Matrix multiplication: AB=BA for every B implies A is of the form cI

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Take as $B$ the $n^2$ matrices $E_{ij}$, whose elements are $1$ in position $ij$ and $0$ elsewhere.

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GraceZ
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GraceZ

Updated on December 22, 2022

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  • GraceZ
    GraceZ 11 months

    If a $n \times n$ matrix $A$ satisfies $AB = BA$ for any $n \times n$ matrix $B$, then $A$ must be of the form $cI$ where $c$ is a scalar and $I$ is the identity matrix. I tried to use the definition of matrix multiplication, but I failed. I am wondering if I should use the inverse to solve the problem, but since now, I have no idea.

    • Sahiba Arora
      Sahiba Arora over 6 years
      Please provide more details, for instance, what are your thoughts about the question, what you have tried, where you are stuck.
    • GraceZ
      GraceZ over 6 years
      Thank you for your reply! I tried to use the definition of matrix multiplication, but I failed. I am wondering if I should use the inverse to solve the problem, but since now, I have no idea.
    • Crostul
      Crostul over 6 years
    • Sahiba Arora
      Sahiba Arora over 6 years
      @GraceZ The matrices are not given to be invertible. So you can't use the inverse.
    • GraceZ
      GraceZ over 6 years
      @Sahiba Arora You are right, thank you for your correction!!!
    • GraceZ
      GraceZ over 6 years
      @Crostul Opps! I did not find the question. Thank you very much!!!
    • snulty
      snulty over 6 years
      Are you looking for something relating to a form of Schur's Lemma?
    • GraceZ
      GraceZ over 6 years
      @snulty Thank you very much! But it seems that they do not have many relation.
  • GraceZ
    GraceZ over 6 years
    Thank you very much!!! I know the solution now. I am sorry that my reputation is not enough, so I cannot change the upvote score.