Answer
$ \left[\begin{array}{c}
-1+\sqrt{2} \\
1
\end{array}\right] $ is eigen vector of the given matrix and the corresponding eigenvalue is $3+\sqrt{2}$
Work Step by Step
Given $$\left[\begin{array}{ll}
2 & 1 \\
1 & 4
\end{array}\right],\ \ \ \ \ \left[\begin{array}{c}
-1+\sqrt{2} \\
1
\end{array}\right]$$
Since
\begin{align*}
\left[\begin{array}{ll}
2 & 1 \\
1 & 4
\end{array}\right]\left[\begin{array}{c}
-1+\sqrt{2} \\
1
\end{array}\right]&= \left[\begin{array}{c}
-1+2 \sqrt{2} \\
3+\sqrt{2}
\end{array}\right]\\
&= (3+\sqrt{2})\left[\begin{array}{c}
-1+\sqrt{2} \\
1
\end{array}\right]
\end{align*}
Then
$ \left[\begin{array}{c}
-1+\sqrt{2} \\
1
\end{array}\right] $ is eigenvector of the given matrix and the corresponding eigenvalue is $3+\sqrt{2}$
