It is said that if matrix multiplication of $n\times n$ matrices is doable in $O(n^\omega)$ for some $\omega\geq 2$, so is Gaussian elimination and vice versa.
Now, this could imply that it is somehow possible to do Gaussian elimination by multiplying some matrices (independent of $n$), but I cannot think of anything how this should be possible. Another possibility could be that we can split the matrix into parts and do Gaussian elimination (somehow by matrix multiplication) on those and then do Gaussian elimination on those 4 matrix parts. But again I can't think of any way this is possible. I am equally confused about the inverse problem.
So, I am curious on how to prove the above statement (both directions). Just the (more or less precise) idea would be awesome.