next up previous
Example Reduce the matrix $A$.

$\displaystyle A = \left(\begin{array}{rrrr} 1&-2&3&-1\\ 2&-1&2&2\\ 3&0&2&3 \end{array}\right) $

Answer If the lower elements of the diagonal are all zeros, then the matrix is called echelon matrix. If the leading elements of all rows are $1$, then the matrix is called reduced and denoted by$A_{R}$．


  $\displaystyle {}$ $\displaystyle \left(\begin{array}{rrrr} 1&-2&3&-1\\ 2&-1&2&2\\ 3&0&2&3 \end{a... ...} \left(\begin{array}{rrrr} 1&-2&3&-1\\ 0&3&-4&4\\ 3&0&2&3 \end{array}\right)$  
  $\displaystyle \stackrel{-3 R_{1} + R_{3}}{\rightarrow }$ $\displaystyle \left(\begin{array}{rrrr} 1&-2&3&-1\\ 0&3&-4&4\\ 0&6&-7&6 \end{... ...{rrrr} 1&-2&3&-1\\ 0&1&-\frac{4}{3}&\frac{4}{3}\\ 0&6&-7&6 \end{array}\right)$  
  $\displaystyle \stackrel{-6 R_{2} + R_{3}}{\rightarrow }$ $\displaystyle \left(\begin{array}{rrrr} 1&-2&3&-1\\ 0&1&-\frac{4}{3}&\frac{4}{... ...gin{array}{rrrr} 1&-2&3&-1\\ 0&1&0&-\frac{4}{3}\\ 0&0&1&-2 \end{array}\right)$  
  $\displaystyle \stackrel{2 R_{2} + R_{1}}{\rightarrow }$ $\displaystyle \left(\begin{array}{rrrr} 1&0&0&\frac{7}{3}\\ 0&1&0&-\frac{4}{3}\\ 0&0&1&-2 \end{array}\right) .$