Question: Show that \(\mathbf{v}_{1}=\left(\begin{array}{c}-1 \\ 1 \\ 0\end{array}\right), \mathbf{v}_{2}=\left(\begin{array}{c}-5 \\ 1 \\ 2\end{array}\right)\), and \(\mathbf{v}_{3}=\left(\begin{array}{c}1 \\ 1 \\ 2\end{array}\right)\) are eigenvectors of the matrix

and find the corresponding eigenvalues. For each \(\mu \in \mathbb{C}\), show that \(\mathrm{v}_{1}, \mathrm{v}_{2}\), and \(\mathrm{v}_{3}\) are eigenvectors of the matrix \(A-\mu I_{3}\) and find the corresponding eigenvalues. If \(T: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}\) is the linear transformation \(T(\mathbf{z})=A \mathbf{z}\left(\right.\) for \(\left.\mathbf{z} \in \mathbb{C}^{3}\right)\), find the matrix of \(T\) with respect to the basis \(\mathscr{B}=\left(\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right)\)

Find the matrix of \((T-(1+i) I)(T+4 I)^{-1}\) with respect to the basis \(\mathscr{B}\).

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