If $\lambda = i$ is an eigenvalue of $A \in \mathbb R^{n \times n}$, explain why the rank of $A^3 + A$ is less than $n$ 
$A$ is a $n \times n$ matrix with real entries and $\lambda = i$ is an eigenvalue of $A$. Explain why the rank of $A^3 + A$ is less than $n$.

I understand completely how to get eigenvalues, as well as information regarding the dimension of null space, rank, etc., but have a really hard time understanding how I am supposed to relate the two, which I can only assume I have to do to answer this question. 

An eigenvalue of $A$ is $i$.
An eigenvalue of $A^3$ is $i^3$
An Eigenvalue of $A^3+A$ is  $i+i^3=0$ which implies $A$ cannot have full rank.
