Obernena matricya matricya poznachayetsya A 1 displaystyle A 1 yaka isnuye dlya kozhnoyi nevirodzhenoyi kvadratnoyi matrici A displaystyle A rozmirnosti n n displaystyle n times n prichomu A A 1 A 1 A I n displaystyle AA 1 A 1 A I n de I n displaystyle I n odinichna n n displaystyle n times n matricya Yaksho dlya matrici A displaystyle A isnuye A 1 displaystyle A 1 to taka matricya nazivayetsya oborotnoyu tobto kozhna nevirodzhena matricya ye oborotnoyu i navpaki kozhna oborotna matricya ye nevirodzhenoyu Zmist 1 Vlastivosti 2 Znahodzhennya obernenoyi matrici 2 1 Tochni metodi 2 2 Iteracijni metodi 3 Prikladi 4 Div takozh 5 DzherelaVlastivosti Redaguvati A 1 1 A displaystyle mathbf A 1 1 mathbf A nbsp operaciya obernennya ye involyuciyeyu A T 1 A 1 T displaystyle mathbf A mathrm T 1 mathbf A 1 mathrm T nbsp obernennya transponovanoyi matrici A 1 A 1 displaystyle mathbf A 1 mathbf A 1 nbsp obernennya spryazhenoyi matrici k A 1 k 1 A 1 displaystyle k mathbf A 1 k 1 mathbf A 1 nbsp dlya dovilnogo koeficiyenta k 0 displaystyle k not 0 nbsp A B 1 B 1 A 1 displaystyle mathbf AB 1 mathbf B 1 mathbf A 1 nbsp det A 1 det A 1 displaystyle det mathbf A 1 det mathbf A 1 nbsp viznachnik obernenoyi matrici rang A n displaystyle operatorname rang A n nbsp rang matrici dorivnyuye rozmiru matrici Vlasni vektori matrici ta yiyi obernenoyi zbigayutsya a vlasni znachennya ye obernenimi Yaksho potribno rozv yazati sistemu linijnih rivnyan A x b displaystyle Ax b nbsp b nenulovij vektor i A 1 displaystyle A 1 nbsp isnuye todi x A 1 b displaystyle x A 1 b nbsp V protilezhnomu vipadku abo rozmirnist prostoru rozv yazkiv bilshe nulya abo yih nemaye zovsim Znahodzhennya obernenoyi matrici RedaguvatiTochni metodi Redaguvati Metod Gausa Zhordana LU rozklad matrici A 1 1 det A A displaystyle A 1 frac 1 det A tilde A nbsp de A displaystyle tilde A nbsp soyuzna matricya Iteracijni metodi Redaguvati Prikladi RedaguvatiA 1 a b c d 1 1 det A d b c a 1 a d b c d b c a displaystyle A 1 begin bmatrix a amp b c amp d end bmatrix 1 frac 1 det A begin bmatrix d amp b c amp a end bmatrix frac 1 ad bc begin bmatrix d amp b c amp a end bmatrix nbsp Obernena matricya isnuye todi i tilki todi koli a d b c det A 0 displaystyle ad bc det A neq 0 nbsp A 1 a b c d e f g h k 1 1 det A e k f h c h b k b f c e f g d k a k c g c d a f d h e g b g a h a e b d 1 a e k b f g c d h c e g b d k a f h e k f h c h b k b f c e f g d k a k c g c d a f d h e g b g a h a e b d displaystyle A 1 begin bmatrix a amp b amp c d amp e amp f g amp h amp k end bmatrix 1 frac 1 det A begin bmatrix ek fh amp ch bk amp bf ce fg dk amp ak cg amp cd af dh eg amp bg ah amp ae bd end bmatrix frac 1 aek bfg cdh ceg bdk afh begin bmatrix ek fh amp ch bk amp bf ce fg dk amp ak cg amp cd af dh eg amp bg ah amp ae bd end bmatrix nbsp Obernena matricya isnuye todi i tilki todi koli a e k b f g c d h c e g b d k a f h det A 0 displaystyle aek bfg cdh ceg bdk afh det A neq 0 nbsp Div takozh RedaguvatiTeoriya matric Psevdoobernena matricya Obernennya blochnoyi matriciDzherela RedaguvatiGantmaher F R Teoriya matric 5 e M Fizmatlit 2010 559 s ISBN 5 9221 0524 8 ros R Horn Ch Dzhonson Matrichnyj analiz M Mir 1989 653 s ros Obernena matricya Visha matematika v prikladah i zadachah Klepko V Yu Golec V L 2 ge vidannya K Centr uchbovoyi literaturi 2009 S 24 594 s nbsp Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Obernena matricya amp oldid 38770078
