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Common form of system of linear equations is A*X = B, X is unknown. But how to find A, if X and B are known?

A is MxN matrix, X is column vector(N), B is column vector(M)

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    If $\mathbf x$ and $\mathbf b$ are vectors, your problem is underdetermined.2010-12-16
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    I have system of equations, such as B1 = A*X1....Bn = A*Xn2010-12-16
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    Then you can treat your set of column vectors as the columns of a matrix. mpiktas's approach then applies.2010-12-16
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    @J.M. I got it. Thank You.2010-12-16
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    @J. M.: ...provided that $n=N$.2010-12-16
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    @Hans, Hmm, I missed that "non-square" condition... in which case, we're back to having an underdetermined problem.2010-12-16

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If all matrices are square, then $A=BX^{-1}$.

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Do it row by row. Row $k$ in $A$ multiplied by the column vector $X$ equals the $k$th entry in the vector $B$. This is a single equation for the $N$ entries in that row of $A$ (so unless $X$ is zero, you get an $(N-1)$-parameter set of solutions for each row).