A^n can be computed many ways, and it is not necessary to actually do any matrix multiplications. I think what you're referring to is diagonalizing the matrix A

A^n can be computed many ways, and it is not necessary to actually do any matrix multiplications. I think what you're referring to is diagonalizing the matrix A=PDP^1, then computing the matrix power of the diagonal matrix A^n = (PDP^-1)^n = P D^n P^-1. I just let numpy figure it out for me, but if you want an analytical solution something like that may be necessary.