25:@0.921538:0.966074:0.940353:0.966074:0.940353:0.949450:0.921538:0.949450:0.000000:0.000000 Resolución de sistemas de ecuaciones lineales mediante :@0.108233:0.097328:0.561603:0.097328:0.561603:0.079645:0.108233:0.079645:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 determinantes:@0.108233:0.115433:0.225913:0.115433:0.225913:0.097750:0.108233:0.097750:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Consideramos el sistema de ecuaciones lineales :@0.108233:0.132140:0.447278:0.132140:0.447278:0.115643:0.108233:0.115643:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ax:@0.447278:0.132256:0.466024:0.132256:0.466024:0.115787:0.447278:0.115787:0.000000:0.000000 :@0.460248:0.132632:0.466663:0.132632:0.466663:0.116814:0.460248:0.116814:0.000000 = ,:@0.466663:0.132140:0.497449:0.132140:0.497449:0.115643:0.466663:0.115643:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.485755:0.132256:0.495022:0.132256:0.495022:0.115787:0.485755:0.115787:0.000000 :@0.488279:0.130752:0.494694:0.130752:0.494694:0.114934:0.488279:0.114934:0.000000 con = .:@0.108233:0.160176:0.243613:0.160183:0.243613:0.143686:0.108233:0.143679:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.139327:0.160292:0.146802:0.160292:0.146802:0.143823:0.139327:0.143823:0.000000 :@0.141023:0.160668:0.147438:0.160668:0.147438:0.144850:0.141023:0.144850:0.000000 :@0.203807:0.159665:0.218460:0.159665:0.218460:0.146291:0.203807:0.146291:0.000000 R:@0.222600:0.159096:0.235905:0.159096:0.235905:0.146234:0.222600:0.146234:0.000000 2:@0.235905:0.153832:0.240858:0.153832:0.240858:0.144213:0.235905:0.144213:0.000000 Sean = :@0.108236:0.205570:0.253174:0.205577:0.253174:0.189079:0.108236:0.189072:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 A:@0.146035:0.205686:0.157305:0.205686:0.157305:0.189217:0.146035:0.189217:0.000000 :@0.234380:0.205058:0.249033:0.205058:0.249033:0.191685:0.234380:0.191685:0.000000 M:@0.253174:0.205693:0.268451:0.205693:0.268451:0.189224:0.253174:0.189224:0.000000 2×2:@0.268451:0.208855:0.284659:0.208855:0.284659:0.199237:0.268451:0.199237:0.000000:0.000000:0.000000 [ ], = .:@0.284659:0.205577:0.421688:0.205577:0.421688:0.189079:0.284659:0.189079:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 R:@0.290516:0.204490:0.303821:0.204490:0.303821:0.191628:0.290516:0.191628:0.000000 b:@0.316574:0.205693:0.325841:0.205693:0.325841:0.189224:0.316574:0.189224:0.000000 :@0.319098:0.204189:0.325513:0.204189:0.325513:0.188371:0.319098:0.188371:0.000000 :@0.381880:0.205058:0.396533:0.205058:0.396533:0.191685:0.381880:0.191685:0.000000 R:@0.400675:0.204490:0.413979:0.204490:0.413979:0.191628:0.400675:0.191628:0.000000 2:@0.413979:0.199225:0.418933:0.199225:0.418933:0.189607:0.413979:0.189607:0.000000 Supongamos que es invertible, lo que es equivalente a decir que :@0.108242:0.240278:0.599861:0.240278:0.599861:0.223781:0.108242:0.223781:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 A:@0.240610:0.240394:0.251880:0.240394:0.251880:0.223925:0.240610:0.223925:0.000000 det( ) ≠ 0, esto es,:@0.108242:0.257629:0.240941:0.257629:0.240941:0.241131:0.108242:0.241131:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 A:@0.138007:0.257745:0.149277:0.257745:0.149277:0.241276:0.138007:0.241276:0.000000 det( ) = = – ≠ 0.:@0.108242:0.292331:0.326941:0.292331:0.326941:0.275833:0.108242:0.275833:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 A:@0.138007:0.292446:0.149277:0.292446:0.149277:0.275978:0.138007:0.275978:0.000000 ad bc:@0.243118:0.292446:0.296599:0.292446:0.296599:0.275978:0.243118:0.275978:0.000000:0.000000:0.000000:0.000000:0.000000 Se prueba que las soluciones e se calculan