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
Updated on December 22, 2022Comments
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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.
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Sahiba Arora over 6 yearsPlease provide more details, for instance, what are your thoughts about the question, what you have tried, where you are stuck.
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GraceZ over 6 yearsThank 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.
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Crostul over 6 yearsPossible duplicate of math.stackexchange.com/questions/1120202/…
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Sahiba Arora over 6 years@GraceZ The matrices are not given to be invertible. So you can't use the inverse.
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GraceZ over 6 years@Sahiba Arora You are right, thank you for your correction!!!
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GraceZ over 6 years@Crostul Opps! I did not find the question. Thank you very much!!!
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snulty over 6 yearsAre you looking for something relating to a form of Schur's Lemma?
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GraceZ over 6 years@snulty Thank you very much! But it seems that they do not have many relation.
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GraceZ over 6 yearsThank you very much!!! I know the solution now. I am sorry that my reputation is not enough, so I cannot change the upvote score.