I have been doing simulation of a system. The underlying system reduces to a linear system $A x = b$. For two systems with matrices $A_1$ and $A_2$, I am getting same solutions for $A_1 x = b $ and $A_2 x = b $ ($b = [0,0,1]$ is same in both cases). I thought that since solutions are the same, row reduced echelon forms of $A_1$ and $A_2$ must be the same (the other way is definitely true)! This is not the case. So I am wondering, given that I have not made any mistake in my simulations (and solutions by maxima CAS), what I can say about $A_1$ and $A_2$? In general I am asking this:
What is the relation between $A_1$ and $A_2$ if the solutions of system $[ A_1 \mid b ]$ and $[ A_2 \mid b ]$ are the same?
Update
Following two matrices have the same solution for $b = [ 0, 0, 1]$, all $a_{ij}$ are randomly chosen between 0 and 0.01. I have only simulated all systems for $b = [ 0, 0, 1]$.
[ a12 - a21 0 ]
[ ]
(%o9) [ - a12 a23 + a21 - a32 ]
[ ]
[ 1 1 1 ]
[ a12 a23 + a12 - a21 - a21 a32 ]
[ ]
(%o10) [ - a12 a23 + a21 - a32 ]
[ ]
[ 1 1 1 ]
Their Echelon forms (as returned by maxima ).
[ 1 1 1 ]
[ ]
[ a12 ]
(%o11) [ 0 1 --------- ]
[ a21 + a12 ]
[ ]
[ 0 0 1 ]
[ 1 1 1 ]
[ ]
[ a32 - a12 ]
(%o12) [ 0 1 - --------------- ]
[ a23 + a21 + a12 ]
[ ]
[ 0 0 1 ]
PS: I have passable knowledge of undergraduate linear algebra. Feel free to comment on vector spaces (or some other mathematical structures) spanned by $A_1$ and $A_2$. A quick glance in relevant chapters of Hoffman and Kunze did not reveal anything about this problem.
I don't think there is necessarily any meaningful relation between $A_1$ and $A_2$. They can be two different systems of linear equations with the same solution that happen to be resulted assuming the same $b$. Assume $b=(1, 1)$ and $x=(1, 0)$. How many $A$ can satisfy $Ax=b$?