como::@0.108242:0.327032:0.476839:0.327032:0.476839:0.310534:0.108242:0.310534:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.315494:0.327148:0.347012:0.327148:0.347012:0.310679:0.315494:0.310679:0.000000:0.000000:0.000000 x:@0.108242:0.379200:0.115717:0.379200:0.115717:0.362731:0.108242:0.362731:0.000000 = .:@0.115717:0.379084:0.203836:0.379084:0.203836:0.362587:0.115717:0.362587:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Recuerda que…:@0.684804:0.161979:0.794564:0.161979:0.794564:0.146506:0.684804:0.146506:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Propiedades de los :@0.685042:0.191029:0.819971:0.191029:0.819971:0.175556:0.685042:0.175556:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 determinantes:@0.638498:0.206871:0.741468:0.206871:0.741468:0.191398:0.638498:0.191398:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Estas propiedades se extienden :@0.638498:0.222290:0.844562:0.222290:0.844562:0.207227:0.638498:0.207227:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a todos los determinantes.:@0.638498:0.238132:0.809542:0.238132:0.809542:0.223069:0.638498:0.223069:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1.:@0.638498:0.261301:0.649703:0.261301:0.649703:0.246040:0.638498:0.246040:0.000000:0.000000 El determinante de una ma-:@0.649702:0.261103:0.836192:0.261103:0.836192:0.246040:0.649702:0.246040:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 triz es igual al de su traspues-:@0.655120:0.276945:0.842524:0.276945:0.842524:0.261882:0.655120:0.261882:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ta: | | = | |.:@0.655120:0.292787:0.739362:0.292786:0.739362:0.277723:0.655120:0.277724:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 A:@0.679166:0.292893:0.689456:0.292893:0.689456:0.277856:0.679166:0.277856:0.000000 A:@0.716034:0.292893:0.726324:0.292893:0.726324:0.277856:0.716034:0.277856:0.000000 T :@0.724850:0.287048:0.732275:0.287048:0.732275:0.278282:0.724850:0.278282:0.000000:0.000000 2.:@0.638501:0.308826:0.649706:0.308826:0.649706:0.293565:0.638501:0.293565:0.000000:0.000000 Si una fila (o columna) tiene :@0.649706:0.308628:0.841699:0.308628:0.841699:0.293565:0.649706:0.293565:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 todos sus elementos nulos, el :@0.655123:0.324470:0.848243:0.324470:0.848243:0.309407:0.655123:0.309407:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 determinante es cero. :@0.655123:0.340312:0.798146:0.340312:0.798146:0.325249:0.655123:0.325249:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3.:@0.638501:0.356352:0.649706:0.356352:0.649706:0.341091:0.638501:0.341091:0.000000:0.000000 Un determinante que tiene :@0.649706:0.356154:0.835753:0.356154:0.835753:0.341091:0.649706:0.341091:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dos filas o dos columnas :@0.655123:0.371996:0.817581:0.371996:0.817581:0.356933:0.655123:0.356933:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 iguales da como resultado :@0.655123:0.387838:0.828682:0.387838:0.828682:0.372775:0.655123:0.372775:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 cero. :@0.655123:0.403680:0.689195:0.403680:0.689195:0.388617:0.655123:0.388617:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 4.:@0.638501:0.419720:0.649706:0.419720:0.649706:0.404459:0.638501:0.404459:0.000000:0.000000 Si en un determinante los :@0.649706:0.419522:0.824638:0.419522:0.824638:0.404459:0.649706:0.404459:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 elementos de una fila o co-:@0.655123:0.435364:0.830758:0.435364:0.830758:0.420301:0.655123:0.420301:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 lumna son múltiplos de los :@0.655123:0.451206:0.835156:0.451206:0.835156:0.436143:0.655123:0.436143:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de