  The Gerdjikov-Ivanov (GI) system of $q$ and $r$ is defined by a quadratic
polynomial spectral problem with $2 \times 2$ matrix coefficients. Each element
of the matrix of n-fold Darboux transformation of this system is expressed by a
ratio of $(n+1)\times (n+1)$ determinant and $n\times n$ determinant of
eigenfunctions, which implies the determinant representation of $q^{[n]}$ and
$r^{[n]}$ generated from known solution $q$ and $r$. By choosing some special
eigenvalues and eigenfunctions according to the reduction conditions
$q^{[n]}=-(r^{[n]})^*$, the determinant representation of $q^{[n]}$ provides
some new solutions of the GI equation. As examples, the breather solutions and
rogue wave of the GI is given explicitly by two-fold DT from a periodic "seed"
with a constant amplitude.
