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# (Solved): 7. (10 points) A real $$2 \times 2$$ matrix $$A$$ has an eigenvalue $$\l ... ??????? 7. (10 points) A real \( 2 \times 2$$ matrix $$A$$ has an eigenvalue $$\lambda_{1}=1+i$$ with corresponding eigenvector $$\mathbf{v}_{1}=\left[\begin{array}{l}1-2 i \\ 3+4 i\end{array}\right]$$. Which of the following is the general REAL solution to the system of differential equations $$\mathbf{x}^{\prime}(t)=A \mathbf{x}(t)$$ ? A. $$c_{1} e^{t}\left[\begin{array}{c}\cos t+2 \sin t \\ 3 \cos t-4 \sin t\end{array}\right]+c_{2} e^{t}\left[\begin{array}{c}\sin t-2 \cos t \\ 3 \sin t+4 \cos t\end{array}\right]$$ B. $$c_{1} e^{t}\left[\begin{array}{c}\cos t-2 \sin t \\ 3 \cos t+4 \sin t\end{array}\right]+c_{2} e^{t}\left[\begin{array}{c}\sin t+2 \cos t \\ 3 \sin t-4 \cos t\end{array}\right]$$ C. $$c_{1} e^{t}\left[\begin{array}{c}\cos t+2 \sin t \\ 3 \cos t+4 \sin t\end{array}\right]+c_{2} e^{t}\left[\begin{array}{c}\sin t-2 \cos t \\ 3 \sin t-4 \cos t\end{array}\right]$$ D. $$c_{1} e^{t}\left[\begin{array}{c}-\cos t+2 \sin t \\ -3 \cos t-4 \sin t\end{array}\right]+c_{2} e^{t}\left[\begin{array}{c}\sin t-2 \cos t \\ 3 \sin t+4 \cos t\end{array}\right]$$ E. $$c_{1} e^{t}\left[\begin{array}{c}\cos t+2 \sin t \\ 3 \cos t-4 \sin t\end{array}\right]+c_{2} e^{t}\left[\begin{array}{c}-\sin t-2 \cos t \\ -3 \sin t+4 \cos t\end{array}\right]$$

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