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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 ???????

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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Given the eigenvalue is 1+i and the eigenvector is [1?2i3+4i] Th
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