According to Wikipedia[1]Matrix Inversion is also O(n^3) and according to the note can be even better by inverting smaller matrices within and multiplying[2].

My linear algebra isn't very good so I'll have to look into LU Factorization but is there vast difference between the computational performance of the two operations? (Assuming you don't need solve Ab=c for different values of b and c)

Also is LU Factorization used often in machine learning instead of inverse?

[1]: http://en.wikipedia.org/wiki/Computational_complexity_of_mat...

[2]: Because of the possibility to blockwise invert a matrix, where an inversion of an n×n matrix requires inversion of two half-sized matrices and 6 mulitplications between two half-sized matrices, and since matrix multiplication has a lower bound of Ω(n2 log n) operations[17], it can be shown that a divide and conquer algorithm that uses blockwise inversion to invert a matrix runs with the same time complexity as the matrix multiplication algorithm that is used internally. source is [1]