I'm wondering about the approach to another problem.
$B = \left( \begin{array}{ccc} a & 2a & 3a \\ 4b & 5b & 6b \\ 7c & 8c & 9c \end{array} \right) $
and I want to find $|B^4|$ i.e. the determinant of B to the 4th.
I was thinking of getting $B^k = S$\Lambda^k$S^{-1}$ where $S$ is the eigenvector matrix. Is that the only approach, or is there a better way to solve this problem?
$B^T$ provides an interesting matrix that I can manipulate more easily, but I'm not sure that helps me get to what I need.
THanks in advance.