If every non-zero vectors be the eigenvector of a real matrix $A$, prove that $A$ is the scalar matrix $\lambda I_n$. 
If every non-zero vectors in $\mathbb{R}^n$ be the eigenvector of a real $n \times n$ matrix $A$ corresponding to a real eigenvalue $\lambda$, prove that $A$ is the scalar matrix $\lambda I_n$.

I have no idea to start with. Please help me to solve it. 

Hint:
$Ax=\lambda x \quad \forall x \in \mathbb{R}^n \quad \iff \quad (A-\lambda I)x=0 \quad \forall x \in \mathbb{R}^n$
