# Hardness result or reference for optimal Gaussian elimination process

I'm wondering if the following problem is NP-Complete or has any hardness result. References on related problem are also welcome.

Input: integers $$n\geq1,k\geq0$$ and an invertible matrix $$M\in\mathbb F_2^{n\times n}$$.

Accept iff: with at most $$k$$ row elimination, $$M$$ can be reduced to identity.

Here, one row elimination is simply adding one row to another (just like Gaussian elimination algorithm).

By Method of Four Russians, we know for any $$n$$, it suffices to have $$k=O\left(\frac{n^2}{\log n}\right)$$. But I'm curious about the minimum number of row eliminations required.