Rank of $A^n$ and $A^{n+1}$ Suppose $A$ is a $n\times n$ real matrix.
Then is it always true that Rank($A^n$) = Rank ($A^{n+1}$) for a matrix $A$?
This doubt came while solving the attached question:
If A is a 10×10 real matrix, then which of the following is true:

*

*rank($A^8$)=rank($A^9$)

*rank($A^9$)= rank($A^{10}$)

*rank($A^{10}$)=rank($A^{11}$)

*rank($A^8$)=rank($A^7$)

Attempt: I can take a nilpotent matrix of maximal index 10 for a matrix of order 10 and therefore option 1,2,4 are rejected but option 3 still is correct.
So I thought is there any generalization or I analyzed the question incorrectly?Please throw some light.

We have
$$
n = \operatorname{rank}(A^0) \ge \operatorname{rank}(A^1) \ge \operatorname{rank}(A^2) \ge \cdots 
$$
This sequence cannot decrease beyond $0$ and each time it decreases it decreases by at least $1$. Therefore, the sequence is constant after $n$ terms. (It may happen earlier of course.)
