Consider the system: Am×nXn×1=Om×1A_{m\times n}X_{n\times 1}=O_{m\times 1} Am×nXn×1=Om×1 A homogenous system is consistent, because X=OX=OX=O is always a solution. Rank A=Rank (A∣B)=n ⟺ unique solution exists\text{Rank }A = \text{Rank }(A|B)=n \iff \text{unique solution exists}Rank A=Rank (A∣B)=n⟺unique solution exists Rank A=Rank (A∣B)<n ⟹ infinitely many solutions\text{Rank }A =\text{Rank }(A|B) <n \implies \text{infinitely many solutions}Rank A=Rank (A∣B)<n⟹infinitely many solutions