otra, el valor del determi-:@0.655123:0.467048:0.838302:0.467048:0.838302:0.451985:0.655123:0.451985:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 nante es cero. :@0.655123:0.482890:0.747117:0.482890:0.747117:0.467827:0.655123:0.467827:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5.:@0.638501:0.498930:0.649706:0.498930:0.649706:0.483669:0.638501:0.483669:0.000000:0.000000 Si a una fila (o columna) se le :@0.649706:0.498732:0.848190:0.498732:0.848190:0.483669:0.649706:0.483669:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 suma otra multiplicada por :@0.655123:0.514574:0.836264:0.514574:0.836264:0.499511:0.655123:0.499511:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 un escalar, el determinante :@0.655123:0.530416:0.831884:0.530416:0.831884:0.515353:0.655123:0.515353:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 no varía. :@0.655123:0.546258:0.713382:0.546258:0.713382:0.531195:0.655123:0.531195:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 6.:@0.638501:0.562298:0.649706:0.562298:0.649706:0.547037:0.638501:0.547037:0.000000:0.000000 Si se cambian entre sí dos :@0.649706:0.562101:0.824162:0.562100:0.824162:0.547037:0.649706:0.547038:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 filas, el valor del determinan-:@0.655123:0.577942:0.840770:0.577942:0.840770:0.562879:0.655123:0.562879:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 te cambia de signo (análoga-:@0.655123:0.593784:0.841664:0.593784:0.841664:0.578721:0.655123:0.578721:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 mente con las columnas). :@0.655123:0.609626:0.825006:0.609626:0.825006:0.594563:0.655123:0.594563:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 7.:@0.638501:0.625666:0.649706:0.625666:0.649706:0.610405:0.638501:0.610405:0.000000:0.000000 Multiplicando a todos los :@0.649706:0.625468:0.825778:0.625468:0.825778:0.610405:0.649706:0.610405:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 elementos de una fila (o :@0.655123:0.641310:0.815349:0.641310:0.815349:0.626247:0.655123:0.626247:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 columna) por , el determi-:@0.655123:0.657152:0.835243:0.657152:0.835243:0.642089:0.655123:0.642089:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 α:@0.748773:0.664034:0.758389:0.664034:0.758389:0.646762:0.748773:0.646762:0.000000 nante queda multiplicado :@0.655121:0.672994:0.827466:0.672994:0.827466:0.657931:0.655121:0.657931:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 por .:@0.655121:0.688836:0.694136:0.688836:0.694136:0.673773:0.655121:0.673773:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 α:@0.682001:0.695718:0.691617:0.695718:0.691617:0.678446:0.682001:0.678446:0.000000 8.:@0.638498:0.704876:0.649703:0.704876:0.649703:0.689615:0.638498:0.689615:0.000000:0.000000 Sean , matrices cuadradas, :@0.649704:0.704678:0.849754:0.704678:0.849754:0.689615:0.649704:0.689615:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 A B:@0.689633:0.704783:0.714611:0.704783:0.714611:0.689747:0.689633:0.689747:0.000000:0.000000:0.000000 entonces el:@0.655122:0.720520:0.728664:0.720520:0.728664:0.705456:0.655122:0.705456:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 det(:@0.663178:0.736362:0.690355:0.736362:0.690355:0.721298:0.663178:0.721298:0.000000:0.000000:0.000000:0.000000 AB:@0.690355:0.736467:0.708930:0.736467:0.708930:0.721431:0.690355:0.721431:0.000000:0.000000 ) = det( ) det( ).:@0.708930:0.736362:0.821996:0.736362:0.821996:0.721298:0.708930:0.721298:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 A:@0.758972:0.736467:0.769262:0.736467:0.769262:0.721431:0.758972:0.721431:0.000000 