Suppose $R$ and $R'$ are two $2\times3$ row reduced echelon matrices if $RX=0$ and $R'X$ have axactly same solutions then prove that $R=R'$.
My try:
Let $x,y,z$ be the same solution of $R$ and $R'$ The most general form of $R$ will be $$R=\begin{pmatrix} 1 & a & b\\ 0 & 1 & c\\ \end{pmatrix}$$ Similarly $$R'=\begin{pmatrix} 1 & p & q\\ 0 & 1 & r\\ \end{pmatrix}$$
Using given Condition we have $$x+ay+bz=0,\\ y+cz=0$$ Now this system has one free variable... For $R'$ will give similar equations but the trouble is free variable
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$\begingroup$I think, this is not true for matrices in row ecelon form. Consider $R=\begin{pmatrix}1&1&1\\0&1&1\end{pmatrix}$ and $R'=\begin{pmatrix}1&0&0\\0&1&1\end{pmatrix}$
EDIT: For matrix in a reduced row echelon form, i.e. $R=\begin{pmatrix}1&0&a\\0&1&b\end{pmatrix}$ and $R=\begin{pmatrix}1&0&p\\0&1&q\end{pmatrix}$ compare the lines in $yz$ space from the first equation $$y+bz=0$$ and from the second equation $$y+qz=0.$$ The lines defined by these equations equal and you can easily conclude $b=q$. You can then continue in a similar way with the first line.
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