From part (a), we know that the eigenvalues for
are
and
, with corresponding eigenvectors
and
, respectively.
Let
denote the matrix of eigenvalues, and
the matrix whose columns are the corresponding eigenvectors. Then
is diagonalized via
, or,
.
This decomposition allows for easy computations of powers of
, as for any integer
, we have
, and
. Therefore, we compute
, and multiply the three matrices to obtain
Then for
, we have