To calculate upper and lower approximations effectively and quickly under covering variation in the covering information systems, a relation matrix was defined by using the concept of characteristic function. Then the expressions for the approximations, positive, boundary and negative regions intuitively from the view of matrix were presented. Then, the expressions for the approximations, positive boundary and negative regions intuitively from the view of matrix were put forward. Furthermore, the idea of matrix was used to research and discuss the approaches for incrementally updating approximations of sets, based on the dynamic number of coverings. The investigations enriched and improved the covering rough set based dynamic learning theory and provided a method for dynamic knowledge update based in covering information systems.