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Find the particular solution to the initial value problem:

$$X'=\left(\begin{array}{ccc}1 & -1 & 0 \\\ 0 & 2 & 2 \\\ 0 & 0 & -1\end{array}\right) X, \ X(0)=\left(\begin{array}{c} 1 \\\ 2 \\\ 3 \end{array}\right)$$

A

$X=\left(\begin{array}{c}-2 \\\ 0 \\\ 0\end{array}\right)e^{t}+\left(\begin{array}{c}1 \\\ -2 \\\ -3\end{array}\right)e^{-t}$

B

$X=\left(\begin{array}{c}2 \\\ 0 \\\ 0\end{array}\right)e^{t}+\left(\begin{array}{c}-1 \\\ 2 \\\ 3\end{array}\right)e^{-t}$

C

$X=\left(\begin{array}{c}-2 \\\ 0 \\\ 0\end{array}\right)e^{t}+\left(\begin{array}{c}1 \\\ -1 \\\ 0\end{array}\right)e^{2t}+\left(\begin{array}{c}1 \\\ -2 \\\ -3\end{array}\right)e^{-t}$

D

$X=\left(\begin{array}{c}1 \\\ 0 \\\ 0\end{array}\right)e^{t}+\left(\begin{array}{c}1 \\\ -1 \\\ 0\end{array}\right)e^{2t}+\left(\begin{array}{c}1 \\\ -2 \\\ -3\end{array}\right)e^{-t}$

E

None of the above

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