If we consider matrix multiplication on $\text{GL}(2,\Bbb C)\times \text{GL}(2,\Bbb C)\to \text{GL}(2,\Bbb C)$, how do I show this is a morphism of varieties?
I know that this is the vanishing of $(x_{11}x_{22}-x_{12}x_{21})q-1 \in K[x_{11},x_{12},x_{21},x_{22},q]$, where the entries are labelled how you probably expect: $$\begin{bmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{bmatrix}$$
How do I check it is a morphism of varieties?
I can see the entries give us: $$\begin{bmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{bmatrix}\begin{bmatrix}x_{11}'&x_{12}'\\x_{21}'&x_{22}'\end{bmatrix}=\begin{bmatrix}x_{11}x_{11}'+x_{12}x_{21}'&\cdot\\\cdot&\cdot\end{bmatrix}$$
So the four entries are just polynomials in the entries of the original matrices, I feel like the it's regular on the entries, but not sure how to say rigorously that this is a morphism of varieties.
Attempt self answer:
$$\text{GL}(2,\Bbb C)\times\text{GL}(2,\Bbb C)\subset \Bbb A^{10},$$ and we see $$\psi(x_{11},x_{12},x_{21},x_{22},x_{11}',\dots,x_{22}',q,q')=(\phi_1(x),\phi_2(x),\phi_3(x),\phi_4(x),w(x)),$$ and $$\phi_1,\cdots,\phi_4,w\in \Bbb{C}[\text{GL}(2,\Bbb C)\times\text{GL}(2,\Bbb C)]$$ so this is a morphism of affine varieties