If $\operatorname{Im}(A)=\operatorname{Im}(A^2)$, what's the relation between $\operatorname{Im}(A^3)$ and $\operatorname{Im}(A^2)$? Let $A$ be a square matrix. If $\operatorname{Im}(A) = \operatorname{Im}(A^2)$, then what is the relation between $\operatorname{Im}(A^3)$ and $\operatorname{Im}(A^2)$? Here $im(A)$ denotes the columnspace of the matrix $A$.


The answer is $\operatorname{Im}(A^2) = \operatorname{Im}(A^3)$.

But I don't know how to obtain the answer. Please anyone help me solve it.  Thanks in advance.
If $v\in\operatorname{Im}(A^3)$, then $v=A^3(w)$ for some vector $w$. But then $v=A^2\bigl(A(w)\bigr)$, and therefore $v\in\operatorname{Im}(A^2)$.

And if $v\in\operatorname{Im}(A^2)$, then $v=A\bigl(A(w)\bigr)$ for some vector $w$. But $A(w)=A^2(u)$ for some vector $u$ (since $\operatorname{Im}(A^2)=\operatorname{Im}(A)$, and therefore $v=A^3(u)\in\operatorname{Im}(A^3)$.
We are given that  $\textrm{im}(A) = \textrm{im}(A^2)$. By definition, this means that $\forall$ $X$ $\in$ $\mathbb{R^n}$, we have $AX = A^2X$. 
This equation implies that $A = A^2$ for any arbitrary $X$. 
Hence, we have that $\forall$ $X$ $\in$ $\mathbb{R^n}$: $ A^3 X = (A A^2) (X) = (AA)(X) = A^2X$. Therefore, $\textrm{im}(A^2) = \textrm{im}(A^3)$, as desired.