Solving a matrix for color manipulation I'm making an application that deals with color transforms. The idea is that if you give it an RGB color and apply a color matrix transform it outputs another color. In this case I'm giving the color [255,0,0] and returns [Rf,Gf,Bf]
$$
\begin{bmatrix} 1&.5&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix}
\begin{bmatrix} 255\\0\\0 \end{bmatrix}=
\begin{bmatrix} Rf\\Gf\\Bf \end{bmatrix}
$$
This is pretty straight forward to solve. 
However, how would I solve this if I don't know what the color matrix is but I know what the start and end colors are? So something like this:
$$
\begin{bmatrix} C1&C2&C3 \\ C4&C5&C6 \\ C7&C8&C9 \end{bmatrix}
\begin{bmatrix} 255\\0\\0 \end{bmatrix}=
\begin{bmatrix} 102\\0\\0 \end{bmatrix}
$$
Thanks!
 A: First, it is better to use 
normalized color components in range $[0,1]$.
Second, we need to handle black color $(0,0,0)$ as well.
Here is one possible way to go: add a fourth component,
$1$ for all colors.
Given four pairs of RGB-colors, 
let $T$ be a transformation matrix, 
$U$ is a non-singular matrix, which columns are the original colors, 
and
$V$ is a non-singular matrix, which columns are the transformed colors:
\begin{align}
T\cdot U&=V.
\end{align}
Then the transformation matrix is just
\begin{align}
T&=V\cdot U^{-1}.
\end{align}
For example, 
\begin{align}
U&=
\begin{bmatrix} 
1&0&0&0 \\ 
0&1&0&0 \\ 
0&0&1&0 \\ 
1&1&1&1 
\end{bmatrix}
\\
V&=
\begin{bmatrix} 
1&0&1&1 \\ 
1&1&0&1 \\ 
0&1&1&1 \\ 
1&1&1&1 
\end{bmatrix}
\\
U^{-1}&=
\begin{bmatrix} 
 1& 0& 0&0\\
 0& 1& 0&0\\
 0& 0& 1&0\\
-1&-1&-1&1
\end{bmatrix}
\\
T=V\cdot U^{-1}
&=
\begin{bmatrix} 
 0&-1& 0&1 \\
 0& 0&-1&1 \\
-1& 0& 0&1 \\
 0& 0& 0&1
\end{bmatrix}
\end{align}
And here is a sample set of colors transformed 
by this matrix $T$:

