How to show that $GL_{n}(\mathbb{C})$ is a algebraic variety

Question

I was reading the Wikipedia page on Algebraic groups.

I understand that algebraic group is a group that is a algebraic variety with some additional conditions (the multiplication and inversion operations are given by regular maps on the variety.)

So first of all I want to show that $GL_{n}(\mathbb{C})$ is a algebraic variety.

I think I have the idea, first of all I have to embed it into some affine space , show that it is the common zero of some set of polynomials( that will show that it is algebraic set) and then show that it is irreducible. That will show that $GL_{n}(\mathbb{C})$ is a variety. ( I hope I am correct!)

But I don't know how to practically apply it.

Answer 1

Let $$V=\{(x,y)\in\mathbb{C}^{n^2}\times\mathbb{C}\mid \det(x)y-1=0\}$$