B:@0.805655:0.736467:0.814046:0.736467:0.814046:0.721431:0.805655:0.721431:0.000000 a b:@0.187750:0.197334:0.220886:0.197334:0.220886:0.180865:0.187750:0.180865:0.000000:0.000000:0.000000 c d:@0.188636:0.218719:0.220828:0.218719:0.220828:0.202250:0.188636:0.202250:0.000000:0.000000:0.000000 a b:@0.182724:0.283375:0.215860:0.283375:0.215860:0.266906:0.182724:0.266906:0.000000:0.000000:0.000000 c d:@0.183610:0.304760:0.215803:0.304760:0.215803:0.288291:0.183610:0.288291:0.000000:0.000000:0.000000 b:@0.355013:0.197330:0.364280:0.197330:0.364280:0.180861:0.355013:0.180861:0.000000 1:@0.364293:0.200496:0.369247:0.200496:0.369247:0.190878:0.364293:0.190878:0.000000 b:@0.355028:0.218718:0.364294:0.218718:0.364294:0.202249:0.355028:0.202249:0.000000 2:@0.364293:0.221879:0.369247:0.221879:0.369247:0.212261:0.364293:0.212261:0.000000 x:@0.180737:0.151417:0.188212:0.151417:0.188212:0.134949:0.180737:0.134949:0.000000 y:@0.180622:0.172802:0.188309:0.172802:0.188309:0.156333:0.180622:0.156333:0.000000 Este método se conoce también como método de Cramer. Un pro-:@0.108231:0.610491:0.595829:0.610491:0.595829:0.593993:0.108231:0.593993:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ceso similar se sigue cuando los determinantes son de orden 3, 4, … :@0.108231:0.627841:0.584407:0.627841:0.584407:0.611344:0.108231:0.611344:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 n.:@0.583886:0.627957:0.595792:0.627957:0.595792:0.611488:0.583886:0.611488:0.000000:0.000000 Ejercicio resuelto:@0.108231:0.662919:0.226209:0.662919:0.226209:0.646132:0.108231:0.646132:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1.:@0.108231:0.680111:0.120503:0.680111:0.120503:0.663396:0.108231:0.663396:0.000000:0.000000 Resolver el sistema de ecuaciones lineales ,:@0.120503:0.679894:0.522219:0.679894:0.522219:0.663396:0.120503:0.663396:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.108231:0.697244:0.112373:0.697244:0.112373:0.680747:0.108231:0.680747:0.000000 mediante el método de Cramer.:@0.131966:0.697244:0.358485:0.697244:0.358485:0.680747:0.131966:0.680747:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.108231:0.731946:0.112373:0.731946:0.112373:0.715448:0.108231:0.715448:0.000000 La matriz del sistema de ecuaciones lineales es y su:@0.131966:0.731946:0.560479:0.731946:0.560479:0.715448:0.131966:0.715448:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.108231:0.766647:0.112373:0.766647:0.112373:0.750150:0.108231:0.750150:0.000000 determinante es = – 3 – 2 = –5. Como el determinante:@0.131966:0.766647:0.585432:0.766647:0.585432:0.750150:0.131966:0.750150:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.108231:0.801349:0.112373:0.801349:0.112373:0.784851:0.108231:0.784851:0.000000 es diferente de cero, sí se puede aplicar el método de Cramer.:@0.131966:0.801349:0.564735:0.801349:0.564735:0.784851:0.131966:0.784851:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.108231:0.853401:0.112373:0.853401:0.112373:0.836904:0.108231:0.836904:0.000000 x:@0.131966:0.853517:0.139441:0.853517:0.139441:0.837048:0.131966:0.837048:0.000000 = = = 2, = = = 3.:@0.139441:0.853401:0.535496:0.853401:0.535496:0.836904:0.139441:0.836904:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.339838:0.853517:0.347525:0.853517:0.347525:0.837048:0.339838:0.837048:0.000000 :@0.108231:0.888103:0.112373:0.888103:0.112373:0.871605:0.108231:0.871605:0.000000 :@0.108231:0.922804:0.112373:0.922804:0.112373:0.906306:0.108231:0.906306:0.000000 La solución es = .:@0.131966:0.922804:0.298534:0.922804:0.298534:0.906306:0.131966:0.906306:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.235438:0.922920:0.242913:0.922920:0.242913:0.906451:0.235438:0.906451:0.000000 :@0.237135:0.923295:0.243551:0.923295:0.243551:0.907477:0.237135:0.907477:0.000000 3 + = 9:@0.443562:0.671117:0.512166:0.671117:0.512166:0.654619:0.443562:0.654619:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.452058:0.671232:0.486312:0.671232:0.486312:0.654764:0.452058:0.654764:0.000000:0.000000:0.000000 2 – = 1:@0.443562:0.688467:0.511858:0.688467:0.511858:0.671970:0.443562:0.671970:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y:@0.452058:0.688583:0.486004:0.688583:0.486004:0.672114:0.452058:0.672114:0.000000:0.000000:0.000000 3:@0.476967:0.722367:0.485463:0.722367:0.485463:0.705869:0.476967:0.705869:0.000000 1:@0.505730:0.722367:0.514226:0.722367:0.514226:0.705869:0.505730:0.705869:0.000000 2 –1:@0.476967:0.743751:0.519485:0.743751:0.519485:0.727254:0.476967:0.727254:0.000000:0.000000:0.000000:0.000000 3 1:@0.262399:0.757836:0.297288:0.757836:0.297288:0.741339:0.262399:0.741339:0.000000:0.000000:0.000000 2 –1:@0.262399:0.779221:0.302547:0.779221:0.302547:0.762723:0.262399:0.762723:0.000000:0.000000:0.000000:0.000000 9 1:@0.171142:0.822223:0.206031:0.822223:0.206031:0.805725:0.171142:0.805725:0.000000:0.000000:0.000000 1 –1:@0.171142:0.843607:0.211290:0.843607:0.211290:0.827110:0.171142:0.827110:0.000000:0.000000:0.000000:0.000000 3 9:@0.380555:0.822223:0.415444:0.822223:0.415444:0.805725:0.380555:0.805725:0.000000:0.000000:0.000000 2 1:@0.380555:0.843607:0.415444:0.843607:0.415444:0.827110:0.380555:0.827110:0.000000:0.000000:0.000000 3 1:@0.171142:0.866236:0.206031:0.866236:0.206031:0.849738:0.171142:0.849738:0.000000:0.000000:0.000000 2 –1:@0.171142:0.887621:0.211290:0.887621:0.211290:0.871123:0.171142:0.871123:0.000000:0.000000:0.000000:0.000000 3 1:@0.380555:0.866236:0.415444:0.866236:0.415444:0.849738:0.380555:0.849738:0.000000:0.000000:0.000000 2 –1:@0.380555:0.887621:0.420704:0.887621:0.420704:0.871123:0.380555:0.871123:0.000000:0.000000:0.000000:0.000000 –9 – 1:@0.244599:0.846126:0.291452:0.846126:0.291452:0.829629:0.244599:0.829629:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –5:@0.258432:0.862465:0.277620:0.862465:0.277620:0.845967:0.258432:0.845967:0.000000:0.000000 3 – 18:@0.455006:0.846126:0.499856:0.846126:0.499856:0.829629:0.455006:0.829629:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –5:@0.467837:0.862465:0.487025:0.862465:0.487025:0.845967:0.467837:0.845967:0.000000:0.000000 2:@0.273739:0.913245:0.282235:0.913245:0.282235:0.896747:0.273739:0.896747:0.000000 3:@0.273739:0.934630:0.282235:0.934630:0.282235:0.918132:0.273739:0.918132:0.000000 En el determinante del nume-:@0.108233:0.448496:0.320247:0.448496:0.320247:0.431999:0.108233:0.431999:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 rador de reemplazamos las :@0.108233:0.465847:0.315178:0.465847:0.315178:0.449349:0.108233:0.449349:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.173195:0.465963:0.180670:0.465963:0.180670:0.449494:0.173195:0.449494:0.000000 componentes del vector:@0.108233:0.483198:0.282794:0.483198:0.282794:0.466700:0.108233:0.466700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.108233:0.518015:0.117500:0.518015:0.117500:0.501546:0.108233:0.501546:0.000000 :@0.110757:0.516511:0.117172:0.516511:0.117172:0.500693:0.110757:0.500693:0.000000 = , por los elementos :@0.117172:0.517899:0.311653:0.517899:0.311653:0.501402:0.117172:0.501402:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de la primera columna del :@0.108233:0.552601:0.298687:0.552601:0.298687:0.536103:0.108233:0.536103:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 determinante.:@0.108233:0.569952:0.207815:0.569952:0.207815:0.553454:0.108233:0.553454:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.149928:0.347833:0.159195:0.347833:0.159195:0.331364:0.149928:0.331364:0.000000 