Let A be a square matrix defined by ( A =egin{pmatrix}-1&-2&-1&3\ -6&-5&1&6\ -6&-4&0&6\ -6&-7&1&8end{pmatrix} )...

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Advance Math

Let A be a square matrix defined by ( A =egin{pmatrix}-1&-2&-1&3\ -6&-5&1&6\ -6&-4&0&6\ -6&-7&1&8end{pmatrix} ) and its characteristics polynomial ( P(lambda)=igg(lambda+1igg)^2igg(lambda-2igg)^2 )
(a) Find the minimal polynomial of A.
(b) Deduce that A is not diagonalizable, but it is triangularizable, then triangularize A.
(c) Write ( A^n ) in terms of n.

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Solution a Find the minimal polynomial of A we have A eginpmatrix1213 6516 6406 6718endpmatrix and Plambdaigglambda1igg2igglambda2igg2 implies mAlambdaigglambda1iggalpha1igglambda2iggalpha2 hspace2mm where alpha1alpha2 are index of lambda1lambda2 respectivdy AIeginpmatrix0213 6516 6406 6718endpmatrixsim eginpmatrix0213 6416 0000 0303endpmatrixsim eginpmatrix0213 6030 0000 0303endpmatrix Let x4timplies x2t x3Simplies x12x23x3 x1fract2 Then See Answer

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Let A be a square matrix defined by ( A =egin{pmatrix}-1&-2&-1&3\ -6&-5&1&6\ -6&-4&0&6\ -6&-7&1&8end{pmatrix} )...