I was reviewing some school notes from many semesters ago and I came across a point which I wish to prove but can't.
Let $F$ be a field (real or complex for example), and we say $\delta : Mat(n,n,F)\to F$ is an alternating multilinear map along the rows if: (1) it is a multilinear map with the n rows being the n coordinates of the map (2) switching any two rows reverses the sign of the result.
My notes then wrote "know the theorem which says that if $\delta (I)=1$, then $\delta(AB)=\delta(A)\delta(B)$". I wish to show that this is true, since now I am not happy with just "knowing" this fact.
[The prof then showed us that $det$ is the unique alternating multilinear map along the rows such that $det(I)=1$]