2x2 Matrix: Calculation of the n-th Power

This work explores a robust method for calculating the n-th power of a 2x2 matrix An without using diagonalization techniques. This approach significantly simplifies complex matrix operations. Firstly, we give here, a single formula for the nth power for any type of 2x2 matrix, that is to say whether it is diagonalizable or not, whether it has two distinct eigenvalues or not and whether these eigenvalues are real or not. We also add a second method.