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Sahithyan's S1
Sahithyan's S1 — Mathematics

Solutions of Homogenous Systems

Consider the system:

Am×nXn×1=Om×1A_{m\times n}X_{n\times 1}=O_{m\times 1}

A homogenous system is consistent, because X=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  ⟹  infinitely many solutions\text{Rank }A =\text{Rank }(A|B) <n \implies \text{infinitely many solutions}