Is there any relation between $Rank(A^2)$ and $Rank(A^3)$ if $Rank(A)=Rank(A^2)$? It is a question from my textbook :

$A$ is a square matrix of order $n\times n$. If $Rank(A)=Rank(A^2)$ then verify whether $Rank(A^2)=Rank(A^3)$ or not.

It is definite that $Rank(A^3)\leq Rank(A^2)$ but after that I cannot proceed.
Please anyone help me solve it. Thanks in advance. 

In terms of triangularisation of $A$ which does not change the rank you may suppose $A$ upper triangular with the last eigenvalues the zero ones, if algebraic multiplicity and geometric multiplicity of $0$ as eigenvalue are equal, $\textrm{rank}(A^n)$ is constant for all $n$, now if not $A$ as upper triangular has its last $m\times m$  block (lower right) of the form $\begin{pmatrix}0&a\\0&0\end{pmatrix}$ $a$ is $1\times m-1$  not zero, elevating for $A^2$ this block turns  to all zero diminishing the rank a contradiction.
