4.1. Hˆe . n phu . o . ng tr`ınh v´o . i n ˆa ’ nc´od i . nh th´u . c kh´ac 0 139 Gia ’ i. 1) Lˆa . p ma trˆa . nmo . ’ rˆo . ng v`a thu . . chiˆe . n c´ac ph´ep biˆe ´ nd ˆo ’ i:  A =    10−2   −3 −21 6   11 −15−4   −4    h 2 +2h 1 → h  2 h 3 + h 1 → h  3 −→    10−2   −3 01 2   5 05−6   −7    −→ h 3 − 5h 2 → h  3    10 −2   −3 01 2   5 00−16   −32    . T`u . d´o suy ra x 1 − 2x 3 = −3 x 2 +2x 3 =5 −16x 3 = −32      ⇒ x 1 =1,x 2 =1,x 3 =2. 2) Lˆa . p ma trˆa . nmo . ’ rˆo . ng v`a thu . . chiˆe . n c´ac ph´ep biˆe ´ ndˆo ’ iso . cˆa ´ p:      2 −13−1   9 11−24   −1 32−13   0 5 −21−2   9      h 1 → h  2 h 2 → h  1 −→      11−24   −1 2 −13−1   9 32−13   0 5 −21−2   9      −→ h 2 − 2h 1 → h  2 h 3 − 3h 1 → h  3 h 4 − 5h 1 → h  4      11−24   −1 0 −37 −9   11 0 −15 −9   3 0 −711−22   14      h 2 → h  3 h 3 → h  2 −→ 140 Chu . o . ng 4. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh −→      11−24   −1 0 −15 −9   3 0 −37 −9   11 0 −711−22   14      h 3 − 3h 2 → h  3 h 4 − 7h 2 → h  4 −→      11 −24   −1 0 −15−9   3 00 −818   2 00−24 41   −7      −→ h 4 − 3h 3 → h  4      11−24   −1 0 −15 −9   3 00−818   2 00 0−13   −13      T`u . d´o suy ra r˘a ` ng x 1 =1,x 2 = −2, x 3 =2,x 4 =1.  B ` AI T ˆ A . P Gia ’ i c´ac hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh sau 1. x 1 − x 2 +2x 3 =11, x 1 +2x 2 − x 3 =11, 4x 1 − 3x 2 − 3x 3 =24.      .(D S. x 1 =9,x 2 =2,x 3 =2) 2. x 1 − 3x 2 −4x 3 =4, 2x 1 + x 2 − 3x 3 = −1, 3x 1 −2x 2 + x 3 =11.      .(DS. x 1 =2,x 2 = −2, x 3 =1) 3. 2x 1 +3x 2 −x 3 =4, x 1 +2x 2 +2x 3 =5, 3x 1 +4x 2 − 5x 3 =2.      .(D S. x 1 = x 2 = x 3 =1) 4.1. Hˆe . n phu . o . ng tr`ınh v´o . i n ˆa ’ nc´od i . nh th´u . c kh´ac 0 141 4. x 1 +2x 2 + x 3 =8, −2x 1 +3x 2 − 3x 3 = −5, 3x 1 − 4x 2 +5x 3 =10.      .(D S. x 1 =1,x 2 =2,x 3 =3) 5. 2x 1 + x 2 − x 3 =0, 3x 2 +4x 3 = −6, x 1 + x 3 =1.      .(D S. x 1 =1,x 2 = −2, x 3 =0) 6. 2x 1 − 3x 2 − x 3 +6 =0, 3x 1 +4x 2 +3x 3 +5 =0, x 1 + x 2 + x 3 +2 =0.      .(D S. x 1 = −2, x 2 =1,x 3 = −1) 7. x 2 +3x 3 +6 =0, x 1 − 2x 2 − x 3 =5, 3x 1 +4x 2 − 2x =13.      .(D S. x 1 =3,x 2 =0,x 3 = −2) 8. 