Let A be a square matrix defined by \( A = \begin{pmatrix}6&2&3\\ -3&-1&-1\\ -5&-2&-2\end{pmatrix} \)L...

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Advance Math

Let A be a square matrix defined by \( A = \begin{pmatrix}6&2&3\\ -3&-1&-1\\ -5&-2&-2\end{pmatrix} \)L be a map from\( \hspace{2mm} \mathbb{R}^3\hspace{2mm} \)into\( \hspace{2mm}\mathbb{R}^3\hspace{2mm} \)by\( \hspace{2mm} L(v) = Av. \)
(a) Show that L is a linear operator on \( \hspace{2mm}\mathbb{R}^3. \)
(b) Find the characteristic polynomial of L with respect to standard basis for \( \mathbb{R}^3 \) Derive the determinant of L then deduce that L is invertible.
(c) Find the eigenvalues and eigenspaces of L.
(d) Show that L is not diagonalizable, but it is triangularizable, then triangularize L.
(e) Write \( L^n \) in term of n, where \( L^n = L(L(...(L)..)) \), the n compositions of L.

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Solution a Show that L is a linear operator on hspace2mmmathbbR3 Let uvin mathbbR3 hspace2mmalpha in mathbbRimplies Lualpha vLualpha Lv Threfore L is linear b Find the characteristic polynomial of L with respect to standard basis for mathbbR3 Derive the determinant of L then deduce that L is invertible we have A beginpmatrix623 311 522endpmatrix implies PlambdaAlambda Ibeginvmatrix 6lambda 2 3 3 1lambda 1 5 2 2lambda endvmatrix lambda33lambda23lambda1bigglambda1bigg3 See Answer

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