Universidad Tecnológica ecnoló gica de El
Asignatura: Matemática I Tema: Modelos matemáticos para resolver sistemas de ecuaciones lineales (método cramer y Gauss-Jordán) Conceptualiz Conceptualización ación básica de determinantes determinantes (método Sarrus y Coactores)
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Contenido!
Índice
"á#$
Introducción _______________________ ___________________________________ _________________________ __________________________ _______________________3 __________3
Objetivos ____________________ _____________________________________________ __________________________________________________ ____________________________4 ___4 Desarrolo __________________________ ________________________________________ ___________________________ __________________________ ____________________5 _______5 Conclusión _____________________________ ____________________________________________ _____________________________ __________________________16 ____________16 ____________________________ __________________ ___________________ ____________________ ___________________ _____________17 ____17 Bibliografía ___________________
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Introducción En el presente presente trabajo trabajo se detalla detalla los diferent diferentes es modelos modelos matemáti matemáticos cos para resolver sistemas de ecuaciones lineales usando el método de Cramer que detalla el trabajo más conocido de Cramer !Introduction á l"anal#se des lignes courb courbes es algé algébri briqu ques! es! $%&'( $%&'() ) es un estud estudio io # una clasi clasific ficac ación ión de curva curvas s algebraicas* la regla de Cramer apareció en el apéndice+ ,unque la regla lleva su nombr nombre e varia variante ntes s de la idea idea básica básica fuero fueron n plant plantead eadas as antes antes por por otros otros matemáticos+ -in embargo la notación superior de Cramer a#udó a aclarar # populari.ar la técnica* # también denota el método de /auss01ordán que detalla de una serie de algoritmos del algebra lineal para determinar los resultados de un sistema de ecuaciones lineales # así 2allar matrices e inversas+ Como también se detallara la conceptuali.ación básica sobre determinantes usan usando do el método método de -arrus -arrus que es
válid válido o 3nica 3nicamen mente te para calcul calcular ar el
determinante de una matri. 454 # recibe su nombre por el matemático francés 6ierre 7réderic -arrus* también se detalla el método cofactor+
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Objetivos
Objetivo General 7ormula 7ormularr ejemplos ejemplos de modelos modelos matemáticos matemáticos simples simples para para la resoluci resolución ón de ecua ecuac cione iones s line lineal ales es por por los los méto método dos s de Cram Cramer er # /auss auss01 01or ordá dán n # la conce conceptu ptuali ali.a .ació ción n básica básica sobre sobre deter determin minant antes es por por el métod método o de -arru -arrus s # Cofactor+
Objetivos Específicos $ 1.
Definir modelos matemáticos simples para la resolución de ecuaciones
lineales por los métodos de Cramer # /auss01ordán+ Enunciarr la concept conceptuali uali.aci .ación ón básica básica sobre sobre determin determinant antes es usando usando el 2. Enuncia método de -arrus # Cofactor+
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Desarrollo Ecuaciones ineales !"todo de #ramer: 8a regla de Cramer es un teorema del álgebra lineal que da la solución de un siste sistema ma linea lineall de ecuaci ecuacion ones es en térmi términos nos de deter determin minant antes es++ 9ecib 9ecibe e este este nombre en 2onor a /abriel Cramer+ 8a regla de Cramer es de importancia teórica porque da una e5presión e5plícita para para la soluc solución ión del del sistem sistema+ a+ -in -in embarg embargo o para para siste sistemas mas de ecuac ecuacion iones es lineales de más de tres ecuaciones su aplicación para la resolución del mismo resul resulta ta e5ces e5cesiva ivamen mente te costo costosa sa:: compu computac tacion ionalm alment ente e es inefic ineficien iente te para para grandes matrices # por ello no es usado en aplicaciones prácticas que pueden implic implicar ar muc2a muc2as s ecua ecuacio ciones nes++ -in -in embarg embargo o como como no es nece necesar sario io pivota pivotar r matr matric ices es es más más efic eficie ient nte e que que la elim elimin inac ació ión n gaus gaussi sian ana a para para matr matric ices es peque;as particularmente cuando son usadas operaciones -I
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#aracterísticas: 8a regla de Cramer se aplica para resolver sistemas de ecuaciones lineales que cumplan las siguientes condiciones: • •
