  In this paper, we introduce the dual index and dual core generalized inverse
(DCGI). By applying rank equation, generalized inverse and matrix
decomposition, we give several characterizations of the dual index when it is
equal to one. And we get that if DCGI exists, then it is unique. We derive a
compact formula for DCGI and a series of equivalent characterizations of the
existence of the inverse. It is worth nothing that the dual index of $\hat{M}$
is equal to one if and only if its DCGI exists. When the dual index of
$\hat{M}$ is equal to one, we study dual Moore-Penrose generalized inverse
(DMPGI) and dual group generalized inverse (DGGI), and consider the
relationships among DCGI, DMPGI, DGGI, Moore-Penrose dual generalized inverse
(MPDGI) and other dual generalized inverses. In addition, we consider symmetric
dual matrix and its dual generalized inverses. At last, two examples are given
to illustrate the application of DCGI in linear dual equations.
