RO-NMO 2016 Problem 112

Consider a natural number, $n\ge 2,$ and three $n\times n$ complex matrices $A,B,C$ such that $A$ is invertible, $B$ is formed by replacing the first line of $A$ with zeroes, and $C$ is formed by putting the last $n-1$ lines of $A$ above a line of zeroes. Prove that:a) $\text{rank} \left( A^{-1} B \right) = \text{rank} \left( \left( A^{-1} B\right)^2 \right) =\cdots =\text{rank} \left( \left( A^{-1} B\right)^n \right)$ b) $\text{rank} \left( A^{-1} C \right) > \text{rank} \left( \left( A^{-1} C\right)^2 \right) >\cdots >\text{rank} \left( \left( A^{-1} C\right)^n \right)$