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?

There are so many good answers in this question that I doubt you'll find anything new here. — krirkrirk, Mar 11 '18 at 13:59

score 0 · Answer 1 · answered Mar 11 '18 at 14:06

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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.

answered Mar 11 '18 at 14:06

José Carlos Santos

427,504

score 0 · Accepted Answer · answered Mar 11 '18 at 14:06

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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.

answered Mar 11 '18 at 14:06

Mohammad Riazi-Kermani

68,728

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

2 Answers2