TITLE: Finding solutions for a linear system of equations
QUESTION [6 upvotes]: Got the following problem where I can't find a way to solve:
Knowing $\begin{pmatrix}5\\ 3\\ 6\end{pmatrix}$ is the unique solution for the system  $Ax=\begin{pmatrix}2\\1\\1\end{pmatrix}$, with $A \in \mathbb{R}^{3\times3}$
and $B=\begin{pmatrix}
1 & 2 & 1 & 2 \\ 
1 & 0 & 4 & -1 \\ 
1 & 3 & -3 & 6
\end{pmatrix}$
Find all solutions for $ABx=\begin{pmatrix}2\\1\\1\end{pmatrix}$
What I've tried:

The problem says that $Ax=b$ got unique solution, so I've tried by getting rid of $A$ by using the inverse matrix but it doesn't work sice I don't know $A$.
Since the constant matrix is $\begin{pmatrix}2\\1\\1\end{pmatrix}$ for both systems, I've tried $ABx = Ax$ but that also doesn't work for me.

Thanks in advance for your help and sorry for my bad English.
Lucas

REPLY [4 votes]: As observed by @GerryMyerson, there is no need for $B$ to be a square matrix.
From
$$
Ax=b \implies x_0=A^{-1}b
$$
and
$$
ABx=b \implies Bx=A^{-1}b
$$
you get
$$
Bx=x_0
$$
where $x_0$ is the known solution of the first system.