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If there is $A B C = D$. and $A$, $B$ and $D$ are given.

How can we solve for $C$?

I'm able to solve when there's two matrices, but not sure when there are three ($A$, $B$, and $C$)

Thank you.

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    Give the matrix A*B the name E. Then you have E*C=D. Does that help?2011-08-10

1 Answers 1

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Assuming $A$ and $B$ are invertible (square and columns are linearly independent, or one of the many thousands (exaggeration) of possibilities for the Invertible Matrix Theorem) which implies $C$ and $D$ are square, then just break it down into steps:

$A \cdot B \cdot C = D$

Now multiply each side by $A^{-1}$

$A^{-1}\cdot A \cdot B \cdot C = A^{-1} \cdot D $

$I$ is the n x n identity matrix.

$I \cdot B \cdot C = A^{-1} \cdot D $

$B \cdot C = A^{-1} \cdot D$

Now, rinse and repeat.

$C = B^{-1} \cdot A^{-1} \cdot D$

And if you look at it as the others mentioned, if you let $E = AB$, then $E^{-1} = B^{-1} A^{-1}$, so:

$E C = D \implies C = E^{-1} D \implies C = B^{-1} A^{-1} D$

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    @Mikael Noted and used.2011-08-10