Let A be a n x n matrix, is X ⊂ Y or is Y ⊂ X

Question

How do I find out where X is subset of Y or the other way around, if Y is a subset of X.

X = im(A) and Y = im(A^3 + A^2)

I simplified the above to Y= im(A^3) + im(A^2) but don't know how to proceed.

Answer 1

To remove comments above and remove this from unanswered queue:

Note two things in particular:

$(A^3+A^2)x = A((A^2+A)x)$ which implies

any arbitrary element of the image of $A^3+A^2$ is in the image of $A$

and there exist examples of matrices like $A=\begin{bmatrix}0&1\\0&0\end{bmatrix}$ where $A^n=0$ for all $n>1$ which implies

the image of $A$ could be potentially strictly larger than the image of $A^3+A^2$