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Equalities and inequalities of ranks

Source: Romanian National Olympiad 2016, grade xi, p.2

August 25, 2019
linear algebrarank

Problem Statement

Consider a natural number, n≥2, n\ge 2, and three n×n n\times n complex matrices A,B,C A,B,C such that A A is invertible, B B is formed by replacing the first line of A A with zeroes, and C C is formed by putting the last n−1 n-1 lines of A A above a line of zeroes. Prove that:
a) rank(A−1B)=rank((A−1B)2)=⋯=rank((A−1B)n) \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) rank(A−1C)>rank((A−1C)2)>⋯>rank((A−1C)n) \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)
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