2x 1 − x 2 + x 3 +2x 4 =5, x 1 +3x 2 − x 3 +5x 4 =4, 5x 1 +4x 2 +3x 3 =2, 3x 1 − 3x 2 − x 3 −6x 4 = −6.          . (D S. x 1 = 1 3 , x 2 = − 2 3 , x 3 =1,x 4 = 4 3 ) 9. x 1 − 2x 2 +3x 3 − x 4 = −8, 2x 1 +3x 2 − x 3 +5x 4 =19, 4x 1 − x 2 + x 3 + x 4 = −1, 3x 1 +2x 2 − x 3 − 2x 4 = −2.          . (D S. x 1 = − 1 2 , x 2 = 3 2 , x 3 = − 1 2 , x 4 =3) 10. x 1 − x 3 + x 4 =3, 2x 1 +3x 2 − x 3 − x 4 =2, 5x 1 − 3x 4 = −6 x 1 + x 2 + x 3 + x 4 =2.          . (D S. x 1 =0,x 2 =1,x 3 = −1, x 4 =2) 142 Chu . o . ng 4. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh 11. 2x 1 +3x 2 +8x 4 =0, x 2 −x 3 +3x 4 =0, x 3 +2x 4 =1, x 1 + x 4 = −24          . (D S. x 1 = −19, x 2 = 26, x 3 = 11, x 4 = −5) 12. 3x 1 + x 2 − x 3 + x 4 =0, 2x 1 +3x 2 − x 4 =0, x 1 +5x 2 − 3x 3 =7, 3x 2 +2x 3 + x 4 =2,          . (D S. x 1 = −1, x 2 =1,x 3 = −1, x 4 =1) 13. x 1 − 2x 2 + x 3 − 4x 4 − x 5 =13, x 1 +2x 2 +3x 3 − 5x 4 =15, x 2 − 2x 3 + x 4 +3x 5 = −7, x 1 − 7x 3 +8x 4 − x 5 = −30, 3x 1 −x 2 − 5x 5 =4.                . (D S. x 1 =1,x 2 = −1, x 3 =2,x 4 = −2, x 5 =0) 14. x 1 + x 2 +4x 3 + x 4 − x 5 =2, x 1 − 2x 2 − 2x 3 +3x 5 =0, 4x 2 +3x 3 − 2x 4 +2x 5 =2, 2x 1 −x 3 +3x 4 − 2x 5 = −2, 3x 1 +2x 2 − 5x 4 +3x 5 =3.                . (D S. x 1 = 2 5 , x 2 = − 3 5 , x 3 = 4 5 , x 4 =0,x 5 =0) 4.2. Hˆe . t`uy ´y c´ac phu . o . ng tr`ınh tuyˆe ´ n t´ınh 143 4.2 Hˆe . t`uy ´y c´ac phu . o . ng tr`ınh tuyˆe ´ n t´ınh Tax´ethˆe . t`uy ´y c´ac phu . o . ng tr`ınh tuyˆe ´ n t´ınh gˆo ` m m phu . o . ng tr`ınh v´o . i n ˆa ’ n a 11 x 1 + a 12 x 2 + ···+ a 1n x n = b 1 , a 21 x 1 + a 22 x 2 + ···+ a 2n x n = b 2 , a m1 x 1 + a m2 x 2 + ···+ a mn x n = b m ,          (4.9) v´o . i ma trˆa . nco . ba ’ n A =    a 11 a 12 a 1n a m1 a m2 a mn    v`a ma trˆa . nmo . ’ rˆo . ng  A =    a 11 a 12 a 1n   b 1   a m1 a m2 a mn   b m    Hiˆe ’ n nhiˆen r˘a ` ng r(A)  r(  A)v`ımˆo ˜ id i . nh th´u . c con cu ’ a A d ˆe ` ul`adi . nh th ´u . c con cu ’ a  A nhu . ng khˆong c´o diˆe ` u ngu . o . . cla . i. Ta luˆon luˆon gia ’ thiˆe ´ t r˘a ` ng c´ac phˆa ` ntu . ’ cu ’ a ma trˆa . n A khˆong dˆo ` ng th`o . ib˘a ` ng 0 tˆa ´ tca ’ . Ngu . `o . i ta quy u . ´o . cgo . idi . nh th´u . c con kh´ac 0 cu ’ amˆo . t ma trˆa . nm`a cˆa ´ pcu ’ an´ob˘a ` ng ha . ng cu ’ a ma trˆa . nd´ol`adi . nh th´u . c con co . so . ’ cu ’ a n´o. Gia ’ su . ’ d ˆo ´ iv´o . i ma trˆa . nd ˜a cho ta d˜acho . nmˆo . tdi . nh th´u . c con co . so . ’ . Khi d ´o c´ac h`ang v`a c´ac cˆo . t m`a giao cu ’ ach´ung lˆa . p th`anh di . nh th´u . c con co . so . ’ d´odu . o . . cgo . il`ah`ang, cˆo . tco . so . ’ . D - i . nh ngh˜ıa. 1 + Bˆo . c´o th´u . tu . . n sˆo ´ (α 1 ,α 2 , ,α n )du . o . . cgo . i l`a nghiˆe . m cu ’ ahˆe . (4.9) nˆe ´ u khi thay x = α 1 ,x= α 2 , ,x= α n v`ao c´ac phu . o . ng tr`ınh cu ’ a (4.9) th`ı hai vˆe ´ cu ’ amˆo ˜ iphu . o . ng tr`ınh cu ’ a (4.9) tro . ’ th`anh d ˆo ` ng nhˆa ´ t. 144 Chu . o . ng 4. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh 2+ Hˆe . (4.9) du . o . . cgo . il`atu . o . ng th´ıch nˆe ´ u c´o ´ıt nhˆa ´ tmˆo . t nghiˆe . mv`a go . il`akhˆong tu . o . ng th´ıch nˆe ´ u n´o vˆo nghiˆe . m. 3 + Hˆe . tu . o . ng th´ıch d u . o . . cgo . il`ahˆe . x´ac d i . nh nˆe ´ u n´o c´o nghiˆe . m duy nhˆa ´ t v`a go . il`ahˆe . vˆo d i . nh nˆe ´ u n´o c´o nhiˆe ` uho . nmˆo . t nghiˆe . m. D - i . nh l´y Kronecker-Capelli. 2 Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh (4.9) tu . o . ng th´ıch khi v`a chı ’ khi ha . ng cu ’ a ma trˆa . nco . ba ’ nb˘a ` ng ha . ng cu ’ a ma trˆa . nmo . ’ rˆo . ng cu ’ ahˆe . ,t´u . cl`ar(A)=r(  A). D ˆo ´ iv´o . ihˆe . tu . o . ng th´ıch ngu . `o . itago . i c´ac ˆa ’ nm`ahˆe . sˆo ´ cu ’ ach´ung lˆa . p nˆen di . nh th´u . c con co . so . ’ cu ’ a ma trˆa . nco . ba ’ nl`aˆa ’ nco . so . ’ , c´ac ˆa ’ n c`on la . id u . o . . cgo . il`aˆa ’ ntu . . do. Phu . o . ng ph´ap chu ’ yˆe ´ udˆe ’ gia ’ ihˆe . tˆo ’ ng qu´at l`a: 1. ´ Ap du . ng quy t˘a ´ c Kronecker-Capelli. 2. Phu . o . ng ph´ap khu . ’ dˆa ` nc´acˆa ’ n (phu . o . ng ph´ap Gauss). Quy t˘a ´ c Kronecker-Capelli gˆo ` m c´ac bu . ´o . c sau. 1 + Kha ’ o s´at t´ınh tu . o . ng th´ıch cu ’ ahˆe . . T´ınh ha . ng r(  A)v`ar(A) a) Nˆe ´ u r(  A) >r(A)th`ıhˆe . khˆong tu . o . ng th´ıch. b) Nˆe ´ u r(  A)=r(A)=r th`ı hˆe . tu . o . ng th´ıch. T`ım di . nh th´u . c con co . so . ’ cˆa ´ p r n`ao d´o (v`a do vˆa . y r ˆa ’ nco . so . ’ tu . o . ng ´u . ng xem nhu . du . o . . c cho . n) v`a thu du . o . . chˆe . phu . o . ng tr`ınh tu . o . ng du . o . ng gˆo ` m r phu . o . ng tr`ınh v´o . i n ˆa ’ nm`a(r ×n)-ma trˆa . nhˆe . sˆo ´ cu ’ an´och´u . a c´ac phˆa ` ntu . ’ cu ’ ad i . nh th ´u . c con co . so . ’ d˜a c h o . n. C´ac phu . o . ng tr`ınh c`on la . i c´o thˆe ’ bo ’ qua. 2 + T`ım nghiˆe . mcu ’ ahˆe . tu . o . ng d u . o . ng thu d u . o . . c a) Nˆe ´ u r = n, ngh˜ıa l`a sˆo ´ ˆa ’ nco . so . ’ b˘a ` ng sˆo ´ ˆa ’ ncu ’ ahˆe . th`ı hˆe . c´o nghiˆe . m duy nhˆa ´ t v`a c´o thˆe ’ t`ım theo cˆong th´u . c Cramer. b) Nˆe ´ u r<n, ngh˜ıa l`a sˆo ´ ˆa ’ nco . so . ’ b´e ho . nsˆo ´ ˆa ’ ncu ’ ahˆe . th`ı ta chuyˆe ’ n n − r sˆo ´ ha . ng c´o ch´u . aˆa ’ ntu . . do cu ’ a c´ac phu . o . ng tr`ınh sang vˆe ´ pha ’ idˆe ’ thu du . o . . chˆe . Cramer dˆo ´ iv´o . i c´ac ˆa ’ nco . so . ’ . Gia ’ ihˆe . n`ay ta thu du . o . . c c´ac biˆe ’ uth´u . ccu ’ a c´ac ˆa ’ nco . so . ’ biˆe ’ udiˆe ˜ n qua c´ac ˆa ’ ntu . . do. 2 L. Kronecker (1823-1891) l`a nh`a to´an ho . cD´u . c, A. Capelli (1855-1910) l`a nh`a to´an ho . c Italia. 4.2. Hˆe . t`uy ´y c´ac phu . o . ng tr`ınh tuyˆe ´ n t´ınh 145 D´o l`a nghiˆe . mtˆo ’ ng qu´at cu ’ ahˆe . . Cho n −r ˆa ’ ntu . . do nh˜u . ng gi´a tri . cu . thˆe ’ t`uy ´y ta t`ım d u . o . . c c´ac gi´a tri . tu . o . ng ´u . ng cu ’ aˆa ’ nco . so . ’ .T`u . d ´o t h u du . o . . c nghiˆe . m riˆeng cu ’ ahˆe . . Tiˆe ´ p theo ta tr`ınh b`ay nˆo . i dung cu ’ aphu . o . ng ph´ap Gauss. Khˆong gia ’ mtˆo ’ ng qu´at, c´o thˆe ’ cho r˘a ` ng a 11 = 0. Nˆo . i dung cu ’ a phu . o . ng ph´ap Gauss l`a nhu . sau. 1 + Thu . . chiˆe . n c´ac ph´ep biˆe ´ ndˆo ’ iso . cˆa ´ p trˆen c´ac phu . o . ng tr`ınh cu ’ a hˆe . dˆe ’ thu du . o . . chˆe . tu . o . ng du . o . ng m`a b˘a ´ tdˆa ` ut`u . phu . o . ng tr`ınh th´u . hai mo . iphu . o . ng tr`ınh d ˆe ` u khˆong ch´u . aˆa ’ n x 1 .K´yhiˆe . uhˆe . n`ay l`a S (1) . 