Exercise1

Consider the matrix and vectors

\begin{equation*} A = \left[\begin{array}{rr} 8 \amp -10 \\ 5 \amp -7 \\ \end{array}\right],\qquad \vvec_1=\twovec{2}{1}, \vvec_2=\twovec{1}{1} \text{.} \end{equation*}
  1. Show that the vectors \(\vvec_1\) and \(\vvec_2\) are eigenvectors of \(A\) and find their associated eigenvalues.

  2. Express the vector \(\xvec = \twovec{-4}{-1}\) as a linar combination of \(\vvec_1\) and \(\vvec_2\text{.}\)

  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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