What is the name of the following property of two equal products?
If $ab = cd$, then $a(b-d)=(c-a)d$
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2I don't suppose this has a name. There are many, many algebraic identities. Most of them lack names. – lulu Dec 20 '18 at 11:58
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It is the multilinearity of the determinant (substracting the first column from the second one does not change the determinant), i.e., $$ ab-cd=\det \begin{pmatrix} a & c \cr d & b \end{pmatrix}=\det \begin{pmatrix} a & c-a \cr d & b-d \end{pmatrix}=a(b-d)-(c-a)d. $$
Dietrich Burde
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$a(b-d)=(c-a)d \iff ab-ad=cd-ad \iff ab=cd$.
The reason ist the distributive property of addition and multiplication in $ \mathbb R$.
Fred
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