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Расчет матриц умножения и их детерминантов, Notas de estudo de Engenharia Química

В данном документе представлены формулы для вычисления матриц умножения и их детерминантов. Здесь рассматриваются матрицы 3 на 3 элементов, а также приведены примеры их вычисления.

Tipologia: Notas de estudo

Antes de 2010

Compartilhado em 27/08/2009

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L<L       LL /! #  #      N-N 3$9 ,# 3& 01F3  8 .GAHHAHH2 # 3 & 3 :  : 82 : .3 #   O I  @ .  0 "<=     X  ! LAL< !     L (L<   LH L< X        ! LM NL !   L<L             LJL 3$9 , & 01F3  8 .GAHH2 & 38  3. : 2  33 : 4 X  OI  @ . #  O I@ I.  I I  "0,0)5( " -,6&)* 6*)* # 0(3  '>%#  1/ ,? 1/ ,3     3        D#     !  9  1A 0A I J AI  J 0IA I !K I2     (      W  /!  D#   Y 9 (      /#   '  )  O  O     (   P 'O       3$9 ,&F3'3'3G9 , 1A  12AI L    3  3IAI    IAI/(7* (I2 1 2 3 4 5 6 7 8 9 10 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 COSENO EJE X coseno de x ,!  9 (D#    ! '     !   !  !     !    ! 9 ,&F3'3'3G9 , 1A  12AI*M* NIAI    IAI/(7* (I2       <  /!  ! D    ! '   1 6%7     ! '      !        #  ! '    D   !   ! $9  ! '9 nombrefuncion = sen(x) + cos(y) + xy2 −√48x ! 9 AQ (A , 1HF6G&    1A02HAH6&12: 102:;0;;-?1< ;;2H2 ,    1<A2  &  --4.<   L9L        LL  L(L       L    L      /!   ! ' 24 CAPÍTULO 2. PRIMEROS PASOS      ) -!& !     X !         !       /!      '   !  6%7   !   !    !            $              !  !Y   X !   3  X !    I  %    !     $  6%7        /!    D   X ! !            /!          $     !        3"+,"*     $              X ' !   D  ! '          !          !    F               !   !        $    /! D   !   '           /! !   !       !    '    !Y     Y    !         &    !             :      #  !   #      ! $!           !  9 (x1, x2, x3 · · · xn) = A            ,&F3'3'8G9 , &F<  < . G9 ,:  & = 8 8 . 33 3 =   /!  !           W          !  ,&F< 8 G  & = < 8  = ,&F3989.G  & = 3 = = 8 = = . = ,: =        3   !      ! !   #  !   !            3 !   ! !  R         Z     /!    !        R9 ka = (ka1, ka2, · · · , kan) 3$9 ,&F< 8 G  & = < 8  = ,8  & = - -8  =     E  @   3 !  !             F     /!    !          #   !   ! !  !  !   * 6%7 ^_OP A ?Q1 ^_OP 1?1AQ1 ^_`  O .  B /!   !  /!  !  !    !        ! !   9   # `   ! =   ! `   # ,;  & 3< ,;  & =  3 3- = = -< <   = = - < <  =    A 3 !  !Y     !     !   '        # ' !    ' 3 6%79 ,&F< 8 4G9 , &F. 3 -G9 ,;  & = -  8 3  = <                 :     ! ‖a‖  !    D          F    1  ! # '  D   9 a = [a1, a2] ‖a‖ = √ a21 + a 2 2    ! Y  !    9 L H& JL 3$9 ,&F< 8 4G9 , 12  & 33<83  :      !    !  !      !        ! !   !  !  ! 8  '  !      3 6%79 ,&F< 3 G9 ,&1 122  & = . <<<8 34333 8 < < =  ! Y    &    ! 9 , 12  & 3 +,* ",0 ,"*            !      !   !   !  '       !    4 /!    !   /!   !     << = . 8  = %      Y !    F  D  9 ,& 01AHH29 ,&F<AA.G9 &F3;;-A8A;;-G9 &F.A8AG9 ,)&F9 9G ) & = -8  . = =    = = 3  3 = = = = - = =  8  = =    = = -= = 3 3  = = = = . 8  = =    = = 3 3 3 =  $       Y9 M =   0,25x 1 6 x 7x 1 x2 1 5x 1 3 x2 7 1 5 1 3 1  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nxm = kMnxm, donde fij = kmij 3$  6%79 ,)&F< 89 < . 9 3  .G9 ,O&9 ,E&O;) E & = 3- -< 38 = = 3- -3 4 = =   -3 =            !         7  /! /!    !     Y NN  !    #   !  Y N7N 3 !V      '  !  Y  ! !  #    ! !  !  7  !   D  !    !  F' <A             -- & = < 4 = =   = , -&F< 89 G  - & = < 8 = =   = , 3&F8 49 < G  3 & = 8 4 = = <  = , -&F< 49 . G  - & = < 4 = = .  = , &F< 89 . <G   & = < 8 = = . < = 4                !    Cij  (Cij)T  * 6%7   Cij  ,33& 1 332;132;;- 33 &   ,3-& 1 3-2;132;; 3- &  ,3& 1 32;132;;< 3 & 3 ,-3& 1 -32;132;; -3 &    <E 4 ,--& 1 --2;132;;< -- &   ,-& 1 -2;132;;8 - & 3 ,3& 1 32;132;;< 3 &  -3 ,-& 1 -2;132;;8 - & 83 ,& 1 2;132;;  &  34 /J&F33 3- 39 -3 -- -9 3 - G /J & =    3 = = 4   3 = = -3 83  34 = * 6%7   $   (Cij)T  ,/J$&/JH /J$ & =  4 -3 = =   83 = = 3 3 34 = * 6%7  (Cij ) T det A  MainvA = CijT de 9 ,)&/J$ ) & = .<.<3 < 3< 38 -84-84 = = .. .<.<. -4-4 = = 3-<8 34.84 -<8.4 =            3 6%7   Y  !   ! ! Y  ! ' L&L        $     9 L&H   '  JL 3$9 ,&F< 8 49. < 9  G9 ,D&12 D & = .<.<3 < 3< 38 -84-84 = = .. .<.<. -4-4 = = 3-<8 34.84 -<8.4 = ,/&1;D2  & = 3   = =  3  = =   3 = , &1D;2  & = 3   = =  3  = =   3 =     $        Y  A−1 = B !   V Y S          !   !    Y          & <   4 )   ' '  '      '  '          <   T   '     &  (                 A   %  Y 6   /! V Y !  !     !         /!   D       In 3$9 In =   1 0 00 1 0 0 0 1      < 3 6%7       !  Y ! Y  ! ' L RHJL 3$  6%79       Y V ,M&F3 -  39-   9 3  8G M & = 3 -  3 = = -    = =  3  8 = ,"1M2  & -         A    Y A = [ a11 a12 a21 a22 ]  H-  -J         Y    9 det A = |A| = a11a22 − a21a12     |A|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a-+ .V- ,             ,&F3 3 <9  - 39 - 3 3G  & = 3 3 < = =  - 3 = = - 3 3 = , 12  & =  = = 3 = = - =  *')5( " "'0)("* )("0"* /%( /&+!  ! '  &  6%7     ! '  !     !     !            ! ' 9 M< RNA  &H< M<NJ      Y  na #  ma !  !    na #     ! ' R    Y   F R !  4    !    &    9 F + A &     '    !   ! '  ! D   !  3$9    9 2x + y − 2z = 10 3x + 2y + 2z = 1 5x + 4y + 3z = 4 !  LL 2x + y − 2z − 10 = 0 3x + 2y + 2z − 1 = 0 5x + 4y + 3z − 4 = 0 A              # :         ;   B         !    !  #! R5 E3 R1 R3 R2 R4 I3 I1 I2E1 E2 − + −+ − + ,! <.9  !   <  * %[ +%   $  [ b-    Y   $ −E1aE2 a+>.a> -i1 V> i2 V>.i3O −E2 V+> -i1 a+> a><a> -i2 V+><-i3O E3 V+>.-i1 V+><-i2 a+>.a><a>=-i3O 5 +>.a> -i1 V> i2 V>.i3 O E1 − E2 V+> -i1 a+> a><a> -i2 V+><-i3 O E2 V+>.-i1 V+><-i2 a+>.a><a>=-i3 O −E3 !  !   9 R1 = 4Ω R2 = 8Ω R3 = 7Ω R4 = 9Ω    E R5 = 2Ω E1 = 9V E2 = 6V E3 = 8V 5Y    AS   12 −8 −4−8 24 7 4 −7 13     i1i2 i3   O   36 −8   3 6%79 ,6&F3-  <9 -< .9< . 3GA C&F99 G 6 & = 3-   < = =   -< . = = < . 3 = C & =  = =  = =   = ,5&162;C 5 & = -<< 3 = = -<< = = <- 48<< = ! '9 i1O  ?<A.  i2O  ? < <  i3O V  AE=  )) .  %   (*+,')5( " )(.)* 3 6%7 !      !   #            ! ' L (L  !   9 =            a = [1 2 3 7 8]  !  !  !       /!       !  !  #    !     !    1 + 2x + 3x2 + 7x3 + 8x4    ! 6%79 ,           ,&F3 -  . G  & = 3 -  .  = ,          A 0         ,       ,& 01AHHAH H2  & 3 : - : 2 : .3 : 4 5     !  !               XY LL    L (L X9 ,& 01AHH2  &  , 3&;;-:8;;;8:.;;;  3 & 2 : .3 : 85 ! /!     !          !    (   !      !  9 ,&F3 -G  & =  3  - = , -& 01AHH2  - & - :  : 2  ,0<"* " '( )(.)      X   !   6%7 ! Y  ! ' N +-N!     $    V  /!  X         Y9    =<  D       !  ! M' <  NA'&H'&'' <'& J 3$9 ,    A     0    ,& 01AHH2  &  , 3&-:;:;;-  3 & - :  : 2 , -&;:3  - & 3 :  ,F  A    G&1 3A -2    &  4 :    & 33333333 8             !   '    L'&L  ",)30 0 " '( )(.) 3 6%7       ! ! ' '  ! Y  ! ' L'& HJL * 6%7 ,        ,& 01AHH2  &  ,&8-;  &  8  - , &-8;;;-:3;;;:;;<   & =            -82 : 33 : 4 ,&   &  8  -  -82 : 33 : 4 ,        , & 12  & -8 : -2 : 3 : 4  -84 : 85 : 386 : -7 : 8    !    ; ,              ,&3'  & = 3 -  < 8  = ,  &  1 A2   & = 4 3< 3 <  3< 8.3 - 3883 = 4      X   ; ,    , & 1 1 22   & =  = = -8 :3<<.8. = = -8 3<<.8. = = 8 =      '     <  &   (  '    <  )      4   &  HJ (      '' <   <    4       !     ! F'  $    F   !V              == ,   & -8 : -2 : 3 : 4  -84 : 85 : 386 : -7 : 8 ,       12 , & 1 2   & -8 : -2 : 3 : 4 ,         12 ,  &  1 2    & -84 : 85 : 386 : -7 : 8 ,          ,  &       & 8 :  : 2  3-83 : .84 : 385 : 6    !    XY  !  !  + - ,           ,  & 1  2    & 8 :  : 2 ,       , 1  2  & =  -8 : 3<<.8. = =  -8  3<<.8. = )+ '  ! 3 6%7  # ! ! '  !      /!   V   # ! ! ' ! Y   'X9 =A            ,        !  @ , @ @ &  , @;;-  &  3 , @;;  &   , @;;<  & 3 80 #"+" (70+"  %0G%("%  6 (70+"  # 01/+9 !   !    Y  !    !     !     NcN ,       !  ,< ; @  & <  ,!3&-; @ !3 & - ,!-& @; @ !- & 3<384-. (  !  $   !  !    !  V  z = x + yi   !  $    (             (< F   !  9     =E Re(z) = x Im(z) = y %   '    Y /!X 9 Re(z) Im(z) z=7+3i ,! < 9   >+Y-  6+Y- %      $  A 9HJ      $ ( A0HJ 3  #! <          $   /!          ! z = 7 + 3i Re(z) = 7 Im(z) = 3 4 ! ! $    ! ! 