  We present necessary and sufficient conditions under which the
anti-triangular matrix $\left(
  \begin{array}{cc}
  a&b
  1&0
  \end{array} \right)$ over a Banach algebra has g-Drazin inverse. New additive
results for g-Drazin inverse are obtained. Then we apply our results to
$2\times 2$ operator matrices and generalize many known results,
e.g.,~\cite[Theorem 2.2]{D}, ~\cite[Theorem 2.1]{YL} and \cite[Theorem 4.1]{Y}.
