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11η διάλεξη Γραμμικής Άλγεβρας

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Αντίστροφοι Πίνακες Θεωρήματα και Ασκήσεις

Text of 11η διάλεξη Γραμμικής Άλγεβρας

  • 1. Extash ProdouGrammik 'AlgebraAntistrofoi PnakecJewrmata kai AskseicTmma Hlektrolgwn Mhqanikn kai Mhqanikn UpologistnPanepistmio Jessalac17 Oktwbrou 2014

2. Extash ProdouGinmeno PinkwnKje stlh tou ginomnou twn do pinkwn A B isotai me toginmeno tou A me thn antstoiqh stlh tou B.A0... [email protected] [email protected] ......AB1 ABn......1CCAKje gramm tou ginomnou twn do pinkwn A B isotai me toginmeno thc antstoiqhc grammc tou A ep ton [email protected] A1 ... An 1CCCAB [email protected] A1B ... AnB 1CCCA 3. Extash ProdouOrismc antistrfouO antstrofoc enc pnaka A enai nac lloc pnakac B ttoiocsteAB BA IO antstrofoc sunjwc sumbolzetai me A1. 4. Extash ProdouAntstrofoc tou antstrofouJerhmaO antstrofoc tou antstrofou enc pnaka enai o dioc opnakac. Dhlad A11A.Apdeixh. 5. Extash ProdouAntstrofoc tou antstrofouJerhmaO antstrofoc tou antstrofou enc pnaka enai o dioc opnakac. Dhlad A11A.Apdeixh.AA1 A1A I. 6. Extash ProdouAntstrofoc ginomnouJerhmaO antstrofoc tou ginomnou do pinkwn isotai me to ginmeno,me antstrofh seir, twn antistrfwn touc. 7. Extash ProdouAntstrofoc ginomnouJerhmaO antstrofoc tou ginomnou do pinkwn isotai me to ginmeno,me antstrofh seir, twn antistrfwn touc. Dhlad(AB)1 B1A1.Apdeixh. 8. Extash ProdouAntstrofoc ginomnouJerhmaO antstrofoc tou ginomnou do pinkwn isotai me to ginmeno,me antstrofh seir, twn antistrfwn touc. Dhlad(AB)1 B1A1.Apdeixh.B1A1(AB) B1A1AB B1IB B1B I. 9. Extash ProdouAntstrofoc ginomnouJerhmaO antstrofoc tou ginomnou do pinkwn isotai me to ginmeno,me antstrofh seir, twn antistrfwn touc. Dhlad(AB)1 B1A1.Apdeixh.B1A1(AB) B1A1AB B1IB B1B I.(AB)B1A1ABB1A1 AIA1 AA1 I. 10. Extash ProdouMonadikthta antistrfouJerhmaAn uprqei o antstrofoc autc enai monadikc.Apdeixh. 11. Extash ProdouMonadikthta antistrfouJerhmaAn uprqei o antstrofoc autc enai monadikc.Apdeixh.'Estw ti uprqoun do antstrofoi tou A o B kai o C. TteB 12. Extash ProdouMonadikthta antistrfouJerhmaAn uprqei o antstrofoc autc enai monadikc.Apdeixh.'Estw ti uprqoun do antstrofoi tou A o B kai o C. TteB BI 13. Extash ProdouMonadikthta antistrfouJerhmaAn uprqei o antstrofoc autc enai monadikc.Apdeixh.'Estw ti uprqoun do antstrofoi tou A o B kai o C. TteB BI B(AC) 14. Extash ProdouMonadikthta antistrfouJerhmaAn uprqei o antstrofoc autc enai monadikc.Apdeixh.'Estw ti uprqoun do antstrofoi tou A o B kai o C. TteB BI B(AC) (BA)C 15. Extash ProdouMonadikthta antistrfouJerhmaAn uprqei o antstrofoc autc enai monadikc.Apdeixh.'Estw ti uprqoun do antstrofoi tou A o B kai o C. TteB BI B(AC) (BA)C IC C. 16. Extash ProdouAntstrofoc kai lseicJerhmaAn uprqei o antstrofoc enc pnaka A tte uprqei monadik lsh tou sustmatoc Ax b giaopoiodpote b 17. Extash ProdouAntstrofoc kai lseicJerhmaAn uprqei o antstrofoc enc pnaka A tte uprqei monadik lsh tou sustmatoc Ax b giaopoiodpote b kai h mnh lsh tou omogenoc