Let $A$ be a square matrix of order $n$ and $ \lambda$ is one of its eigenvalues. Let $X$ be an eigenvector associated to eigenvalue $ \lambda $ then we must have equation

$AX = \lambda X$ --------- Equation $(1)$

In Question, It is given that $AX=R$ ----------Equation $(2)$

Now, From Equation $(1)$ and $(2),$ $\lambda X = R$

So, $\lambda \begin{Bmatrix} 2\\ 1\\ 4 \end{Bmatrix} = \begin{Bmatrix} 32\\ 16\\64 \end{Bmatrix}$

$\Rightarrow$ $\lambda \begin{Bmatrix} 2\\ 1\\ 4 \end{Bmatrix} =16 \begin{Bmatrix} 2\\ 1\\4 \end{Bmatrix}$

$\Rightarrow$ $\lambda = 16$

So, Answer should be $(D)$