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I have added a separate LU decomposition class that extends Matrix to take advantage of the fact that the LU decomposition can be stored in a single matrix, and that matrix can use the built-in Matrix methods of the parent. I've also modified the Matrix class to use the LU Decomposition class to calculate the determinant. Some basic tests are there, but I plan to add a more comprehensive test for the decomposition itself. I left the old LU decomposition code in the Matrix class (I actually didn't see it until I had already started this one) because I don't know which one will be faster. The algorithm I implemented is generally considered to be one of the fastest - but I plan to write a benchmarking script to compare, and then choose and optimize accordingly.