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How to show $\det(AB) =\det(A)\det(B)$

Consider the map $q:\operatorname{GL}(n,R)→R^*$ given by $q(A)=\det(A)$. I know that $$q(AB)=q(A)q(B)=\det(A)\det(B)$$ does not hold if determinants of $A$ and $B$ are not unity. I would like to know why the map is homomorphic?

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    the multiplicativity of the determinant holds independently of any other conditions: http://en.wikipedia.org/wiki/Determinant#Multiplicativity_and_matrix_groups2011-12-28
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    What do you mean by $\text{det}(AB)=\text{det}(A)*\text{det}(B)$ [doesn't hold](http://en.wikipedia.org/wiki/Determinants#Multiplicativity_and_matrix_groups)?2011-12-28
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    See the answers to [this question](http://math.stackexchange.com/q/60284/660).2011-12-28
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    @Pierre: Should we close as a duplicate, then?2011-12-28
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    @ZevChonoles: I think so.2011-12-28

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