Reverse engineering a matrix in RREF

Question

I know that multiple matrices can have the same RREF, but a matrix has a unique RREF, I am trying to reverse engineer a 3x3 RREF matrix to get an abstract matrix of the same size that has 2 columns known, but am really confused how to do so, as I am only able to get one solution when multiple solutions exist. For example what I am trying to do if that given an RREF and 2 columns find the third one.

Answer 1

For example, suppose the RREF is

$$ \pmatrix{1 & 0 & 1\cr 0 & 1 & 1\cr 0 & 0 & 0\cr}$$

and your matrix with first two columns known is supposed to be $$ \pmatrix{1 & 2 & x\cr 3 & 4 & y\cr 5 & 6 & z\cr} $$

Now the RREF has the same null space as the original matrix, in this case the span of $$\pmatrix{1\cr 1\cr -1}$$ That says $$ \pmatrix{1 & 2 & x\cr 3 & 4 & y\cr 5 & 6 & z\cr} \pmatrix{1\cr 1\cr -1\cr} = \pmatrix{3 - x\cr 7 - y\cr 11 - z\cr} = \pmatrix{0\cr 0\cr 0\cr} $$ from which you can read out $x,y,z$.

Here there is just one solution.