  We show that all invertible $n \times n$ matrices over any finite field
$\mathbb{F}_q$ 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 $\mathbb{F}_q$. 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 $n\ge 3$, or $n=2$ and $q>2$. In other words, the corresponding flip
graph on all invertible $n \times n$ matrices over $\mathbb{F}_q$ is Hamilton
connected if it is not a cycle.
