How to prove $\det(AB)=\det(A)\det(B)$ using group theory? $A,B\in GL(n,R)$

Question

I know $\det(AB)=\det(A)\det(B)$. How can i prove using group theory if $A,B$ square matrices and I know $A, B\in GL(n,R)$. If the matrices were not square, it would be correct to say that it is valid?

Answer 1

Note that $$ det(AB)=det(A)det(B)$$ makes sense only for square matrices.

As you know, $det(M)$ is defined only when M is a square matrix M.

Answer 2

If the matrices are not square, then their determinants are not defined. Therefore, it definitely would not be correct to say that it is valid.