Let $A$ be a non-zero linear transformation on a real vector space $V$ of dimension $n$. 
Let $A$ be a non-zero linear transformation on a real vector space $V$ of dimension $n$. Let the subspace $V_0 \subset V$ be the image of $V$ under $A$. Let $k = \dim (V_0) \lt n$ and suppose that for some $\lambda \in \mathbb{R}$, $A^2 = \lambda A$. Then which of the following are true?
  
  
*
  
*$\lambda = 1$
  
*$\det(A) = |\lambda|^{1/n}$
  
*$\lambda$ is the only eigenvalue of $A$
  
*There is a nontrivial subspace $V_1 \subset V$ such that $Ax = 0$ for all $x \in V_1$
  

I am able to say that 4 is true as $\operatorname{rank}(A) < n$. 2 is false by using the determinant function on both sides. Using the argument used for 4, 0 is also an eigenvalue. So 3 is also false.
But I want to know what we can say about $\lambda$. Is is an eigenvalue? Is it equal to 1?

Let $A$ be a matrix of order $2 \times 2$ with all entries equal to $1$. Then $A^2 = 2A$ and the eigenvalues of A are $0$ and $2$. Using this example, options $1$, $2$ and $3$ can be discarded. 
