A Universal Formula of the n-th Power of 2 × 2 Matrix and Its Applications

A complete universal formula of the n-th power of matrix,which has not been proposed in the literature ever before,is deduced in this study,by means of matrix transformation and complex representation.With the help of this formula,both the whole solutions for the difference equations with fractions,second order and linear difference equations and difference equation systems,and the universal formula for the equivalent resistance of laddered networks have been derived.Results obtained here can make calculation of the related problems easier.