Let A be a square matrix defined by \( A =\begin{pmatrix}-8&-3&-6\\ 4&0&4\\ 4&2&2\end{pmatrix} \)
(a) Find...
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Advance Math
Let A be a square matrix defined by \( A =\begin{pmatrix}-8&-3&-6\\ 4&0&4\\ 4&2&2\end{pmatrix} \)
(a) Find the characteristic polynomial of A.
(b) Find the eigenvalues and eigenspaces of A.
(c) Show that A is not diagonalizable, but it is triangularizable, then triangularize A.
(d) Write \( A^n \) in terms of \( I, A,A^2 \) and n.
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Solution a Find the characteristic polynomial of A we have A beginpmatrix836 404 422endpmatrix implies PlambdaAlambda Ilambda36lambda212lambda8bigglambda2bigg3 So Plambdabigglambda2bigg3 b Find the eigenvalues and eigenspaces of A forall lambdain spAiff Plambda0implies lambda2 hspace2mmwith hspace2mmam23 bullet hspace2mmFindhspace2mm eigenspace A2Ibeginpmatrix636 424 424endpmatrixsim beginpmatrix636 000 000endpmatrix implies 2x1x22x30 Thus
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