In this paper, we propose a new method for investigating the stability of discrete-time switched linear systems by means of linear algebra techniques. Exponential stability of such systems is equivalent to the existence of solutions of linear matrix inequalities indexed by matrix words, i.e., products of matrices of the sub-systems). Our method consists in using the link between this characterization of exponential stability and linear dependency among matrix words. In particular, we introduce a criterion to reduce drastically the number of linear matrix inequalities, by removing redundant ones. This is achieved by eliminating matrix words that depend on the others. From this criterion, we also relate exponential stability to quadratic stability of another switched system. An example is given to illustrate our methods.