Let $U_n$ and $V_n$ denote the $n$-th generalized Fibonacci and generalized Lucas numbers, respectively. In this paper, we calculate the $n$-th powers of some square matrices by diagonalizing them with their eigenvalues and eigenvectors. We find some terms of these matrices as generalized Fibonacci numbers and generalized Lucas numbers. Using matrix powers and the binomial theorem, we give some new identities. Finally, we show again that the known identity $(k^{2}+4t)(-t)^{n-1}=V_{n+1}V_{n-1}-V_{n}^{2}$ where is satisfied with the help of the obtained matrices.