역행렬

역행렬(inverse (matrix))이란 행렬의 곱에 대한 역원으로, 이것이 존재할 때 그 행렬을 가역행렬(invertible matrix) 또는 정칙행렬(nonsingular)이라 하고, A의 역행렬은 A^-1으로 나타낸다.

공식

역행렬의 공식은 모두 다음과 같다.

[math]\displaystyle{ A^{-1} = \frac{1}{\det A} \operatorname{adj}A }[/math]

여기서 adj A는 A의 수반행렬, det A는 A의 행렬식을 말한다.

이차 정사각행렬

[math]\displaystyle{ \displaystyle A_{2\times 2}^{-1}=\begin{bmatrix}a & b\\ c & d\end{bmatrix}^{-1} = \frac{1}{\det A}\operatorname{adj} A=\frac{1}{ad-bc} \begin{bmatrix}d & -b\\ -c & a\end{bmatrix} }[/math]

삼차 정사각행렬

[math]\displaystyle{ \displaystyle A_{3\times 3}^{-1}=\begin{bmatrix} a & b & c \\ d & e & f \\ g& h & i\end{bmatrix}^{-1} = \frac{1}{\det A} \operatorname{adj}A = \frac{1}{aei+bfg+cdh-afh-bdi-ceg}\begin{bmatrix} ei-fh & ch-bi & bf-ce \\ fg-di & ai-cg & cd-af \\ dh-eg& bg-ah & ae-bd\end{bmatrix} }[/math]

사차 정사각행렬

[math]\displaystyle{ \displaystyle A_{4\times 4}^{-1}=\begin{bmatrix}a&b&c&d \\ e&f&g&h \\ i&j&k&l \\ m&n&o&p \end{bmatrix}^{-1} = \frac{1}{\det A} \operatorname{ adj} A$ }[/math]

[math]\displaystyle{ =\displaystyle\frac{1}{a f k p + a g l n + a h j o + b e l o + b g i p + b h k m + c e j p + c f l m + c h i n + d e k n + d f i o + d g j m - a f l o - a g j p - a h k n - b e k p - b g l m - b h i o - c e l n - c f i p - c h j m - d e j o - d f k m - d g i n} \begin{bmatrix}fkp+gln+hjo-flo-gjp-hkn&blo+cjp+dkn-bkp-cln-djo&bgp+chn+dfo-bho-cfp-dgn&bhk+cfl+dgj-bgl-chj-dfk \\ elo+gip+hkm-ekp-glm-hip&akp+clm+djo-alo-cjp-dkm&aho+cep+dgm-agp-chm-deo&agl+chi+dek-ahk-cel-dgi \\ ejp+flm+hin-eln-fip-hjm&aln+bip+djm-ajp-blm-din&afp+bhm+den-ahn-bep-dfm&ahj+bel+dfi-afl-bhi-dej \\ ekn+fio+gjm-ejo-fkm-gin&ajo+bkm+cin-akn-bio-cjm & agn+beo+cfm-afo-bgm-cen & afk + bgi + cej - agj - bek - cfi \end{bmatrix} }[/math]

오차 정사각행렬

[math]\displaystyle{ \displaystyle A_{5\times 5}^{-1}=\begin{bmatrix}a&b&c&d&e \\ f&g&h&i&j \\ k&l&m&n&o \\ p&q&r&s&t \\ u&v&w&x&y \end{bmatrix}^{-1} = \frac{1}{\det A} \operatorname{ adj} A }[/math]

