Centre of $GL_n(\mathbb R)$

Question

How to prove that center of $GL_n(\mathbb R)$ is scalar matrices?

Again what can you say about the center of set of all upper triangular matrices and center of set of all diagonal matrices.

Answer 1

Hint: Let $E_{ij}$ be the matrix with a one in the $ij$ position and zeros elsewhere. Note that $I+E_{ij}$ is invertible, and anything which commutes with it commutes with $E_{ij}$. Now compare the entries of $AE_{ij}$ with those of $E_{ij}A$.

For the upper triangular matrices, restrict yourself to $i\le j$.

I'll assume the question on diagonal matrices is a joke.