Product property

Question

What is the name of the following property of two equal products?

If $ab = cd$, then $a(b-d)=(c-a)d$

Answer 1

$a(b-d)=(c-a)d \iff ab-ad=cd-ad \iff ab=cd$.

The reason ist the distributive property of addition and multiplication in $ \mathbb R$.

Answer 2

It is the multilinearity of the determinant (substracting the first column from the second one does not change the determinant), i.e., $$ ab-cd=\det \begin{pmatrix} a & c \cr d & b \end{pmatrix}=\det \begin{pmatrix} a & c-a \cr d & b-d \end{pmatrix}=a(b-d)-(c-a)d. $$