Show that when $BA = I$, the solution of $Ax=b$ is unique I'm just getting back into having to do linear algebra and I am having some trouble with some elementary questions, any help is much appreciated.
Suppose that $A = [a_{ij}]$ is an $m\times n$ matrix and $B = [b_{ij}]$ and is an $n\times m$ matrix and $BA = I$ the identity matrix that is $n \times n$.
Show that if for some $b \in \Bbb{R}^m$ the equation $Ax = b$ has a solution, then the solution is unique.

The usual method for solving uniqueness problems is generally this: assume you have two solutions, say $x$ and $y$. Then do manipulation, use theorems, whatever, and somehow show that $x=y$.
In your case, if $Ax=b$ and $Ay=b$, then multiply both sides on the left by $B$. Then $B(Ax)=Bb$ and $B(Ay)=Bb$. Can you take it from here?
