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• Notes

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

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

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 $\lambda_{2,3}=1$

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

$(I-A) x=0$

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

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

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

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

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

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

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