Ivan Shishkin, Rye (1878)

Problems/General algebraReviewed

A classical matrix equation

by araucaria araucana·
26
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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Find all matrices A,B∈Mn(R)A,B \in M_{n}(\R) such that
AB−BA=In.AB - BA = I_{n}.

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Solution by araucaria araucana

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Suppose there exists two matrices A,B∈Mn(R)A,B \in M_{n}(\R) such that AB−BA=InAB - BA = I_{n}. By linearity and the cyclic property of the Trace, we get
n=Tr(In)=tr(AB−BA)=tr(AB)−tr(BA)=0n=\mathrm{Tr}(I_n)=\text{tr}(AB - BA) = \text{tr}(AB) - \text{tr}(BA) = 0an absurdity.

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