Why don't you simply use normal matrix arithmetic to get the "factorization" you need - if you need the result at all. You have different ways to do so:
Why do you want to solve for 21 variables when you only have 12 equations???????????????
The way to do it is exactly the same as I have shown above (2nd), just do it symbolically. the 3x3 matrix can be anything as long as the matrix isn't singular.