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Is there an easy way to get the matrix of coefficients from a product of a matrix and a vector?

I have a matrix multiplication of the form

$$ B = A x $$


$$ \begin{pmatrix} a_{11} x_1 + a_{12} x_2 + a_{13} x_3 \\ a_{21} x_1 + a_{22} x_2 + a_{23} x_3 \\ a_{31} x_1 + a_{32} x_2 + a_{33} x_3 \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix} \cdot \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} $$

Is there a way in Sage to factor $B$ in a way where I give it $x$ and it returns $A$?

Edited from a question posted by someone else at the Mathematica Stackexchange