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?