Mixing time of a matrix random walk generated by elementary transvections

We consider a Markov chain on invertible $n\times n$ matrices with entries in ${\mathbb{Z}}_2$ which moves by picking an ordered pair of distinct rows and add the first one to the other, modulo $2$. We establish a logarithmic Sobolev inequality with constant $n^2$, which yields an upper bound of $O(n^2\log n)$ on the mixing time.