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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. –  J. M. Dec 16 '10 at 13:04
    
I have system of equations, such as B1 = A*X1....Bn = A*Xn –  qutron Dec 16 '10 at 13:07
    
Then you can treat your set of column vectors as the columns of a matrix. mpiktas's approach then applies. –  J. M. Dec 16 '10 at 13:09
    
@J.M. I got it. Thank You. –  qutron Dec 16 '10 at 13:11
    
@J. M.: ...provided that $n=N$. –  Hans Lundmark Dec 16 '10 at 13:16
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2 Answers 2

up vote 3 down vote accepted

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).

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