A 2x2 matrix has (in this case) 2 distinct eigenvalues. The matrix to the power n can be expressed as a coefficient times the matrix plus a second coefficient times the identity matrix. Can someone point me to a simple derivation of a formula that computes the two coefficients? That is, find the two coefficients in terms of the two eigenvalues and n. Yes, I know the problem can also be soved by diagonalizing the matrix and then raising the diagonal elements to the power n, but I want to avoid computing and then multiplying by the pre and post-matrices that are required for that method. There are (at least) two ways to skin this cat, and I'm asking for help on the method that does not involve explicit diagonalization. Many thanks.