Generating all invertible matrices by row operations

We show that all invertible n × n matrices over any finite field Fq can be generated in a Gray code fashion. More specifically, there exists a listing such that (1) each matrix appears exactly once, and (2) two consecutive matrices differ by adding or subtracting one row from a previous or subsequent row, or by multiplying or diving a row by the generator of the multiplicative group of Fq. This even holds if the addition and subtraction of each row is allowed to some specific rows satisfying a certain mild condition. Moreover, we can prescribe the first and the last matrix if n≥ 3, or $n=2$ and $q>2$. In other words, the corresponding flip graph on all invertible n × n matrices over Fq is Hamilton connected if it is not a cycle.