En ejemplos de páginas anteriores se ha observado que un sistema lineal es compatible en aquellos casos en los que el rango de la matriz de coeficientes coincide con el de la ampliada, y sólo en esos casos. Este resultado, válido para cualquier sistema, se recoge en el teorema de Rouché-Fröbenius.

La condición necesaria y suficiente para que el sistema S:{ a 11 x 1 + a 12 x 2 + ········ + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + ········ + a 2n x n = b 2 ········ ·· ········ ·· ········ ·· ········ ·· ··· a m1 x 1 + a m2 x 2 + ········ + a mn x n = b m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8aspq0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@C8C7@   tenga solución es que el rango de la matriz de los coeficientes A MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8aspq0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaWGbbaaaa@38F0@   coincida con el rango de su matriz ampliada A ˜ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8aspq0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaceWGbbGbaGaaaaa@38FF@ , siendo ( a 11 a 12 ···· a 1n a 21 a 22 ···· a 2n ···· ···· ···· ···· a m1 a m2 ···· a mn ︸ A matriz   de   coeficientes     |    b 1 b 2 ··· b m ) ︸ A ˜  matriz   ampliada MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8aspq0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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aiaaysW7aiaawIa7aiaaysW7faqabeabbaaaaeaacaWGIbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamOyamaaBaaaleaacaaIYaaabeaaaOqaaiaacElacaGG3cGaai4TaaqaaiaadkgadaWgaaWcbaGaamyBaaqabaaaaaGccaGLOaGaayzkaaaaleaaceWGbbGbaGaacaaMf8UaaeyBaiaabggacaqG0bGaaeOCaiaabMgacaqG6bGaaGjbVlaabggacaqGTbGaaeiCaiaabYgacaqGPbGaaeyyaiaabsgacaqGHbaakiaawIJ=aaaa@AFBF@

Demostración     

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