Ivan Shishkin, Rye (1878)

Problems/Linear algebraReviewed

Familles de matrices inversibles dans Mn(Z)\mathcal{M}_n(\mathbb{Z}), le retour

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50
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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Suite à ce problème. Soient n⩾4n \geqslant 4, A,B∈Mn(Z)A, B \in \mathcal{M}_n(\mathbb{Z}) telles que {k∈Z,A+kB∈GLn(Z)}\left\{k \in \mathbb{Z}, A+k B \in \mathrm{GL}_n(\mathbb{Z})\right\} possède au moins n+1n+1 éléments. Montrer que A+kBA+k B est inversible dans Mn(Z)\mathcal{M}_n(\mathbb{Z}) pour tout k∈Zk \in \mathbb{Z}.

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