I've just seen a proof of the statement: "Given $\alpha$ in a commutative ring $K$ there is a unique alternating multilinear function $f$ with $f(Id)=\alpha$."
The determinant is defined as the unique $f$ such that $f(Id)=1$. I don't understand why for each alternating multilinear function $f$ we have $$f(A)=\det(A)f(Id)$$
I would appreciate if anyone could explain me why this is true. Thanks in advance.