Question 62
Let $A \in \mathcal{M}_n(\mathbb{C})$ be an invertible matrix, with lines $L_1, L_2, \dots, L_n$. Consider the matrices $B \in \mathcal{M}_n(\mathbb{C})$ with lines $O, L_2, \dots, L_n$ and $C \in \mathcal{M}_n(\mathbb{C})$ with lines $L_2, \dots, L_n, O$, where $O$ denotes a line all of whose entries are zero. Let $D = A^{-1} \cdot B$ and $E = A^{-1} \cdot C$. Prove that:
a) $\operatorname{rank}(D) = \operatorname{rank}(D^2) = \dots = \operatorname{rank}(D^n);$
b) $\operatorname{rank}(E) \gt \operatorname{rank}(E^2) \gt \dots \gt \operatorname{rank}(E^n).$
a) $\operatorname{rank}(D) = \operatorname{rank}(D^2) = \dots = \operatorname{rank}(D^n);$
b) $\operatorname{rank}(E) \gt \operatorname{rank}(E^2) \gt \dots \gt \operatorname{rank}(E^n).$