[math]\displaystyle{ =\displaystyle\frac{1}{agmsy+agntw+agorx+ahltx+ahnqy+ahosv+ailry+aimtv+aioqw+ajlsw+ajmqx+ajnrv+bfmtx+bfnry+bfosw+bhksy+bhntu+bhopx+biktw+bimpy+bioru+bjkrx+bjmsu+bjnpw+cflsy+cfntv+cfoqx+cgktx+cgnpy+cgosu+cikqy+ciltu+ciopv+cjksv+cjlpx+cjnqu+dfltw+dfmqy+dforv+dgkry+dgmtu+dgopw+dhktv+dhlpy+dhoqu+djkqw+djlru+djmpv+eflrx+efmsv+efnqw+egksw+egmpx+egnru+ehkqx+ehlsu+ehnpv+eikrv+eilpw+eimqu-agmtx-agnry-agosw-ahlsy-ahntv-ahoqx-ailtw-aimqy-aiorv-ajlrx-ajmsv-ajnqw-bfmsy-bfntw-bforx-bhktx-bhnpy-bhosu-bikry-bimtu-biopw-bjksw-bjmpx-bjnru-cfltx-cfnqy-cfosv-cgksy-cgntu-cgopx-ciktv-cilpy-cioqu-cjkqx-cjlsu-cjnpv-dflry-dfmtv-dfoqw-dgktw-dgmpy-dgoru-dhkqy-dhltu-dhopv-djkrv-djlpw-djmqu-eflsw-efmqx-efnrv-egkrx-egmsu-egnpw-ehksv-ehlpx-ehnqu-eikqw-eilru-eimpv} \begin{bmatrix}gmsy+gntw+gorx+hltx+hnqy+hosv+ilry+imtv+ioqw+jlsw+jmqx+jnrv-gmtx-gnry-gosw-hlsy-hntv-hoqx-iltw-imqy-iorv-jlrx-jmsv-jnqw&bmtx+bnry+bosw+clsy+cntv+coqx+dltw+dmqy+dorv+elrx+emsv+enqw-bmsy-bntw-borx-cltx-cnqy-cosv-dlry-dmtv-doqw-elsw-emqx-enrv&bhsy+bitw+bjrx+cgtx+ciqy+cjsv+dgry+dhtv+djqw+egsw+ehqx+eirv-bhtx-biry-bjsw-cgsy-citv-cjqx-dgtw-dhqy-djrv-egrx-ehsv-eiqw&bhox+bimy+bjnw+cgny+ciov+cjlx+dgow+dhly+djmv+egmx+ehnv+eilw-bhny-biow-bjmx-cgox-cily-cjnv-dgmy-dhov-djlw-egnw-ehlx-eimv&bhnt+bior+bjms+cgos+cilt+cjnq+dgmt+dhoq+djlr+egnr+ehls+eimq-bhos-bimt-bjnr-cgnt-cioq-cjls-dgor-dhlt-djmq-egms-ehnq-eilr \\fmtx+fnry+fosw+hksy+hntu+hopx+iktw+impy+ioru+jkrx+jmsu+jnpw-fmsy-fntw-forx-hktx-hnpy-hosu-ikry-imtu-iopw-jksw-jmpx-jnru&amsy+antw+aorx+cktx+cnpy+cosu+dkry+dmtu+dopw+eksw+empx+enru-amtx-anry-aosw-cksy-cntu-copx-dktw-dmpy-doru-ekrx-emsu-enpw&ahtx+airy+ajsw+cfsy+citu+cjpx+dftw+dhpy+djru+efrx+ehsu+eipw-ahsy-aitw-ajrx-cftx-cipy-cjsu-dfry-dhtu-djpw-efsw-ehpx-eiru&ahny+aiow+ajmx+cfox+ciky+cjnu+dfmy+dhou+djkw+efnw+ehkx+eimu-ahox-aimy-ajnw-cfny-ciou-cjkx-dfow-dhky-djmu-efmx-ehnu-eikw&ahos+aimt+ajnr+cfnt+ciop+cjks+dfor+dhkt+djmp+efms+ehnp+eikr-ahnt-aior-ajms-cfos-cikt-cjnp-dfmt-dhop-djkr-efnr-ehks-eimp \\flsy+fntv+foqx+gktx+gnpy+gosu+ikqy+iltu+iopv+jksv+jlpx+jnqu-fltx-fnqy-fosv-gksy-gntu-gopx-iktv-ilpy-ioqu-jkqx-jlsu-jnpv&altx+anqy+aosv+bksy+bntu+bopx+dktv+dlpy+doqu+ekqx+elsu+enpv-alsy-antv-aoqx-bktx-bnpy-bosu-dkqy-dltu-dopv-eksv-elpx-enqu&agsy+aitv+ajqx+bftx+bipy+bjsu+dfqy+dgtu+djpv+efsv+egpx+eiqu-agtx-aiqy-ajsv-bfsy-bitu-bjpx-dftv-dgpy-djqu-efqx-egsu-eipv&agox+aily+ajnv+bfny+biou+bjkx+dfov+dgky+djlu+eflx+egnu+eikv-agny-aiov-ajlx-bfox-biky-bjnu-dfly-dgou-djkv-efnv-egkx-eilu&agnt+aioq+ajls+bfos+bikt+bjnp+dflt+dgop+djkq+efnq+egks+eilp-agos-ailt-ajnq-bfnt-biop-bjks-dfoq-dgkt-djlp-efls-egnp-eikq \\fltw+fmqy+forv+gkry+gmtu+gopw+hktv+hlpy+hoqu+jkqw+jlru+jmpv-flry-fmtv-foqw-gktw-gmpy-goru-hkqy-hltu-hopv-jkrv-jlpw-jmqu&alry+amtv+aoqw+bktw+bmpy+boru+ckqy+cltu+copv+ekrv+elpw+emqu-altw-amqy-aorv-bkry-bmtu-bopw-cktv-clpy-coqu-ekqw-elru-empv&agtw+ahqy+ajrv+bfry+bhtu+bjpw+cftv+cgpy+cjqu+efqw+egru+ehpv-agry-ahtv-ajqw-bftw-bhpy-bjru-cfqy-cgtu-cjpv-efrv-egpw-ehqu&agmy+ahov+ajlw+bfow+bhky+bjmu+cfly+cgou+cjkv+efmv+egkw+ehlu-agow-ahly-ajmv-bfmy-bhou-bjkw-cfov-cgky-cjlu-eflw-egmu-ehkv&agor+ahlt+ajmq+bfmt+bhop+bjkr+cfoq+cgkt+cjlp+eflr+egmp+ehkq-agmt-ahoq-ajlr-bfor-bhkt-bjmp-cflt-cgop-cjkq-efmq-egkr-ehlp \\flrx+fmsv+fnqw+gksw+gmpx+gnru+hkqx+hlsu+hnpv+ikrv+ilpw+imqu-flsw-fmqx-fnrv-gkrx-gmsu-gnpw-hksv-hlpx-hnqu-ikqw-ilru-impv&alsw+amqx+anrv+bkrx+bmsu+bnpw+cksv+clpx+cnqu+dkqw+dlru+dmpv-alrx-amsv-anqw-bksw-bmpx-bnru-ckqx-clsu-cnpv-dkrv-dlpw-dmqu&agrx+ahsv+aiqw+bfsw+bhpx+biru+cfqx+cgsu+cipv+dfrv+dgpw+dhqu-agsw-ahqx-airv-bfrx-bhsu-bipw-cfsv-cgpx-ciqu-dfqw-dgru-dhpv&agnw+ahlx+aimv+bfmx+bhnu+bikw+cfnv+cgkx+cilu+dflw+dgmu+dhkv-agmx-ahnv-ailw-bfnw-bhkx-bimu-cflx-cgnu-cikv-dfmv-dgkw-dhlu&agms+ahnq+ailr+bfnr+bhks+bimp+cfls+cgnp+cikq+dfmq+dgkr+dhlp-agnr-ahls-aimq-bfms-bhnp-bikr-cfnq-cgks-cilp-dflr-dgmp-dhkq\end{bmatrix}. }[/math]

고만해, 미친놈들아!

n차 정사각행렬에 대한 역행렬 구하기

작성중