Let A be the 2by 2 matrix , first row (2,-1) second row second(-1, 2)

Find A power 25?
I'll select best answer today.

If A is diagonizable, then A^k=P*D^k*P^-1 where P is a matrix of eigenvectors and D is a matrix with eigenvalues on the main diagonal. So, I see that 1,3 are eigenvalues with [1;1] and [-1;1] for eigenvectors respectively. So, taking D to the kth power is like taking the eigenvalues on the main diagonal to the kth power. Then, just multiply P * D^k * P^-1 (inverted).