El n3mero de ecuaciones es igual al n3mero de incógnitas+ El determinante de la matri. de los coeficientes es distinto de cero+
$ormas de trabajarlas: =ales sistem sistemas as son son sistem sistemas as compa compatib tibles les deter determin minado ados s # se denomi denomina nan n sistemas de Cramer+
-ea > el determinan te de la matri. de coef icientes+
=odo sistema de Cramer tiene una sola solución $es decir es un sistema compatible determinado) que viene dada por las siguientes e5presiones:
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% & % ' % + + + % n s o n l o s d e t e r m i n a n t e s q u e s e o b t i e n e a l
s us u s t it i t u ir i r l o s c o ef e f i ci c i e nt n t e s d e l ? @ m i em e m b ro r o $ l os o s t é rm r m i no no s independientes) en la %A columna en la ?A columna en la 4A columna # en la enésima columna respectivamente+
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Ejercicio ?:
El determinante de la matri. es
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!"todo de Gauss%&ord'n: En matemáticas la eliminación de /auss01ordán llamada así debido a Carl 7ried 7riedric ric2 2 /aus /auss s # il2e il2elm lm 1ordá 1ordán n es un algor algoritm itmo o del del álgebr álgebra a linea lineall para para determin determinar ar las solucion soluciones es de un sistema sistema de ecuacio ecuaciones nes lineale lineales s encont encontrar rar matrices e inversas+ n sistema de ecuaciones se resuelve por el método de /auss cuando se obtienen sus soluciones mediante la reducción del sistema
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• • • • •
•
•
•
na na varia variante nte inter interes esant ante e de la elimin eliminac ación ión de /auss /auss es la que que llamam llamamos os elimi elimina naci ción ón de /auss /auss01o 01ord rdan an $debi $debido do al menci mencion onado ado /auss /auss # a il2e il2elm lm 1ordan) esta consiste en ir obteniendo los % delanteros durante los pasos uno al cuatro $llamados paso directo) así para cuando estos finalicen #a se obtendrá la matri. en forma escalonada reducida+
Ejercicios: Ejercicio %
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-olución:
#onceptuali(ación )'sica sobre determinantes* !"todo de +arrus* 8a regl regla a de -arr -arrus us es un méto método do fác fácil para para memo memori ri.a .arr # calc calcul ular ar un determinante 44+ 9ecibe su nombre del matemático francés 6ierre 7rédéric -arrus que la introdujo en el artículo Fouvelles mét2odes pour la résolution
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,egla de +arrus:
8a regla de -arrus: las diagonales a.ules se suman # las diagonales rojas se resta
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•
-i la dimensión es 3 la matri. es de la forma
-e define su determinante como:
8a fórmula anterior se conoce como la Regla de Sarrus +
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Ejercicio %:
El determinante de la matri. es
Ejercicio 2:
,plicamos ,plicamos la regla regla de -arrus -arrus
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8leva
signo
negativo porque la suma de los valores del subíndice es impar ?%L4
DETE,!I-A-TE DE .-A !AT,I/: El resultado de la matri. es la suma de los cofactores de una columna o renglón+ Matriz singular. - Cuando el determinante de una matri. es igual a cero se4 dice que es singular+ Matriz transpuesta. - Es intercambiar filas por columnas+
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En el presente trabajo se detalla problemas matemáticos simples como por ejemplo en el método de cramer* que nos permite obtener directamente los montos e intereses en distintas situaciones: simple compuesta # continua* # para cada una de estas se aplicaran aplicaran distintas propiedades+ propiedades+ En nuestras nuestras vidas necesitamos dic2as propiedades #a que nos facilita la resolución de problemas ma#ormente bancarios+ =ambién podemos decir que la eliminación de /auss0 1ordá 1ordán n nos son son 3tile 3tiles s en la resolu resolució ción n de ecuac ecuacio iones nes lineal lineales es brind brinda a el resultado más preciso # permite 2asta más de %' ecuaciones simultáneas+ 8as diferentes formas de resolución nos llevó a un enfoque muc2o más amplio en la resolución del determinante por ejemplo el método de cofactores se usa muc2o en la resolución de el determinante de ?5? por el método de -arrus+
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Método de Cramer +en l,nea. /isponible en! 0ttps!11222$uoc$edu1in1emat01docs1S3456esolucion$pd 2ttp:PPP+vitutor+comalgebrasistemasQ?(Icramer+2tml 0ttps!11222$matesacil$com17C1e*ercicios-resueltos-C6M36$0tml
Método de Gauss-Jordan +en l,nea /isponible en!