Consider the sequence defined recursively by a1=−2, a2=2, an+1=8an−1−2an. we can use matrix diagonalization to find an explicit formula for an. a. find a matrix that satisfies [anan+1]=m[an−1an]

From the given starting values, we have



and more generally, the -th term pair is captured by



so that



We notice that



which is to say that the explicit formula for is given by the dot product of the second row of with .

First we diagonalize . It has eigenvalues , where



with corresponding eigenvectors , where




Then we have



where is the matrix whose columns are and and is the diagonal matrix whose non-zero entries are the eigenvalues of . So



which gives



From here, you would just need to determine the bottom row of the final matrix product.