Inverting a vector If I have $Ax=b$ where $A$ is $n$ by $n$ while $x$ and $b$ are $n$ by 1, is it possible to find $A$ given $x$ and $b$. The idea would be some sort of $x^{-1}$ operation on the right of both equations but I'm not sure how to go about it. If necessary, we can consider a specific case of $n=2$.
Also, is the $A$ unique or not?

An easy answer is pick a component of $x$, say $j$, that is not equal to zero (If $x = 0$, then $b$ better be zero; otherwise no solution). Then set the $j$th column of A, $A_j = b/x_j$. Choose the rest of $A$ matrix to be zero. 
