Solution for a particular column vector implies solution for all column vectors.

Let $\displaystyle A$ be a square matrix. Show that if the system $\displaystyle AX=B$ has a unique solution for some particular column vector $\displaystyle B$, then it has a unique solution for all $\displaystyle B$.

So I'm not really sure how to go about this since there's no assumption that $\displaystyle A$ is invertible, and I'm assuming I have to utilize row operations or the like, but I just can't make any connections. Any help would be appreciated, thanks.