In Wikipedia it says that "A $n\times n$ matrix of a ring $R$ is commutative if and only if $n=1$ and $R$ is commutative". Could someone please provide me with a proof/reference to a proof of that ?
Criterion for commutativity of matrices
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1I cannot find your exact quote there. How is a single matrix commutative? – 2012-12-08
1 Answers
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All this is saying is that if you have the group of invertible $n\times n$ matrices $G = GL_n(R)$ where the matrices have entries from a ring $R$, then for $G$ to be commutative you need
- $R$ to be commutative
- $n=1$.
You probably already know that the group of $2\times 2$ matrices isn't commutative even over a (non-trivial) field. And if $n=1$ then $G \simeq R^{\times}$ and so $\dots$
Now, you can of course just consider $M_n(R)$ and say that this as a set is commutative if the elements commute, but then you still have the same thing.