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.
Best Answer
Hint: In general, if $V$ is a vector space and $U$ is a subspace of $V$ such that $dim(U)\geq dim(V)$, then $U=V$ (the proof is simply based on the fact that any basis of $U$ will necessarily be linearly independent in $V$).
$rank(A)$ is the dimension of $range(A)$. Can you show that $range(A^2)$ is a subspace of $range(A)$? How then can you strengthen the relationship between $range(A)$ and $range(A^2)$? Finally, what does this allow you to conclude about the relationship between $range(A^2)$ and $range(A^3)$?