Hello,

the task is to prove that center of a matrix ring $\displaystyle R=Mat_2(\mathbb{R})$ is the scalar matrices $\displaystyle \alpha I, \alpha \in \mathbb{R}$.

I found two more general proofs:

Center of an Algebra « Abstract Nonsense

The center of a matrix ring over a commutative ring is precisely the scalar matrices « Project Crazy Project

However I didn't understand those totally, so I coudn't reduce them to this simpler case. But I think the main point is:

Let's assume that $\displaystyle M \in Z(Mat_2(\mathbb{R}))$ and$\displaystyle E=\left(\begin{array}{cc}1&0\\0&0\end{array}\right )$.

Matrix multiplication $\displaystyle ME$ is commutative ($\displaystyle ME=EM$) iff $\displaystyle M$ is in form $\displaystyle \left(\begin{array}{cc}a_1&0\\0&a_2\end{array}\rig ht)$.

I don't know how to continue etc., so any help is appreciated. Thanks!