Calculating the transform matrix

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I have 2 sets of known 3x1 vectors A = [ x y 1 ] and B =[ x' y' 1 ] which represent points on two coordinates calculated by some MATLAB algorithm. I was wondering what I could do to find the 2x3 transform matrix that turns the x y set into the x' y' set.

[x'] [ ] [x]
[y']=[?] [y]
[1] [1]

B = QA

Q = 2x3 unknown transform matrix

What I've tried:
multiplying both sides by A-1, and I am left with A-1*B which is just a 1x1 scalar value. I've tried multiplying the variables out and I'm left with 2 equations and 6 unknowns. I also have many different values of xy & x'y' pairs given by the MATLAB algorithm but I don't see how that could help me. Any help is appreciated.
 
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The system QA has two equations and six unknowns, the elements of Q. We can solve the system for these unknowns, which gives a solution with four parameters.
 

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