2 + C˜ung khˆong mˆa ´ ttˆo ’ ng qu´at, c´o thˆe ’ cho r˘a ` ng a  22 = 0. La . i thu . . c hiˆe . n c´ac ph´ep biˆe ´ ndˆo ’ iso . cˆa ´ p trˆen c´ac phu . o . ng tr`ınh cu ’ ahˆe . S (1) (tr `u . ra phu . o . ng tr`ınh th´u . nhˆa ´ tdu . o . . cgi˜u . nguyˆen!) nhu . d˜a l`am trong bu . ´o . c 1 + ta thu du . o . . chˆe . tu . o . ng du . o . ng m`a b˘a ´ tdˆa ` ut`u . phu . o . ng tr`ınh th´u . ba mo . iphu . o . ng tr`ınh d ˆe ` u khˆong ch´u . aˆa ’ n x 2 , 3 + Sau mˆo . tsˆo ´ bu . ´o . ctac´othˆe ’ g˘a . pmˆo . t trong c´ac tru . `o . ng ho . . p sau dˆa y . a) Thˆa ´ yngaydu . o . . chˆe . khˆong tu . o . ng th´ıch. b) Thu du . o . . cmˆo . thˆe . “tam gi´ac”. Hˆe . n`ay c´o nghiˆe . m duy nhˆa ´ t. c) Thu d u . o . . cmˆo . t“hˆe . h`ınh thang” da . ng a 11 x 1 + a 12 x 2 + + a 1n x n = h 1 , b 22 x 2 + + b 2n x n = h 2 , b rr x r + ···+ b rn x n = h r , 0=h r+1 , 0= h m .                          Nˆe ´ u c´ac sˆo ´ h r+1 , ,h m kh´ac 0 th`ı hˆe . vˆo nghiˆe . m. Nˆe ´ u h r+1 = ··· = h m =0th`ıhˆe . c´o nghiˆe . m. Cho x r+1 = α, ,x m = β th`ı thu du . o . . chˆe . Cramer v´o . iˆa ’ nl`ax 1 , ,x r . Gia ’ ihˆe . d´o ta thu du . o . . c 146 Chu . o . ng 4. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh nghiˆe . m x 1 = x 1 ; x 2 = x 2 , ,x r = x r v`a nghiˆe . mcu ’ ahˆe . d˜achol`a (x 1 , x 2 , ,x r ,α, ,β). Lu . u´yr˘a ` ng viˆe . c gia ’ ihˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh b˘a ` ng phu . o . ng ph´ap Gauss thu . . cchˆa ´ t l`a thu . . chiˆe . n c´ac ph´ep biˆe ´ nd ˆo ’ iso . cˆa ´ p trˆen c´ac h`ang cu ’ a ma trˆa . nmo . ’ rˆo . ng cu ’ ahˆe . du . an´ovˆe ` da . ng tam gi´ac hay da . ng h`ınh thang. C ´ AC V ´ IDU . V´ı d u . 1. Gia ’ ihˆe . phu . o . ng tr`ınh 3x 1 − x 2 + x 3 =6, x 1 −5x 2 + x 3 =12, 2x 1 +4x 2 = −6, 2x 1 + x 2 +3x 3 =3, 5x 1 +4x 3 =9.                Gia ’ i. 1. T`ım ha . ng cu ’ a c´ac ma trˆa . n A =         3 −11 1 −51 240 213 504         ,  A =         3 −11   6 1 −51   12 240   −6 213   3 504   9         Ta thu d u . o . . c r(  A)=r(A) = 3. Do d´ohˆe . tu . o . ng th´ıch. Ta cho . nd i . nh th´u . c con co . so . ’ l`a ∆=        1 −51 240 213        v`ı∆=36=0v`ar(A) = 3 v`a c´ac ˆa ’ nco . so . ’ l`a x 1 ,x 2 ,x 3 . 