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  • MAT223 Readings 01

Rank in Linear Algebra

• A rank is a matrix of the number of leading $1$s in it's Reduced Row Echelon Form.
• Noted as $\mathrm{rank}\left(A\right)$
• Ex:
  • $A=\left[\begin{array}{cccc}0& 2& 0& 2\\ 3& 3& 6& 0\\ 2& 1& 4& -1\end{array}\right]$
  • Reduced Row Echelon Form
  • $R=\left[\begin{array}{cccc}1& 0& 2& -1\\ 0& 1& 0& 1\\ 0& 0& 0& 0\end{array}\right]$
  • See in the Reduced Row Echelon Form that there are two leading 1s.
  • $\mathrm{rank}\left(A\right)=2$
• If we have an $m\times n$ system.
  • It's either inconsistent
  • Consistent with one solution $n=r$
  • Consistent, infinite solutions.
  • $r<n$