Exercise2

Consider the matrix and vectors

\begin{equation*} A = \left[\begin{array}{rrr} -5 \amp -2 \amp 2 \\ 24 \amp 14 \amp -10 \\ 21 \amp 14 \amp -10 \\ \end{array}\right],\qquad \vvec_1=\threevec{1}{-2}{-1}, \vvec_2=\threevec{2}{-3}{0}, \vvec_3=\threevec{0}{-1}{-1} \end{equation*}
  1. Show that the vectors \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\) are eigenvectors of \(A\) and find their associated eigenvalues.

  2. Express the vector \(\xvec = \threevec{0}{-3}{-4}\) as a linear combination of the eigenvectors.

  3. Use this expression to compute \(A\xvec\text{,}\) \(A^2\xvec\text{,}\) and \(A^{-1}\xvec\) as a linear combination of eigenvectors.

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