 − − 3 3 3 1 1 0 − 4 4 ` T` d´ suy r˘ng u o a 1 1 −... 1 −2 2 1) λ 3 0; 2 1 1 λ 2 0 2) 2 λ 1 0 1 λ √ 9 (DS 1) λ = ; 2) λ = 0, λ = ± 5) 4 ’ ı 24 T` ma trˆn X thoa m˜n c´c phu.o.ng tr`nh ım a a a 1) 2 1 1 1 X= 3 1 0 1 (DS 1 1 0 ) 5 −3 5 3 1 −2 1 = 2 1 3 1 (DS 5 2 ) 13 5 2) X 3) 2 1 3 1 1 1 6 −3 X = (DS ) 1 0 1 2 3 0 11 −2 4) AX + B = 2C, trong d´ o 1 1 2 1 1 1 A = 0 1 1 , B = 0 3 4 , −2 0 1 0 0 1 ... 5 7 1 2 −3 0 1 2 0 0 1 129 1 −3 11 −38 0 1 −2 7 (DS ) 0 0 1 −2 0 0 0 1 0 0 a22 0 , a11a12 · · · ann = 0 ann 0 1 0 0 a 11 1 0 0 a22 (DS ) 1 0 0 ann a 11 0 20 0 1 0 21 0 1 1 0 1 1 1 0 0 0 a a2 1 a 0 1 0 0 0 1 0 22 0 1 1 1 1 1. .. 0 1 0 2 5 2 1 0 2 − − − − 0 1 4 − − − → 0 2 5 h2 ( 1) →h2 1 0 2 −→ 0 1 4 0 0 −3 1 0 0 2 1 −→ 1 0 2 1 h − 2h2 → h3 0 1 3 2 1 0 0 2 1 −→ 1 0 2 1 1 h × (− 3 ) → h3 2 1 3 − 2 Chu.o.ng 3 Ma trˆn D nh th´.c a -i u 12 2 1 0 2 0 1 4 0 0 1 1 0 0 1 0 0 h − 2h3 → h1 2 1 1 1 0 − − − − − 0 1 0 −−− −→ 2 2 1 h2 − 4h3 → h2 1 − − 0 0 1 6 3 3... a 12 1 a v´.i ma trˆn o 2 0 4 1) A = 1 1 −2 ; 1 2 3 ’ Giai 1) Ta lˆp ma trˆn a a 2 0 4 M = 1 1 −2 1 2 3 2 3 4 2) A = 2 6 8 2 6 12 1 0 0 0 1 0 0 0 1 1 ´ a o Nhˆn h`ng th´ nhˆt v´.i ta thu du.o.c a a u 2 1 1 0 2 0 0 2 M −→ 1 1 −2 0 1 0 h2 − h1 → h2 −→ 1 2 3 0 0 1 h3 + h1 → h3 1 0 0 2 1 0 2 1 −→ 0 1 −4 1 0 − 2 1 0 1 0... trong d´ o 1 1 2 1 1 1 A = 0 1 1 , B = 0 3 4 , −2 0 1 0 0 1 5 16 −8 (DS 4 −7 5 ) 4 −2 1 2 3 0 C = 4 −3 5 1 1 0 o 5) XA − 2B = E, trong d´ 1 1 3 1 3 −2 A = −2 5 7 , B = 1 2 0 1 1 2 3 1 4 − 21 45 15 6 1 (DS − 21 15 − 21 ) 15 51 20 −79 ´ ´ ’ ’ e 25 Gia su A l` ma trˆn cˆp n v` (E + A)k = O v´.i sˆ tu nhiˆn k o o a a a a ` ´ n`o . 1 ) 17 . 1 100 10 11 0 011 1 110 .(D S. 0 11 0 11 10 10 01 1 01 1 ) 18 . 12 12 25 35 00 5 4 00 6 5 .(D S. 5 −2 54 − 21 11 9 0 05 4 00− 65 ) 3.4 = 13 2 05 2 6 9 712 −2 52 4 5 1 4 8420 ∼ 13 20 5 0 057 2 016 4 15 016 4 15 ∼ 13 20 5 016 4 15 0 057 2 016 4 15 ∼ 13 20 5 016 4 15 0 057 2 0000. th´u . c 12 . 15 2 14 1 12 1 .(D S. − 1 3 1 6 1 2 0 1 2 − 1 2 1 3 − 7 6 3 2 ) 13 . 14 1 11 4 12 2 .(D S. 2 9 2 9 − 5 9 2 9 − 1 9 1 9 − 1 9 2 9 1 9 ) 14 . 1 −3 1 14 1 19−2 .(D S. 17 15 1 11 0 13 12 1 ) 15 . 12