I'm trying to write a proof for the following problem, but my proof is not strong. Assuming the law, det(AB) = det(A) det(B)[for any n×n matrices A and B, prove that det(ABC) = det(A) det(B) det(C)[for any n×n matrices A, B, and C].
How would I go about solving this problem? I was thinking about trying to argue because the numbers of a given matrix multiply as scalars, the determinant is the product of them all and because the order of the multiplication of det(ABC) stays the same, det(ABC) = det(A) det(B) det(C) holds true. However, I don't think is a good enough proof and would greatly appreciate some insight.
Simply $$\det(ABC)=\det(AB\cdot C)=\det(AB)\det(C)=\det(A)\det(B)\det(C).$$ Note that the order in which we write the terms $\det(A),\det(B),\det(C)$ doesn't matter since these are just real numbers, and multiplication of real numbers is commutative.