【正文】 ? ?????????????? ?????? ? Theorem : The inverse of a matrix expressed by its adjoint matrix If A is an n n invertible matrix, then 1 1 a d j ( )d e t( )AAA? ?Consider the product A[adj(A)] 1 1 2 2de t ( ) i f 0 i f i j i j i n j nA i ja C a C a Cij??? ? ? ? ???The entry at the position (i, j) of A[adj(A)] Pf: Consider a matrix B similar to matrix A except that the jth row is replaced by the ith row of matrix A 11 12 1121212de t ( ) 0ni i ini i inn n nna a aa a aBa a aa a a? ? ?1 1 2 2de t ( ) 0i j i j in jnB a C a C a C? ? ? ? ? ?Perform the cofactor expansion along the jth row of matrix B (Note that Cj1, Cj2,…, and Cjn are still the cofactors for the entries of the jth row) 1[ a dj ( ) ] de t ( ) a dj ( )de t ( )A A A I A A IA??? ? ? ?????A1 ※ Since there are two identical rows in B, according to Theorem , det(B) should be zero det( ) ,A ad bc? ? ????????dcbaA1 1 2 11 2 2 2a d j ( ) CC dbA CC ca ??? ?????? ???????? ?1 1 a d j ( )d e tAAA???? Ex: For any 22 matrix, its inverse can be calculated as follows ?????????? acbdbcad1 ? Ex 2: ??????????????201120231A(a) Find the adjoint matrix of A (b) Use the adjoint matrix of A to find A–1 1121 4,02C?? ? ? ??Sol: 1201 1,12C ? ?? 1302 2,10C?? ? ?ijjiij MC ??? )1(?2132 6,02C ? ? ?? 2212 0,12C?? ? ?? 2313 3,10C?? ? ?3132 7,21C ? ? ?? 3212 1,01C?? ? ?3313 2.02C?? ? ?? cofactor matrix of A ?? ????????????217306214ijCadjoint matrix of A ?467a dj ( ) 1 0 1232TijAC?????????? ?????????????????23210176431? inverse matrix of A ? ?1 1 a d j ( )d e tAAA? ? ? ?( de t 3 )A ????????????323231313734102? Check: IAA ??1※ The putational effort of this method to derive the inverse of a matrix is higher than that of the . E. (especially to pute the cofactor matrix for a higherorder square matrix) ※ However, for puters, it is easier to implement this method than the . E. since it is not necessary to judge which row operation should be used and the only thing needed to do is to calculate determinants of matrices ? Theorem : Cramer’s Rule 1 1 1 1 2 2 1 12 1 1 2 2 2 2 21 1 2 2nnnnn n n n n na x a x a x ba x a x a x ba x a x a x b? ? ? ?? ? ? ?? ? ? ?A??xb( 1 ) ( 2 ) ( )w h e r e ,nijnnA a A A A? ???????? ??12,nxxx?????????????x12nbbb?????????????b11 12 121 22 212S upp ose this sy ste m ha s a un ique sol uti o n, ., de t ( ) 0nnn n nna a aa a aAa a a??A(i) represents the ith column vector in A ( 1 ) ( 2 ) ( 1 ) ( 1 ) ( )B y d e f in in g j j njA A A A A A????? ?? b11 1 ( 1 ) 1 1 ( 1 ) 121 2 ( 1 ) 2 2 ( 1 ) 21 ( 1 ) ( 1 )j j nj j nn n j n n j nna a b a aa a b a aa a b a a???????????????1 1 2 2( i. e . , d e t( ) )j j j n n jA b C b C b C? ? ? ?d e t( ) , 1 , 2 , ,d e t( )jjAx j nA? ? ? ? Pf: ( det( ) 0)A ? Ax = b bx 1??? A 1a d j( )d e t( ) AA? b11 21 1 112 22 2 2121det( )nnn n nn nC C C bC C C bAC C C b? ? ? ?? ? ? ?? ? ? ??? ? ? ?? ? ? ?? ? ? ?1 11 2 21 11 12 2 22 21 1 2 21de t ( )nnnnn n n nnb C b C b Cb C b C b CAb C b C b C? ? ?????? ? ????? ? ??? (according to Thm. ) 1 1 11 2 21 12 1 12 2 22 21 1 2 2121 de t ( )de t ( ) / de t ( )de t ( ) / de t ( ) de t ( ) / de t ( )nnnnn n n n nnnx b C b C b Cx b C b C b CAx b C b C b CAAAAAA? ? ?? ? ? ?? ? ? ?? ? ?? ? ? ???? ? ? ?? ? ? ?? ? ?? ? ? ??????????????d e t( ) , 1 , 2 , ,d e t( )jjAx j nA? ? ?1 1 2 2( On S l i de 9, i t i s al r e a dy de r i v e d t h a t de t ( ) )j j j n njA b C b C b C? ? ? ? ? Ex 6: Use Cramer’s rule to solve the system of linear equation 244302132?????????zyxzxzyxSol: 8442100321)d et( 1 ????A10443102321)d e t ( ?????A,15423102311)d et( 2 ?????A 16243002121)d et( 3 ?????A54)d e t ()d e t ( 1 ??AAx23)d et ()d et ( 2 ??AAy58)d et ()d et ( 3 ???AAz Keywords in Section : ? matrix of cofactors: 餘因子矩陣 ? adjoint matrix: 伴隨矩陣 ? Cramer’s rule: Cramer 法則