  We discuss the algebra of $N\times N$ matrices as a reduced quantum plane. A
$3-$nilpotent deformed differential calculus involving a complex parameter $q$
is constructed. The two cases, $q$ $3^{rd}$ and $N^{th}$ root of unity are
completely treated. As application, a gauge field theory for the particular
cases $n=2$ and $n=3$ is established.
