Let A be a square matrix defined by\( A =\begin{pmatrix}2&-3&1\\ 1&-2&1\\ 1&-3&2\end{pmatrix} \)
(a) Find the...
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Let A be a square matrix defined by\( A =\begin{pmatrix}2&-3&1\\ 1&-2&1\\ 1&-3&2\end{pmatrix} \)
(a) Find the characteristic polynomial of A.
(b) Show that A is diagonalizable then diagonalize it.
(c) Write $A^n$ \hspace{2mm} in term of n.
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Solution a Find the characteristic polynomial of A we have A beginpmatrix231 121 132endpmatrix implies PlambdaAlambda Ibigg1bigg3bigglambda3S1lambda2S2lambdaS3bigg bullet hspace2mmS12222 bullet hspace2mmS2beginvmatrix21 32endvmatrixbeginvmatrix11 12endvmatrixbeginvmatrix12 12endvmatrix bigg43biggbigg21biggbigg32bigg1 bullet hspace2mmS3beginvmatrix231 121
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