. ▁ ▁More ▁specifically , ▁according ▁to ▁the ▁R ouch é – Cap elli ▁theorem , ▁any ▁system ▁of ▁linear ▁equations ▁( und erd et erm ined ▁or ▁otherwise ) ▁is ▁incons istent ▁if ▁the ▁rank ▁of ▁the ▁augment ed ▁matrix ▁is ▁greater ▁than ▁the ▁rank ▁of ▁the ▁coefficient ▁matrix . ▁If , ▁on ▁the ▁other ▁hand , ▁the ▁ranks ▁of ▁these ▁two ▁matrices ▁are ▁equal , ▁the ▁system ▁must ▁have ▁at ▁least ▁one ▁solution ; ▁since ▁in ▁an ▁under det erm ined ▁system ▁this ▁rank ▁is ▁necessarily ▁less ▁than ▁the ▁number ▁of ▁unknown s , ▁there ▁are ▁indeed ▁an ▁inf init ude ▁of ▁solutions , ▁with ▁the ▁general ▁solution ▁having ▁k ▁free ▁parameters ▁where ▁k ▁is ▁the ▁difference ▁between ▁the ▁number ▁of ▁variables ▁and ▁the ▁rank . ▁ ▁There ▁are ▁algorithms ▁to ▁decide ▁whether ▁an ▁under det erm ined ▁system ▁has ▁solutions , ▁and ▁if ▁it ▁has ▁any , ▁to ▁express ▁all ▁solutions ▁as ▁linear ▁functions ▁of ▁k ▁of ▁the ▁variables ▁( same ▁k ▁as ▁above ). ▁The ▁simplest ▁one ▁is ▁Gaussian ▁elim ination . ▁See ▁System ▁of ▁linear ▁equations ▁for ▁more ▁details . ▁ ▁Hom ogeneous ▁case ▁The ▁hom ogeneous ▁( with ▁all ▁constant ▁terms ▁equal ▁to ▁zero ) ▁under det erm ined ▁linear ▁system ▁always ▁has ▁non - t rivial ▁solutions ▁( in ▁addition ▁to ▁the ▁trivial ▁solution ▁where ▁all ▁the ▁unknown s ▁are ▁zero ). ▁There ▁are ▁an ▁infinity ▁of ▁such ▁solutions , ▁which ▁form ▁a ▁vector ▁space , ▁whose ▁dimension ▁is ▁the ▁difference ▁between ▁the ▁number ▁of ▁unknown