1:@0.159194:0.351063:0.163900:0.351063:0.163900:0.341461:0.159194:0.341461:0.000000 b:@0.181049:0.347833:0.190316:0.347833:0.190316:0.331364:0.181049:0.331364:0.000000 b:@0.149936:0.369218:0.159202:0.369218:0.159202:0.352749:0.149936:0.352749:0.000000 2:@0.159194:0.372447:0.163900:0.372447:0.163900:0.362846:0.159194:0.362846:0.000000 d:@0.181098:0.369217:0.190268:0.369217:0.190268:0.352748:0.181098:0.352748:0.000000 a b:@0.409545:0.347833:0.442701:0.347833:0.442701:0.331364:0.409545:0.331364:0.000000:0.000000:0.000000 1:@0.442710:0.351063:0.447416:0.351063:0.447416:0.341461:0.442710:0.341461:0.000000 c b:@0.410441:0.369217:0.442711:0.369217:0.442711:0.352748:0.410441:0.352748:0.000000:0.000000:0.000000 2:@0.442710:0.372447:0.447416:0.372447:0.447416:0.362846:0.442710:0.362846:0.000000 a b:@0.150050:0.391838:0.185576:0.391838:0.185576:0.375370:0.150050:0.375370:0.000000:0.000000:0.000000 c d:@0.150956:0.413223:0.185537:0.413223:0.185537:0.396755:0.150956:0.396755:0.000000:0.000000:0.000000 a b:@0.409545:0.391838:0.445070:0.391838:0.445070:0.375370:0.409545:0.375370:0.000000:0.000000:0.000000 c d:@0.410450:0.413223:0.445032:0.413223:0.445032:0.396755:0.410450:0.396755:0.000000:0.000000:0.000000 y:@0.368580:0.379115:0.376267:0.379115:0.376267:0.362646:0.368580:0.362646:0.000000 = .:@0.376267:0.378999:0.464386:0.378999:0.464386:0.362501:0.376267:0.362501:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 En el determinante del nume-:@0.368580:0.448402:0.580595:0.448402:0.580595:0.431904:0.368580:0.431904:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 rador de reemplazamos las :@0.368580:0.465753:0.575738:0.465753:0.575738:0.449255:0.368580:0.449255:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.433543:0.465868:0.441230:0.465868:0.441230:0.449400:0.433543:0.449400:0.000000 componentes del vector:@0.368580:0.483103:0.543141:0.483103:0.543141:0.466606:0.368580:0.466606:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.368580:0.517921:0.377847:0.517921:0.377847:0.501452:0.368580:0.501452:0.000000 :@0.371104:0.516417:0.377519:0.516417:0.377519:0.500599:0.371104:0.500599:0.000000 = , por los elementos :@0.377519:0.517805:0.572000:0.517805:0.572000:0.501307:0.377519:0.501307:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de la segunda columna del :@0.368580:0.552506:0.563025:0.552506:0.563025:0.536009:0.368580:0.536009:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 determinante.:@0.368580:0.569857:0.468162:0.569857:0.468162:0.553360:0.368580:0.553360:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.146433:0.508016:0.155699:0.508016:0.155699:0.491547:0.146433:0.491547:0.000000 1:@0.155693:0.511184:0.160646:0.511184:0.160646:0.501566:0.155693:0.501566:0.000000 b:@0.146427:0.529407:0.155693:0.529407:0.155693:0.512939:0.146427:0.512939:0.000000 2:@0.155693:0.532568:0.160646:0.532568:0.160646:0.522950:0.155693:0.522950:0.000000 b:@0.407764:0.508023:0.417031:0.508023:0.417031:0.491554:0.407764:0.491554:0.000000 1:@0.417032:0.511184:0.421985:0.511184:0.421985:0.501566:0.417032:0.501566:0.000000 b:@0.407764:0.529407:0.417031:0.529407:0.417031:0.512939:0.407764:0.512939:0.000000 2:@0.417032:0.532568:0.421985:0.532568:0.421985:0.522950:0.417032:0.522950:0.000000 ©:@0.079553:0.731952:0.079553:0.720342:0.061505:0.720342:0.061505:0.731952:0.000000:0.000000 maya:@0.080774:0.720340:0.080774:0.684191:0.055786:0.684191:0.055786:0.720340:0.000000:0.000000:0.000000:0.000000 ®EDUCACIÓN – Libro resuelto solo para fines didácticos – Prohibida su reproducción :@0.079553:0.684191:0.079553:0.268044:0.061505:0.268044:0.061505:0.684191:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000