Let A be a square matrix defined by \( A = \begin{pmatrix}-3&-1&-3\\ 5&2&5\\ -1&-1&-1\end{pmatrix} \) (a)...

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Advance Math

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.
\( A=PDP^{-1},D=\begin{pmatrix}0&0&0\\ 0&-1-i&0\\ 0&0&-1+i\end{pmatrix},P=\begin{pmatrix}-1&-1-2i&-1+2i\\ 0&1+3i&1-3i\\ 1&1&1\end{pmatrix} \)

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Solution a Find the characteristic polynomial of A implies PlambdaAlambda Ibeginvmatrix 3lambda 1 3 5 3lambda 5 1 1 1 endvmatrix 3lambdabeginvmatrix 2lambda 5 1 1lambda endvmatrixbeginvmatrix 5 5 1 1lambda endvmatrix3beginvmatrix 5 2lambda 1 1 endvmatrix Therefore Plambdalambdabigglambda22lambda2bigg b Find the eigenvalues of A Show that A is not diagonalizable over mathbbR since Plambdahspace2mmishspace2mm nothspace2mm See Answer

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