Matrix multiplication and matrix inversion have the same asymptotic complexity (see Handbook of Linear Algebra, chapter 47, for references, or numerous places online or look in a convenient numerical linear algebra textbook).  Let's say that matrix multiplication is complexity O(n^w).  In the 60s it was proved that w<2.81 (Strassen), and more recent advancements have shown w<2.38.  In the last few years, there are claims that w<2.7323.  It's an open question how close matrix inversion and multiplication is to w=2.  In short, it's not right to say that matrix inversion is O(n^3).