We know that AX=⋋X where A is the matrix given, X is the eigen vector and ⋋ is the eigen value.
Our A is a upper triangular matrix and the diagonal elements are the eigen values i.e. 5,5,2,3.
In such questions try matching with the options because it takes lesser time.
Starting with Option A,
AX=⋋X and we put X=(1 -2 0 0)T as it is the eigen vector.
$\begin{bmatrix} 5 & 0& 0& 0\\ 0 & 5& 5& 0\\ 0 & 0& 2& 1\\ 0 & 0& 3& 1 \end{bmatrix} \begin{bmatrix} 1\\ -2\\ 0\\ 0 \end{bmatrix} = \begin{bmatrix} 5\\ -10\\ 0\\ 0 \end{bmatrix} => \begin{bmatrix} 5 & 0& 0& 0\\ 0 & 5& 5& 0\\ 0 & 0& 2& 1\\ 0 & 0& 3& 1 \end{bmatrix} \begin{bmatrix} 1\\ -2\\ 0\\ 0 \end{bmatrix} = 5\begin{bmatrix} 1\\ -2\\ 0\\ 0 \end{bmatrix}$
This is of the form AX=⋋X where ⋋=5 (one of the eigen values).
If you put any other options and calculate you won't be able to take out anything common.
For eg: Take option D where X=(1 -1 2 1)T
$\begin{bmatrix} 5 & 0& 0& 0\\ 0 & 5& 5& 0\\ 0 & 0& 2& 1\\ 0 & 0& 3& 1 \end{bmatrix} \begin{bmatrix} 1\\ -1\\ 2\\ 1 \end{bmatrix} = \begin{bmatrix} 5\\ 5\\ 5\\ 7 \end{bmatrix}$
From RHS you cannot take anything common such that it gets in the form of ⋋(1 -1 2 1)T