Consider the First-Order Homogeneous System of Linear Differential Equations

Multiple Choice

Consider the first-order homogeneous system of linear differential equations

What is the general solution of this system?

A) $\mathbf{x}(t) =C_{1}\left(\begin{array}{l}-1 \\ 2 \\ -4\end{array}\right) e^{2 t}+C_{2}\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right) e^{-t}+C_{3}\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right) e^{t}$
B) $\mathbf{x}(t) =C_{1}\left(\begin{array}{c}1 \\ -2 \\ 4\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right) e^{-t}+C_{3}\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right) e^{t}$
C) $\mathbf{x}(t) =C_{1}\left(\begin{array}{l}1 \\ 1 \\ -1\end{array}\right) +C_{2}\left(\begin{array}{c}1 \\ -1 \\ 1\end{array}\right) e^{-t}+C_{3}\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right) e^{t}$
D) $x(t) =C_{1}\left(\begin{array}{c}1 \\ -2 \\ 4\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{c}2 \\ 1 \\ -1\end{array}\right) e^{-t}+C_{3}\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right) e^{t}$
E) $\mathbf{x}(t) =C_{1}\left(\begin{array}{l}1 \\ -2 \\ 4\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{l}1 \\ -1 \\ 1\end{array}\right) e^{-t}+C_{3}\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)$