<s>[INST] Complete the Matrix The eigenvectors of A (corresponding to the eigenvalue 2) = \begin{bmatrix}
    1 & 2  \\
    -1 & 4 \\
    \end{bmatrix}
can be solved by solving the matrix equation:

Here is my working:

My answers were $4$ and $-1/2$ which were apparently wrong, though I am wondering if I made a typo when I put in $-1/2$ (I am unable to check further).
 [/INST] I don't know where the matrix $\begin{pmatrix} 4 & -2 \\ 1 & -1/2 \end{pmatrix}$ comes from.
The way I have seen it done is that $\begin{pmatrix} x \\ y \end{pmatrix}$ is a $\lambda$-eigenvector if
$\begin{pmatrix} 1-\lambda & 2 \\ -1 & 4-\lambda \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \end{pmatrix}$.
So when $\lambda = 2$ we want:
$\begin{pmatrix} -1 & 2 \\ -1  & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$.
Of course $\begin{pmatrix} -1 & 2 \\ -1  & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -x+2y \\ -x+2y \end{pmatrix}$, so the vector works when $x=2y$.
So we want vectors $\begin{pmatrix} 2y \\ y \end{pmatrix}$ with $y\neq 0$.
</s>