6%7  FV  !     !    X9 ,        !       J ,3&3-8:; @ 3 & ?            3-8 :  , 132  & 3-8 ,!132  &  ,  J  ,-& @:-; @; @ - & 3<384-. : - 3 8 , 1-2  & 3<384-. ,!1-2  & - 3 8        Y !        !  #           D!   /!         $      ?< = - : - = =   = ,    ,H  & = -  = =   -  =       Y  $!    Y  !   !     R +- ,  J! 0    , 3& J12  3 & = -  - = =   = , 3H  & = -  = = : -  = , J!        Q , Q12  & = -  = = : -  =  * =      : ,       J  ,&3: @  & 3 :  ,-&:; @ - &  :  ,&:-  & < : < ?            ,        J  , &-   & - : -     =  #       : ,        J  ,-&<:; @9 &:<; @9 , &-;  & 3- : < ,        J  , 3&-  3 & 4-.. :  <38         : 3 X  '          $   ! V !       '          ! ' F  $ ejθ ejθ = cosθ + jsenθ ,  !   '!  3!      !  3!  6%7      !  !                !     /!  D  ! D! !  5o   85o     ?= Re(z) Im(z) r=2 r=5 r=8 θ=14 θ=45ο ο θ=74ο ,! < 9        !          /!  D   !       /!  <  !9 - @    D#  ! < !  <        !   D!   !        !  !     /!  D     !      /!  !    !     D !  ! X /!    . #  A !  !   .U        ! ,        !  -A 8 0 ,&F-;  13A-.2A 8;  13A-.2A ;  13A-2G9 , 12  & = 3  =    !               /!   '    =     ' # A=    '     !      . #  A ! /!    !    ,   8 0 8 !     ?A            0.00 2.14 4.29 6.43 8.57 10.71 12.86 15.00 0 1 2 3 4 5 6 7 8 9 ,! <?9 (D#         5  )     '  O   '  -' (  -'Q         :9Y,01#3 %/ 310*S )2 !   ! '!&! ! && 3! 4 6%7 ! Y  ! ' L 'HJL  !  !    !  V    3 !    9 (A 'H(<<< J &9 ( 9  !    !  Y         ! '  9     9    /!    ! 9 F'  ! '  !         ! ?E 3$9    !       /!     ! '  9 t0 = 0; 1 < t < 81 y0 = 11 dy dt = y − ycos(t) ! '9 ! Y L'XHJL !      t0ty0       ! ' L 'HJL  /!   !   ! D#  ! Y L -'HJL * 6%79 , 1IF   G&1A02IAI   &00; 12I29 ,0&39&9&3'8' 9 ,0&  10AAA29 , -1A0A-29 , 1H7 B    0&0  0; 12H AH  HAH012H29 >!    ,! <@9 @            1 2 3 4 5 6 7 8 0 200 400 600 800 1000 1200 Soluciˆ‡n a la ode y=y - y*cos(t) tiempo y(t) ,! <@9 ! ' )        @< 0 40 80 120 160 200 240 280 3.0 3.4 3.8 4.2 4.6 5.0 5.4 5.8 6.2 ,!  9 (D#   =a+F-  a +F-    !   ! ! '  =a+F-!  a +F-9 !  8: 12  !  <: 12 ,&'8'<; @9 ,0&8:129 ,+&<: 129 , 1F0A+G2 3 !     !    (D#  < @         0 100 200 300 400 500 600 3.0 3.4 3.8 4.2 4.6 5.0 5.4 5.8 6.2 ,! <9 (D#   =a+F- !  a +F- + *       567#  !    D#   6%7   !         D#   N   + -N  !     #!   =  ,!  9 4X !     D#   6%7  %6)*: ,! =9 4X !     D#   6%7  2 ;6)&5;  "   "     @= X /!  D    D#   !  /!   !V Y '   D  ! D#     $!     /!    !       !  $    ' !    ,      !      ,&' @'-; @9 ,0&129 03&129 0-& 129 ,9  1A029 9  1A0329 9  1A0-29 3 !        !    #! ? @  A ,! ?9   ) . @A         ,&<'3'<9 , 1A>-<AHHAH0HAH H29 ,!182 3 !        ! !   $  !    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