Fastest algorithm for matrix inversion
Gaussian elimination requires $O(n^3)$ operations, not $O(n^2)$.

In general, matrix inversion has the same exponent as matrix multiplication (any matrix multiplication algorithm faster than $O(n^3)$ gives a matrix inversion algorithm faster than $O(n^3)$), see for example P.Burgisser, M.Clausen, M.A.Shokrollahi "Algebraic complexity theory", Chapter 16 "Problems related to matrix multiplication".