Let $A$ be $10 \times 10$ real matrix. then which of the following is correct?

Question

Let $A$ be $10 \times 10$ real matrix. then which of the following is correct? [$\rho(A)=Rank(A)]$

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

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

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

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

(A),(B),(D) are false. since if I take an upper triangular matrix $A$ with all zero entries in the diagonal and 1 in all $i<j$ region. Then, (A),(B),(D) are false. Hence, $C$ is the correct answer. Is there any shorter way to solve this?

Answer 1

If there is $v$ such that $A^{11} v = 0$ but $A^{10} v \ne 0$, then the null spaces of $A, A^2, \ldots, A^{11}$ are all distinct. But these are nested, so their dimensions are all different. $\mathfrak N(A)$ has dimension at least $1$, ...., $\mathfrak N(A^{11})$ has dimension at least $11$, but that's impossible.