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[Community Question] Linear-algebra: Building matrices from linearly independent rows

One of our user asked:

Let $A\in M_n(\mathbb C) $. Denote its rows by $L_1,L_2,...,L_n$. We know that $L_1,L_2,...,L_n$ are linearly independent. Construct the matrices $B\in M_n(\mathbb C)$ with the rows $O,L_2,...,L_n$($O$ denotes a row of zeroes) and $C$ with the rows $L_2,...,L_n,O$. Let $D=A^{-1}B$ and $E=A^{-1}C$.
a)Prove that $rank(D) =rank(D ^2)=...=rank(D^n)$. b) Prove that $rank(E) >rank(E^2)>...>rank(E^n) $. My approach : $rank(A) =n$ since all its rows are linearly independent. Also $rank D=rank B=n-1$ and $rank E=rank C=n-1$ by the same reasoning and using the fact that if a matrix is multiplied by an invertible matrix its rank doesn't change.
I don't know how to do the same thing for $D^2,...,D^n$ and $E^2,...,E^n$


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