Regular Matrix A with $AB=BA$

Asked Dec 14 '20 at 16:06

Active Dec 14 '20 at 16:15

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I need help with an exercise, probably I think too difficult, so it would be nice to have just a beginning:

$A\in GL_n(\mathbb{R})$, with $A*B=B*A$ for all $B\in GL_n(\mathbb{R}) $. Show that there exists a $r\in \mathbb{R}$ without $0$ with $A=r*E_n$ where $E_n$ is the $n \times n$ unit matrix.

It is obvious that this statement is true but probably I'm too dumb to show it correctly, so it would be great to have a beginning.

Thank you very much, have a nice day and stay healthy!

edited Dec 14 '20 at 16:15

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asked Dec 14 '20 at 16:06

jonas_01

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take $E_{p,k}=(e_{ij})$ with $e_{ij}=1$ if $(i,j)=(p,k)$ and $e_{ij}=0$ otherwise. now replace $B$ with every $E_{p,k}$ and see what it gives you. – Laassila souhayl Dec 14 '20 at 16:14
See this post and this post. – Ben Grossmann Dec 14 '20 at 20:31

Regular Matrix A with $A*B=B*A$

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Regular Matrix A with $AB=BA$