To paraphrase the video (and just by the by, if anyone has a better way to represent matrices here, please share it):

Take the 2*2 matrix A ((0,1),(1,1)). Computing powers of this matrix manually can quickly become tedious. However, as shown in the video, if a 2*2 matrix has two eigenvectors which are linearly independent, these vectors can be used as the basis vectors for a new basis. Given that two eigenvectors of this matrix are v

_{1}= (2,1+sqrt(5)) and v

_{2}= (2,1-sqrt(5)), can you find a way to simply compute any arbitrary A

^{n}by first changing to an eigenbasis, computing the new representation of A

^{n}in this basis, then converting back to our standard basis?

On a conceptual level, I understand what needs to be done here. I need some change of basis matrix B and its inverse B

^{-1}, and I need to compute B

^{-1}A

^{n}B. Where I'm sticking is in computing B

^{-1}, and possibly also in computing AB. My method, at the moment, is to compute AB for now, since I know how to do that, and then apply A to this repeatedly, and I'll apply B

^{-1}at the end. Is it going to be simpler to compute B

^{-1}AB, and take the output A' and then compute A'

^{n}? If so, is there some way to identify an easier path to computing the inverse of a matrix, apart from the standard "write B and I next to each other, and then perform row operations until you have turned B into I, and then whatever sits in place of where I was is now B

^{-1}"? Because doing that is resulting in headbanging and terms like (-1/1-sqrt(5))(1-((1+sqrt(5))/(1-sqrt(5))) and I'd rather not.