Difference equations are recursive relations defined for sequences. They are considered as the discrete counterpart of differential equations due to various similarities shared. Their applications are widespread across various scientific, engineering and economics disciplines. For example, in biology, difference equations may be used to study the population dynamics of organisms. The thesis shuld deal with systems of linear difference equations and systems of first order difference equations. The first is closely related to the latter in the manner that linear difference equations may be transformed into a system of first-order equations. The need to compute the matrix powers A^n arises naturally in the finding of solutions to a system of first-order linear difference equations. Apart from the brute force method, one may calculate A^n through Jordan canonical form or inspection in relation to the eigenvectors and eigenvalues. Likewise, the matrix exponential \exp(At) arises in the finding of solutions to a system of linear differential equations. Several aids are available for the calculation of \exp(At): Jordan canonical form, interpolation, the Putzer method, the Kirchner method, the Rootselaar method, the Leonard method, and the Liz method. This can involve solving related differential equations. The purpose of this thesis is to seek the discrete analogues of the methods for calculating \exp(At), with the help of difference equations, in order to calculate A^n.