  We have employed the relativistic coupled-cluster theory to calculate the
electric dipole matrix elements in the singly ionized barium (Ba$^+$) and
radium (Ra$^+$). Matrix elements of Ba$^+$ are used to determine the
light-shift ratios for two different wavelengths on which recent experiments
are carried out. By combining the measured light-shift ratios and our
calculations, we are able to estimate possible errors associate with the used
matrix elements in Ba$^+$. Static polarizabilities of the low-lying states and
lifetimes of the first excited P-states are also determined using these matrix
elements in Ba$^+$. A similar approach has been followed to estimate the
lifetimes and light-shift ratios in Ra$^+$.
