• ↑
  • MAT223 Readings 01

Row Reduction in Linear Algebra

• Reduced Row Echelon Form
• To reduce a matrix $A$ to RREF, you have to:
  • Find the first column with a non-zero entry. Move the row with the non-zero entry to the top call it $a$.
  • Divide the top row by $a$ to obtain a leading $1$.
  • Add or subtract multiples of the resulting row from all other rows in the entire matrix $A$. To make each entry above and below have leading $1$s.
• Example:
  • Find the RREF of:
  • $\left[\begin{array}{cccc}0& 2& 0& 2\\ 3& 3& 6& 0\\ 2& 1& 4& -1\end{array}\right]$
  • Move (2) to the top
  • $\left[\begin{array}{cccc}3& 3& 6& 0\\ 0& 2& 0& 2\\ 2& 1& 4& -1\end{array}\right]$
  • Divide (1) by $a_{11}$
  • $\left[\begin{array}{cccc}1& 1& 2& 0\\ 0& 2& 0& 2\\ 2& 1& 4& -1\end{array}\right]$
  • (3) - 2(1)
  • $\left[\begin{array}{cccc}1& 1& 2& 0\\ 0& 2& 0& 2\\ 0& -1& 0& -1\end{array}\right]$
  • (2)/2
  • $\left[\begin{array}{cccc}1& 1& 2& 0\\ 0& 1& 0& 1\\ 0& -1& 0& -1\end{array}\right]$
  • (3)+(2)
  • $\left[\begin{array}{cccc}1& 1& 2& 0\\ 0& 1& 0& 1\\ 0& 0& 0& 0\end{array}\right]$
  • (1)-(2)
  • $\left[\begin{array}{cccc}1& 0& 2& -1\\ 0& 1& 0& 1\\ 0& 0& 0& 0\end{array}\right]$
• Ex:
  • Reduce:
  • $\left[\begin{array}{cc}1& 3\\ 4& 12\\ 3& 1\end{array}\right]$
  • 4(1)-(2)
  • $\left[\begin{array}{cc}1& 3\\ 0& 0\\ 3& 1\end{array}\right]$
  • Move (2) -> (3)
  • $\left[\begin{array}{cc}1& 3\\ 3& 1\\ 0& 0\end{array}\right]$
  • 3(1)-(2)
  • $\left[\begin{array}{cc}1& 3\\ 0& 8\\ 0& 0\end{array}\right]$
  • (2)/8
  • $\left[\begin{array}{cc}1& 3\\ 0& 1\\ 0& 0\end{array}\right]$
  • (2) = 3(2)-(1)
  • $\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]$