We fix a natural number $p$ greater than or equal to 2. We denote $\mathcal { V } _ { p } = \left\{ \mathrm { e } ^ { \frac { 2 \mathrm { i } k \pi } { p } } ; k \in \llbracket 1 , p - 1 \rrbracket \right\}$, the set of $p$-th roots of unity different from 1. Let $A = \left( a _ { i , j } \right)$ be a matrix in $\mathcal { M } _ { n } ( \mathbb { C } )$ that is upper triangular and invertible. Let $\lambda$ be a non-zero complex number. Assume that, for all $i \in \llbracket 1 , n \rrbracket , \frac { a _ { i , i } } { \lambda } \notin \mathcal { V } _ { p }$. Prove that the matrix $\sum _ { j = 0 } ^ { p - 1 } \lambda ^ { p - 1 - j } A ^ { j }$ is invertible.
We fix a natural number $p$ greater than or equal to 2. We denote $\mathcal { V } _ { p } = \left\{ \mathrm { e } ^ { \frac { 2 \mathrm { i } k \pi } { p } } ; k \in \llbracket 1 , p - 1 \rrbracket \right\}$, the set of $p$-th roots of unity different from 1. Let $A = \left( a _ { i , j } \right)$ be a matrix in $\mathcal { M } _ { n } ( \mathbb { C } )$ that is upper triangular and invertible. Let $\lambda$ be a non-zero complex number. Assume that, for all $i \in \llbracket 1 , n \rrbracket , \frac { a _ { i , i } } { \lambda } \notin \mathcal { V } _ { p }$. Prove that the matrix $\sum _ { j = 0 } ^ { p - 1 } \lambda ^ { p - 1 - j } A ^ { j }$ is invertible.