The full vibrational energy spectra for the A1Σ+ states of alkali hydride diatomic molecules 6LiH, 7LiH, NaH, KH, RbH and CsH have been studied using algebraic method (AM), and the dissociation energies De of these molecules have also been studied using algebraic energy method (AEM). The results show that the AM produces accurate full vibrational energy spectra for these molecules for the first time so far as we know, and based on these AM spectra, the AEM can give correct molecular dissociation energies De for these systems.