Trying to write a program to check orderings

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I'm currently helping a professor research something in Linear Algebra involving multiplying a bunch of matrices together and seeing how order influences the LU decomposition of the product, however, there are lots of different ways to multiply these matrices together, and there are only a few good orderings, IE, ones that produce purely polynomial entries in the L and U parts.

I want to be able to have a program create these matrices and try all of these orderings and have them determine if the ordering is good or not. The way I can check if the ordering is good or not is if the entry contains a "/", is there any way for me to write a java program or in some other language that allows me to put variables in a matrix, I want to be able to have the program look through the entries and tell me an ordering is bad if say, it looks like this: (abc)/(1+ab), but good if it looked like: a^2(bc+da)+3abcd

Can anyone help?