sustmatoc enai h mhdenik.Apdeixh.Ax b 18. Extash ProdouAntstrofoc kai lseicJerhmaAn uprqei o antstrofoc enc pnaka A tte uprqei monadik lsh tou sustmatoc Ax b giaopoiodpote b kai h mnh lsh tou omogenoc sustmatoc enai h mhdenik.Apdeixh.Ax b)A1Ax A1b 19. Extash ProdouAntstrofoc kai lseicJerhmaAn uprqei o antstrofoc enc pnaka A tte uprqei monadik lsh tou sustmatoc Ax b giaopoiodpote b kai h mnh lsh tou omogenoc sustmatoc enai h mhdenik.Apdeixh.Ax b)A1Ax A1b)x A1b. 20. Extash Prodou'Uparxh antistrfouJerhmaO antstrofoc enc pnaka A uprqei ann la ta odhg stoiqeamet thn apaloif me odghsh tou A enai mh mhdenik.Apdeixh.Gia na uprqei prpei na mporome na upologsoume lec ticstlec tou.Prpei dhlad ta sustmata Avj ej gia j 1,2, . . . ,n na qounla lsh. 21. Extash ProdouAntstrofoc trigwnikoJerhmaO antstrofoc enc nw(ktw) trigwniko pnaka enainw(ktw) trigwnikc pnakac.Apdeixh.Ekolh all jlei ton qrno thc kai enai baret. 22. Extash Prodou'AskhshEA 24351 0 00 1 0 0 12435 ??2 2 40 1 32 7 4A)2435 B)2 2 42 17 342 7 4242 2 40 1 322 27 4435G)2435 D)2 2 420 1 32 7 4724352 2 40 1 32 7 43 23. Extash Prodou'AskhshO antstrofoc tou pnaka1 32 4enai o2 321 12. 24. Extash Prodou'AskhshO antstrofoc tou pnaka1 32 4enai o2 321 12.Poi enai h lsh tou sustmatoc2x1 4x2 2x1 3x2 1 25. Extash Prodou'AskhshO antstrofoc tou pnaka1 32 4enai o2 321 12.Poi enai h lsh tou sustmatoc2x1 4x2 2x1 3x2 1A)1 23 1B)10G)03D) 12 00 1 26. Extash Prodou'AskhshO antstrofoc tou pnaka1 32 4enai o2 321 12.Poi enai h lsh tou sustmatoc2x1 4x2 2x1 3x2 1A)1 23 1B)10G)03D) 12 00 1Dikaiologste thn apnths sac 27. Extash Prodou'AskhshO antstrofoc tou pnaka1 32 4enai o2 321 12.Poi enai h lsh tou sustmatoc2x1 4x2 2x1 3x2 1A)1 23 1B)10G)03D) 12 00 1Dikaiologste thn apnths sacApnthsh: To ssthma se morf pinkwn1 32 4x 12ralsh enai h B):2 321 121210 28. Extash Prodou'AskhshApodexte ti gia kje antistryimo pnaka A gia kjepragmatik arijm r6 0 isqei(rA)1 1rA1 29. Extash Prodou'AskhshApodexte ti gia kje antistryimo pnaka A gia kjepragmatik arijm r6 0 isqei(rA)1 1rA1(1rA1)rA (r(1rA1))AA1A I 30. Extash Prodou'AskhshEnai o pnakacA241 2 31 2 41 2 535Antistryimoc?A Nai.B 'Oqi.G 'Iswc.D Ta qw qamna. 31. Extash Prodou'AskhshEnai o pnakacB 24351 1 12 2 23 4 5antistryimoc?A Nai.B 'Oqi.G 'Iswc. 32. Extash Prodou'AskhshAn gnwrzoume ti to ssthma241 1 12 1 03 4 535x 2435000qei san lsh mnon thn x ~0 ti isqei gia to ssthma241 1 12 1 03 4 535x 2435?173 33. Extash Prodou'AskhshAn gnwrzoume ti to ssthma241 1 12 1 03 4 535x 2435000qei san lsh mnon thn x ~0 ti isqei gia to ssthma241 1 12 1 03 4 535x 2435?173A Uprqei toulqiston ma lsh x.B Uprqei to pol mia lsh x.G Kai ta do apo ta parapnwD Tpote apo ta parapnw. 34. Extash Prodou'AskhshH isthta (AB)T AT BT isqeiA Gia kje zegoc nn pinkwn A kai B.B Gia kanna zegoc nn pinkwn A kai B.G Gia merik mnon zegh nn pinkwn A kai B en gia lladen isqei 35. Extash Prodou'AskhshH isthta (AB)1 A1 B1 isqeiA Gia kje zegoc nn antistryimwn pinkwn A kai B.B Gia kanna zegoc nn antistryimwn pinkwn A kai B.G Gia merik mnon zegh nn antistryimwn pinkwn A kai Ben gia lla den isqei