4.2. Hˆe . t`uy ´y c´ac phu . o . ng tr`ınh tuyˆe ´ n t´ınh 147 2. Hˆe . phu . o . ng tr`ınh d ˜a cho tu . o . ng d u . o . ng v´o . ihˆe . x 1 − 5x 2 + x 3 =12, 2x 1 +4x 2 = −6, 2x 1 + x 2 +3x 3 =3.      Sˆo ´ ˆa ’ nco . so . ’ b˘a ` ng sˆo ´ ˆa ’ ncu ’ ahˆe . nˆen hˆe . c´o nghiˆe . m duy nhˆa ´ tl`ax 1 =1, x 2 = −2, x 4 =1.  V´ı du . 2. Gia ’ ihˆe . phu . o . ng tr`ınh x 1 +2x 2 − 3x 3 +4x 4 =7, 2x 1 +4x 2 +5x 3 − x 4 =2, 5x 1 +10x 2 +7x 3 +2x 4 =11.      Gia ’ i. T`ım ha . ng cu ’ a c´ac ma trˆa . n A =    12−34 24 5 −1 510 7 2    ,  A =    12−34   7 24 5 −1   2 510 7 2   11    Tathud u . o . . c r(  A)=r(A) = 2. Do d´ohˆe . tu . o . ng th´ıch. Ta c´o thˆe ’ lˆa ´ ydi . nh th´u . c con co . so . ’ l`a ∆=      2 −3 45      v`ı∆=22= 0 v`a cˆa ´ pcu ’ ad i . nh th´u . c=r(A) = 2. Khi cho . n ∆ l`am di . nh th´u . c con, ta c´o x 2 v`a x 3 l `a ˆa ’ nco . so . ’ . Hˆe . d˜a cho tu . o . ng du . o . ng v´o . ihˆe . x 1 +2x 2 − 3x 3 +4x 4 =7, 2x 1 +4x 2 +5x 3 − x 4 =2 hay 2x 2 − 3x 3 =7−x 1 − 4x 4 , 4x 2 +5x 3 =2−2x 1 + x 4 . 148 Chu . o . ng 4. Hˆe . phu . o . ng tr`ınh tuyˆe ´ n t´ınh 2. Ta c´o thˆe ’ gia ’ ihˆe . theo quy t˘a ´ c Cramer. D˘a . t x 1 = α, x 4 = β ta c´o 2x 2 −3x 3 =7−α − 4β, 4x 2 +5x 3 =2−2α + β. Theo cˆong th´u . c Cramer ta t`ım du . o . . c x 2 =      7 −α − 4β −3 2 −2α + β 5      22 = 41 −11α −17β 22 , x 3 =      27− α −4β 42−2α + β      22 = −24 + 18β 22 · Do d´otˆa . pho . . p c´ac nghiˆe . mcu ’ ahˆe . c´o da . ng  α; 41 −11α −17β 22 ; 9β − 12 11 ; β   ∀α,β ∈ R   V´ı d u . 3. B˘a ` ng phu . o . ng ph´ap Gauss h˜ay gia ’ ihˆe . phu . o . ng tr`ınh 4x 1 +2x 2 + x 3 =7, x 1 − x 2 + x 3 = −2, 2x 1 +3x 2 −3x 3 =11, 4x 1 + x 2 −x 3 =7.          Gia ’ i. Trong hˆe . d ˜a cho ta c´o a 11 =4=0nˆen dˆe ’ cho tiˆe . ntadˆo ’ ichˆo ˜ hai phu . o . ng tr`ınh d ˆa ` u v`a thu du . o . . chˆe . tu . o . ng d u . o . ng x 1 − x 2 + x 3 = −2, 4x 1 +2x 2 + x 3 =7, 2x 1 +3x 2 −3x 3 =11, 4x 1 + x 2 −x 3 =7.          [...]