$Ax = y$, possible to isolate matrix $A$? I have the matrix equation of real coefficients
$Ax = y$
where $y$ and $x$ are of different size. Is it possible to do a simple operation which will turn $A$ into a matrix function of both $y$ and $x$? I guess I can multiply on the right side with $x^{\top}$ and divide by the norm to yield the identity-matrix, but is there a smoother way to proceed?

Does this help?!
Let $ A\begin{bmatrix} 2  \\ 3  \end{bmatrix} =\begin{bmatrix} 10 \\ -2 \\ 5 \end{bmatrix}$, then we can find $A$ by assuming
$$A=\begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix}$$ we get 3 equations
$$2a+3b=10,~ 2c+3d=-2,~ 2e+3f=5$$ as there are 6 unknowns one will get many solutions and hence many matrices.
