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