On a real square matrix of order $10$

Question

Let $M_{10}$ be the set of $10×10$ real matrices; if $U\in M_{10}$, then let $\rho(U)=rank(U)$.

Which of the followings are true for every $A\in M_{10}$?

$(1)\rho(A^8)=\rho(A^9)$

$(2)\rho(A^9)=\rho(A^{10})$

$(3)\rho(A^{10})=\rho(A^{11})$

$(4)\rho(A^8)=\rho(A^7)$

I am able to discard options $(1),(2)$ &$ (4)$ by taking nilpotent matrices of order $9,10$ & $8$ respectively. This leaves $(3)$ as true but I am finding it difficult to write a general proof.

I think characteristics polynomial can be used.

Can you give any suggestions? Thanks for your time.

Answer 1

The fact that $\rho(A^{10})$ is different from $0$ implies that there exists $k$ such that $\rho(A^{k}) = \rho(A^{k-1})$. The reason is that the rank can only decrease, and cannot be negative.

Then $\rho(A^{k+1})$ cannot be lower than $\rho(A^{k}).$

If $\rho(A^{10}) = 0$ then the problem is directly solved.