Answer
$ \left[\begin{array}{l}
1 \\
4
\end{array}\right]$ is not eigen vector for the given matrix
Work Step by Step
Given $$\left[\begin{array}{ll}
-3 & 1 \\
-3 & 8
\end{array}\right],\ \ \ \ \ \ \left[\begin{array}{l}1 \\ 4\end{array}\right] $$
Since
\begin{align*}
\left[\begin{array}{cc}
-3 & 1 \\
-3 & 8
\end{array}\right]\left[\begin{array}{l}
1 \\
4
\end{array}\right]&=\left[\begin{array}{l}
1 \\
29
\end{array}\right]\\
&\neq \lambda \left[\begin{array}{l}
1 \\
29
\end{array}\right]
\end{align*}
Then $ \left[\begin{array}{l}
1 \\
4
\end{array}\right]$ is not eigenvector for the given matrix
