Show the center of the general linear group are the proportional diagnol matrices.

Meaning, $\displaystyle Z(\mbox{GL}_n(\mathbb{R})) = \{ k I| k\in \mathbb{R}^{*} \mbox{ and }I \mbox{ identity matrix} \}$.

I try to explain this so it makes it more elementary. The set $\displaystyle \mbox{GL}_n(\mathbb{R})$ is the set of all $\displaystyle n\times n$ invertible matrices having $\displaystyle \mathbb{R}$ as their entries. This set is called the "general linear group". The "center" of this set are all the matrices which commute with everything else. So you need to show if a matrix commute with all invertible matrices then it must be a proportional diagnol matrix.