Why is the map: $GL_n(K)\times GL_n(K) \to GL_n(K)$ regular?

Question

Let $K$ be a field and $GL_n(K)$ the set of all invertible $n$ by $n$ matrices over $K$. Let $m: GL_n(K)\times GL_n(K) \to GL_n(K)$ be the usual multiplication of matrices. Why the map $m$ is regular? Thank you very much.

Answer 1

First forget $GL_n(K)$ and work in $M_n(K)$. The multiplication map $$ M_n(K)\times M_n(K)\to M_n(K)$$ is polynomial in the entries: $$((x_{ij})_{ij}, (y_{kl})_{kl})\mapsto (\sum_{r} x_{ir}y_{rl})_{il}, $$ so it is a regular map. When you restrict to $GL_n(K)$, you get a regular map $$ GL_n(K)\times GL_n(K)\to M_n(K).$$ As the multiplication lands in $GL_n(K)$, you get the statement you want to prove.