... trˆn mo rˆng e e o a   1 1 1 −2 4 2 1 7  h2 − 4h1 → h2   A=  2 3 −3 11  h3 − 2h1 → h3 7 h4 − 4h1 → h4 4 1 1   −2 1 1 1 0 6 −3 15    −→  →  0 5 −5 15  0 5 −5 15 h4 − h3 → h4   1 1 1 −2 0 6 −3 15  h2 × 5 → h2   −→ −→   0 5 −5 15  h3 × 6 → h3 0 0 0 0    −2 h3 − h2 → h3 1 1 1 1 1 1 0 30 15 0 30 15 75     −→   −→  0 30 −30 0 0 15 90  0 0 0 0 0 0 0 T`... thˆ l` t` r(A) khi ı e e e e u a a e a ım 21 λ = Ta c´ o 2  21 1 2 3 h1 × 2 → h   1 2   21 A = 3 1 − 2  h2 × 2 → h2 −→   2 µ 2 1 3   2 4 21 6   −→ 6 −2 − 21 4  h2 − 3h1 → h2 −→ µ h3 − h1 → h3 2 1 3   6 2 4 21 1   −→ 0 14 −84 14  h2 × → h2 −→ 14 0 −3 18 µ 6   2 4 21  −→ 0 1 6 0 −3 18   2 4 21 6   −→ 0 1 6 1  µ − 6 h3 + 3h1 → h3 0 0 0 ´ ´ ’ ’ e o T` kˆt qua biˆn dˆi... Dx2 , Dx3 : Dx1 1 1 1 = λ λ 1 = −(λ − 1) 2 (λ + 1) , λ2 1 λ Dx 2 λ 1 1 = 1 λ 1 = (λ − 1) 2 , 1 λ2 λ λ 1 1 Dx3 = 1 λ λ = (λ − 1) 2 (λ + 1) 2 1 1 λ2 o u T` d´ theo cˆng th´.c Cramer ta thu du.o.c u o λ +1 1 (λ + 1) 2 , x2 = , x3 = · x1 = − λ+2 λ+2 λ+2 Ta c`n x´t gi´ tri λ = 1 v` λ = −2 o e a a ’ a Khi λ = 1 hˆ d˜ cho tro th`nh e a  x1 + x2 + x3 = 1,   x1 + x2 + x3 = 1,   x1 + x2 + x3 = 1 ´ ´ ´ Hˆ n`y... e u ´ a ınh e ınh 3 x1 + x2 + 2x3 + x4 = 1, x1 − 2x2 − x4 = −2 1 (DS x3 = (−2x1 + x2 − 1) , x4 = x1 − 2x2 + 2, 2 u ´ x1 , x2 t`y y)  x1 + 5x2 + 4x3 + 3x4 = 1,   4 2x1 − x2 + 2x3 − x4 = 0,   5x1 + 3x2 + 8x3 + x4 = 1 14 2 1 6 7 2 , x2 = − x3 − x4 + , (DS x1 = − x3 + x4 + 11 11 11 11 11 11 u ´ x3 , x4 t`y y)  3x1 + 5x2 + 2x3 + 4x4 = 3,  5 2x1 + 3x2 + 4x3 + 5x4 = 1,   5x1 + 9x2 − 2x3 + 2x4 = 9... 1, x2 = 2, x3 = −2) 29  2x1 + 3x2 + 4x3 + 3x4 = 0,   4x1 + 6x2 + 9x3 + 8x4 = −3,   6x1 + 9x2 + 9x3 + 4x4 = 8 7 3x2 − , x3 = 1, x4 = 1, x2 t`y y) u ´ 2 2  3x1 + 3x2 − 6x3 − 2x4 = 1,      = −2, 6x1 + x2 − 2x4  6x1 − 7x2 + 21x3 + 4x4 = 3,   = 3,  9x1 + 4x2 + 2x4    12 x1 − 6x2 + 21x3 + 2x4 = 1  (DS x1 = 30 (DS x1 = 7 11 16 , x2 = −4, x3 = − , x4 = ) 5 5 5 16 1 ´ Chu.o.ng 4 Hˆ phu.o.ng...   −x1 + 2x2 − 3x3 + 4x4 = 10 , x2 − x3 + x4 = 3,     = 10 x1 + x2 + x3 + x4 (DS x1 = 11 (DS x1 = 1, x2 = 2, x3 = 3, x4 = 4) 12 5x1 + x2 − 3x3 2x1 − 5x2 + 7x3 4x1 + 2x2 − 4x3 5x1 − 2x2 + 2x3  = 6,    = 9,  = −7,    = 1 1 1 3 (DS x1 = − , x2 = , x3 = ) 3 6 2 13 x1 − x2 + x3 − x4 x1 + x2 + 2x3 + 3x4 2x1 + 4x2 + 5x3 + 10 x4 2x1 − 4x2 + x3 − 6x4  = 4,    = 8,  = 20,    = 4 3 1 a u... tu  2x1 − x2 + 3x3 = 3,    3x1 + x2 − 5x3 = 0,  26 4x1 − x2 + x4 = 3,     x1 + 3x2 − 13 x3 = 6 (DS x1 = 1, x2 = 2, x3 = 1) 27 2x1 − x2 + x3 − x4 2x1 − x2 − 3x4 3x1 − x3 + x4 2x1 + 2x2 − 2x3 + 5x4  = 1,    = 2,  = −3,    = 6 5 4 , x4 = − ) 3 3  = 2,    = 5,  = −7,    = 14 (DS x1 = 0, x2 = 2, x3 = 28 2x1 + x2 + x3 x1 + 3x2 + x3 x1 + x2 + 5x3 2x1 + 3x2 − 5x3 (DS x1 = 1, x2 =... nghiˆm) 6 x1 + 2x2 + 3x3 3x1 + 2x2 + x3 x1 + x2 + x3 2x1 + 3x2 − x3 x1 + x2  = 14 ,     = 10 ,  = 6,   = 5,     = 3  (DS x1 = 1, x2 = 2, x3 = 3) 7  x1 + 3x2 − 2x3 + x4 + x5 = 1,   x1 + 3x2 − x3 + 3x4 + 2x5 = 3,   = 2 x1 + 3x2 − 3x3 − x4 e o e (DS Hˆ vˆ nghiˆm) 8 5x1 + x2 − 3x3 2x1 − 5x2 + 7x3 4x1 + 2x2 − 4x3 5x1 − 2x2 + 2x3  = 6,    = 9,  = −7,    = 1 1 1 3 (DS x1 = − ,... = 21 , 2 r(A) = r(A) = 2 ⇔ µ = 3 Khi d´ hˆ d˜ cho tu.o.ng du.o.ng v´.i hˆ o e a o e 2x1 + 4x2 = 6 − 21 , α = x3, 6x1 − 2x2 = 4 + 21 3 ’ o a v` nghiˆm cua n´ l` 1 + α, 1 − 6 , α ∀ α ∈ R a e 2 ` ˆ BAI TAP ´ ’ a e Giai c´c hˆ phu.o.ng tr`nh tuyˆn t´ ı e ınh 1 6x1 + 3x2 + 4x3 = 3; 3x1 − x2 + 2x3 = 5 7 18 − 15 x1 (DS x2 = − , x3 = , x1 t`y y) u ´ 5 10 2 x1 − x2 + x3 = 1, 2x1 + x2 − x3 = 5 (DS x1... = 8 x1 − 2x2 + 3x3 − 4x4 3x1 + 3x2 − 5x3 + x4 −2x1 + x2 + 2x3 − 3x4 3x1 + 3x3 − 10 x4 13 7 , x4 = − ) 9 18  = 1,    = 1,  = 5,     = 4 = u ´ (DS x1 = 17 x3 + 29x4 + 5, x2 = 10 x3 − 17 x4 − 2, x3, x4 t`y y)  x1 − 5x2 − 8x3 + x4 = 3,    3x1 + x2 − 3x3 − 5x4 = 1,  17 = −5, x1 − 7x3 + 2x4    11 x2 + 20x3 − 9x4 = 2 e o e (DS Hˆ vˆ nghiˆm)   x2 − 3x3 + 4x4   x − 2x3 + 3x4 1 18 3x1 + . x 4 =0, 5x 1 +3x 2 +8x 3 + x 4 =1.      (D S. x 1 = − 14 11 x 3 + 2 11 x 4 + 1 11 , x 2 = − 6 11 x 3 − 7 11 x 4 + 2 11 , x 3 ,x 4 t`uy ´y) 5. 3x 1 +5x 2 +2x 3 +4x 4 =3, 2x 1 +3x 2 +4x 3 +5x 4 =1, 5x 1 +9x 2 −. 15 3 D x 1 , D x 2 , D x 3 : D x 1 =        11 1 λ 1 λ 2 1 λ        = −(λ 1) 2 (λ +1) , D x 2 =        λ 11 1 λ 1 1 λ 2 λ        =(λ − 1) 2 , D x 3 =        λ 11 1. =      1 11   −2 42 1   7 23−3   11 41 1   7      h 2 − 4h 1 → h  2 h 3 − 2h 1 → h  3 h 4 − 4h 1 → h  4 −→      1 11   −2 06 3   15 05−5   15 05−5   15      h 4 −