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I want center of $GL_n(\Bbb R)$.

Artin, in his book Algebra, hints that, 'You are asked to determine the invertible matrices $A$ that commute with every invertible matrix $B$. Do not test with general matrix $B$, test with elementary matrices.'

I know that Row-multiplying transformations and Row-addition transformations generate $GL_n(\Bbb R).$ I also see that Row-multiplying transformations (unless all diagonal entries are 1's) and Row-addition transformations (unless all off-diagonal entries are 0's) don not commute with each other, so only elenentary matrix in center is $I$.

What I don't get is:

1. How do we derive from this that $cI$ is in center for any $c\in \Bbb R$? (I can see $cI$ in center, but, it did not click me, because I thought this way: since center is subgroup, $\langle I\rangle$ is in center, but $\langle I\rangle=\{I\}$. Where did I not think through?)

2. No other matrix is in center?

Silent
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  • Yes, no other matrix is in the center. For this we need the elementary matrices, see the duplicates. All of them do the same. See, for example, another duplicate here.
    • – Dietrich Burde Apr 07 '18 at 19:04
  • No, no, I know that why $cI$ is in center, i don't know how did we guess that in first place if all we knew was only elementary matrix in center is $I$. Also, what do you mean by duplicates? @DietrichBurde – Silent Apr 07 '18 at 19:07
  • The question you have has been answered several times. This is called a "duplicate". Did you see the answers? And why "No,no"? Of course we guess that the identity commutes with everything, hence also $cI$. We guess this, because we know that the identity commutes with all $A$, and not because of elementary matrices. – Dietrich Burde Apr 07 '18 at 19:10
  • I am asking: only elementary matrix in center is $I$, right? From that information, how did we guess that $cI$ is in center? (I am sorry if I seem harsh, but believe me, i am a polite person, in fact i respect your decision to mark this as duplicate.) @DietrichBurde – Silent Apr 07 '18 at 19:15