How is the general linear group a topological group?

Question

How to see if the general linear group GL($n$), of non-singular $n$-square matrices over the real (or complex) numbers under matrix multiplication, is a topological group? How to show that matrix multiplication and inversion are continuous mappings?

Answer 1

Hint: Matrix multiplication, entry wise, is just a polynomial! Inversion can be seen to be continuous using the same logic and Cramer's theorem: $\displaystyle A^{-1}=\frac{1}{\det(A)}\text{ adj}(A)$.

Answer 2

Along with Alex's answer I think this might help you too. Try to think of these,$\phi: M_n(\mathbb{R})\rightarrow \mathbb{R}$ we define $\phi(A)=\det(A)$ is continuous map because $1.$ Polynomials are continuous $2.$ Sum of Two Continuous function is again a continuous. $3.$ Product of two continuous function is again a continuous function.