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October 9th, 2015, 11:59 AM  #1 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  Reduced Row Echelon Form is Unique Proof
Let R1 and R2 be reduced row echelon forms of A. Then R1<>R2 by elementary row operations. Therefore: R1 and R2 have same number of nonzero rows, same location of the (0,..0,1,0,..0)columns, and same elements in the nonzero rows. Examples $\displaystyle \begin{vmatrix} 1 &0 & 2 &1 \\ 0 & 1&3 &5 \\ 0 &0 & 0 & 0 \end{vmatrix} and\begin{vmatrix} 1 &4 &0 &0 \\ 0& 0 &1 &0 \\ 0 &0 &0 &1 \end{vmatrix}$ $\displaystyle \begin{vmatrix} 1 &0 & 2 &5 \\ 0 & 1&7 &2 \\ 0 &0 & 0 & 0 \end{vmatrix} and\begin{vmatrix} 1 &0 &4 &6 \\ 0& 1 &1 &3 \\ 0 &0 &0 &0 \end{vmatrix}$ 

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