grandes-ecoles 2025 Q15

grandes-ecoles · France · mines-ponts-maths1__pc Matrices Linear System and Inverse Existence

For every $\lambda \in \mathbb{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, where $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.
Prove that $J_n(\lambda)$ is invertible and determine in terms of $N$ and $\lambda$ the matrix $N'$ such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$.

For every $\lambda \in \mathbb{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, where $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.

Prove that $J_n(\lambda)$ is invertible and determine in terms of $N$ and $\lambda$ the matrix $N'$ such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$.

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