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$$A=\left(\begin{array}{ccc}{4} & {0} & {0} \\ {-2} & {1} & {0} \\ {-2} & {1} & {1}\end{array}\right)$$

$$\lambda_{1}=4, \lambda_{2,3}=1$$

det $$(\lambda I-A)=\left|\begin{array}{ccc}{\lambda_{-} 4} & {0} & {0} \\ {2} & {\lambda_{-1}} & {0} \\ {2} & {-1} & {\lambda_{-1}}\end{array}\right|=0$$

$$=(\lambda-4)(\lambda-1)(\lambda-1)=0$$

$$=(\lambda-4)(\lambda-1)(\lambda-1)=0$$

For $$\lambda_{1}=4$$

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

$$(\lambda I-A) x=0$$

$$(I-A) x=0$$

$$\left(\begin{array}{ccc|c}{-3} & {0} & {0} & {0} \\ {2} & {0} & {0} & {0} \\ {2} & {-1} & {0} & {0}\end{array}\right) {R}_{1} \Rightarrow-1 / 3 R_{1} $$

$$\left(\begin{array}{ccc|c}{1} & {0} & {0} & {0} \\ {2} & {0} & {0} & {0} \\ {2} & {-1} & {0} & {0}\end{array}\right) \begin{array}{c}{R_{2} \rightarrow R_{2}-2 R_{1}} \\ {R_{3} \rightarrow R_{3}-2 R_{1}} \end{array} $$

$$\left(\begin{array}{ccc|c}{1} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {0} & {-1} & {0} & {0}\end{array}\right) R_{3} \Rightarrow-R_{3} $$

$$\left(\begin{array}{lll|l}{1} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0}\end{array}\right)$$

$$x_{3}=t$$
$$x_{1}=0$$
$$x_{2}=0$$

$$X=\left(\begin{array}{l}{0} \\ {0} \\ {1}\end{array}\right) t$$

$$X_{2,3}=\left(\begin{array}{l}{0} \\ {0} \\ {1}\end{array}\right) t$$

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