###### 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 linear 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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