Si A ¯ MathType@MTEF@5@5@+=feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeqabaWaaqaafaaakeaaceWGbbGbaebaaaa@3982@    es la matriz ampliada de un sistema lineal de ecuaciones, al hacer operaciones elementales sobre ella obtenemos matrices que corresponden a sistemas equivalentes al dado.

EJEMPLO.

x 2x x +y −y −y +z −3z +z +t +4t −3t = = = 3 0 1      ↔     ( 1 2 1    1 −1 −1    1 −3    1    1    4 −3 )   ( x y z t )=( 3 0 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@825E@

Operaciones elementales:

x 2x x +y −y −y +z −3z +z +t +4t −3t = = = 3 0 1 ( 1 2 1    1 −1 −1    1 −3    1    1    4 −3 | 3 0 1 ) ↓ ↓ x  +y    +z  +   t=   3    −3y−5z+2t=−6⊲    −2y        −4t=−2⊲ Segunda-2Primera Tercera-Primera ⊳ ⊳ ( 1 0 0    1 −3 −2    1 −5    0    1    2 −4 | 3 −6 −2 ) ↓ ↓ x  +y    +z  +   t=     3      y        +2t=     1⊲     −3y−5z+2t=−6⊲ -(1/2)Tercera Segunda ⊳ ⊳ ( 1 0 0    1  1 −3    1    0    −5    1    2  2 | 3 1 −6 ) ↓ ↓ x  +y    +z  +   t=     3      y        +2t=     1            −5z+8t=−3⊲ Tercera+3Segunda ⊳ ( 1 0 0    1 1 0    1 0    −5 1    2  8 | 3 1 −3 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@D79F@

El Método de Gauss para obtener un sistema escalonado se corresponde, paso a paso, con el proceso de obtener una matriz escalonada mediante operaciones elementales. Por ello, llamamos Método de Gauss tanto al proceso de escalonamiento de un sistema como al de una matriz. Puedes ampliar información sobre el método de Gauss en el tema de Sistemas Lineales

A continuación veremos la relación que existe entre el rango de las matrices de coeficientes y ampliada  y el carácter del sistema (compatible-incompatible; determinado-indeterminado).