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Solve the following system of linear equation

$$\left(\begin{array}{c}{x_{1}^{\prime}} \\ {x_{2}^{\prime}} \\ {x_{3}^{\prime}}\end{array}\right)=\left(\begin{array}{ccc}{3} & {0} & {0} \\ {0} & {-2} & {0} \\ {0} & {0} & {4}\end{array}\right)\left(\begin{array}{l}{x_{1}} \\ {x_{2}} \\ {x_{3}}\end{array}\right)$$

$$x_{1}^{\prime}=3 x_{1}$$
$$x_{2}^{\prime}=-2 x_{2}$$
$$x_{3}^{\prime}=4 x_{3}$$

$$x_{1}=b_{1} e^{3 t}$$
$$x_{2}=b_{2} e^{-2 t}$$
$$x_{3}=b_{3} e^{4 t}$$

$$X(t)=\left(\begin{array}{c}{b_{1}} {e^{3t}} \\ {b_{2}} {e^{-2t}} \\ {b_{3}} {e^{4t}}\end{array}\right)$$

$$X(t)=b_{1}\left(\begin{array}{l}{1} \\ {0} \\ {0}\end{array}\right) e^{3 t}+b_{2}\left(\begin{array}{l}{0} \\ {1} \\ {0}\end{array}\right) e^{-2 t}+b_{3}\left(\begin{array}{l}{0} \\ {0} \\ {1}\end{array}\right) e^{4 t} $$

ex: $$x^{\prime}=\left(\begin{array}{cc}{1} & {-1} \\ {2} & {4}\end{array}\right) x$$

$$A=\left(\begin{array}{cc}{1} & {-1} \\ {2} & {4}\end{array}\right) \Rightarrow$$

$$\lambda_{1}=2$$
$$\lambda_{2}=3$$

$$P_{1}=\left(\begin{array}{c}{1} \\ {-1}\end{array}\right)$$

$$P_{2}=\left(\begin{array}{c}{1} \\ {-2}\end{array}\right)$$

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

$$x_{1}(t)=b_{1} e^{2 t}+b_{2} e^{3t} $$

$$x_{2}(t)=-b_{1} e^{2 t}-2 b_{2} e^{3 t} $$

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