Let A be a square matrix defined by \( A = \begin{pmatrix}-3&-1&-3\\ 5&2&5\\ -1&-1&-1\end{pmatrix} \)
(a) Find the characteristic polynomial of A.
(b) Find the eigenvalues of A. Show that A is not diagonalizable over \( \mathbb{R} \)
(c) Show that A is diagonalizable over\( \mathbb{C} \). Find the eigenspaces. Diagonalize A.
(d) Express \( A^n \) in the form of \( a_nA^2+b_ nA+c_nI_n \) where \( (a_n), (b_n) \) and \( (c_n) \) are real sequences to be specified.
\( A=PDP^{-1},D=\begin{pmatrix}0&0&0\\ 0&-1-i&0\\ 0&0&-1+i\end{pmatrix},P=\begin{pmatrix}-1&-1-2i&-1+2i\\ 0&1+3i&1-3i\\ 1&1&1\end{pmatrix} \)
