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1. Classificació d'un sistema d'equacions: Teorema de Rouché-Frobenius

Donem un criteri per classificar un sistema d'equacions lineals.

Donat un sistema d'equacions lineals, sigui

    M la matriu associada al sistema

    M' la matriu ampliada

Donem un criteri per classificar un sistema d'equacions lineals.

Teorema de Rouché-Frobenius:

box enclose times Si bold space bold italic r bold italic a bold italic n bold italic g bold space bold M bold space bold not equal to bold space bold italic r bold italic a bold italic n bold italic g bold space bold M bold apostrophe space rightwards double arrow space sistema bold space bold incompatible times Si space bold italic r bold italic a bold italic n bold italic g bold space bold M bold space bold equals bold italic r bold italic a bold italic n bold italic g bold space bold M bold apostrophe equals straight r space space rightwards double arrow open curly brackets table attributes columnalign left end attributes row cell Si bold space bold r bold equals bold nombre bold space bold d bold apostrophe bold incògnites space space rightwards double arrow space bold compatible bold space bold determinat space end cell row cell Si space bold r bold less than bold nombre bold space bold d bold apostrophe bold incògnites space space rightwards double arrow space bold compatible bold space bold indeterminat end cell end table close end enclose  Exemples:   

Exemple space 1

Classifiqueu el sistema: 

open table row cell x minus 2 y plus 3 z equals 3 end cell row cell 2 x plus y minus z equals 1 end cell row cell negative x minus 3 y plus 4 z equals negative 1 end cell end table close curly brackets

Esglaonem la matriu ampliada per tal de calcular els rangs de la matriu associada i de l'ampliada: 

 open parentheses right enclose table row 1 cell negative 2 end cell 3 row 2 1 cell negative 1 end cell row cell negative 1 end cell cell negative 3 end cell 4 end table end enclose table row 3 row 1 row cell negative 1 end cell end table close parentheses rightwards arrow space space table row blank row cell negative bold 2 f subscript 1 plus f subscript 2 space end subscript end cell row cell bold space bold space bold space f subscript 1 plus f subscript 3 end cell end table space open parentheses right enclose table row 1 cell negative 2 end cell 3 row 0 5 cell negative 7 end cell row 0 cell negative 5 end cell 7 end table end enclose table row 3 row cell negative 5 end cell row 2 end table close parentheses space space space space rightwards arrow space space space space table row blank row blank row cell f subscript 2 plus f subscript 3 end cell end table space space open parentheses right enclose table row 1 cell negative 2 end cell 3 row 0 5 cell negative 7 end cell row 0 0 0 end table end enclose table row 3 row cell negative 5 end cell row cell negative 3 end cell end table close parentheses space

Recordem que el rang d'una matriu, un cop esglaonada, és el nombre de files no nul·les:

   open table attributes columnalign right end attributes row cell rang space straight M equals 2 space end cell row cell rang space straight M apostrophe equals 3 end cell end table close curly brackets space rightwards double arrow space bold space bold Sistema bold space bold incompatible space

Exemple space 2

Classifiqueu el sistema: 

open table row cell x minus 2 y plus 3 z equals 3 end cell row cell 2 x plus y minus z equals 1 end cell row cell negative x minus 3 y plus 4 z equals 2 end cell end table close curly brackets

Esglaonem la matriu ampliada per tal de calcular els rangs de la matriu associada i de l'ampliada: 

 open parentheses right enclose table row 1 cell negative 2 end cell 3 row 2 1 cell negative 1 end cell row cell negative 1 end cell cell negative 3 end cell 4 end table end enclose table row 3 row 1 row 2 end table close parentheses rightwards arrow space space table row blank row cell negative bold 2 f subscript 1 plus f subscript 2 space end subscript end cell row cell bold space bold space bold space f subscript 1 plus f subscript 3 end cell end table space open parentheses right enclose table row 1 cell negative 2 end cell 3 row 0 5 cell negative 7 end cell row 0 cell negative 5 end cell 7 end table end enclose table row 3 row cell negative 5 end cell row 5 end table close parentheses space space space space rightwards arrow space space space space table row blank row blank row cell f subscript 2 plus f subscript 3 end cell end table space space open parentheses right enclose table row 1 cell negative 2 end cell 3 row 0 5 cell negative 7 end cell row 0 0 0 end table end enclose table row 3 row cell negative 5 end cell row 0 end table close parentheses space

Recordem que el rang d'una matriu, un cop esglaonada, és el nombre de files no nul·les:

   open table attributes columnalign right end attributes row cell rang space straight M equals 2 space end cell row cell rang space straight M apostrophe equals 2 end cell end table close curly brackets space rightwards double arrow space bold space Sistema space compatible


     i el nombre de incógnites, x, y, z és 3  

   Per tant: 

     rango 2 < 3 nombre incógnites      Sistema compatible indeterminat

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