Thanks. Yeah, I knew there was some qualifier missing that would make it true, I just couldn’t intuit exactly what it was.
Edited to add:
Actually I would say that the determinant distributes through multiplication. Commutativity: a⨁b=b⨁a. Distributivity: a⨀(b⨁c)=(a⨀b)⨁(a⨀c). Neither is a perfect analog, because the determinant is a unary operation, but distributivity at least captures that there are two operations involved. But unlike my other comment, this one doesn’t actually impair comprehension, as there’s not really a different thing you could be trying to say here using the word “commutes”.
The assignment has to commute with multiplication, and your proposed assignment would not. Just consider, say, (12)(23)=(123).
I’ve edited the post to make this clearer, thanks for the comment.
Thanks. Yeah, I knew there was some qualifier missing that would make it true, I just couldn’t intuit exactly what it was.
Edited to add: Actually I would say that the determinant distributes through multiplication. Commutativity: a⨁b=b⨁a. Distributivity: a⨀(b⨁c)=(a⨀b)⨁(a⨀c). Neither is a perfect analog, because the determinant is a unary operation, but distributivity at least captures that there are two operations involved. But unlike my other comment, this one doesn’t actually impair comprehension, as there’s not really a different thing you could be trying to say here using the word “commutes”.