We construct an algebraic method for detecting bifurcations of discrete dynamical systems using the Conley index theory. We define the transition matrix pair for discrete semidynamical systems, which detects bifurcations of connecting orbits. As an application, we give a sufficient condition for the occurrence of homoclinic and heteroclinic tangencies by showing that the occurrence of a heterodimensional cycle in the projectivization of a dynamical system implies the tangency of the original system.