I would like them to learn the Commutative Law of Multiplication before worrying about higher Calculus.
For the record, I took Calc 140s (mid level) in a college known to have a good program, and also known for NOT invoking grade inflation, and these matrices never came up. I also took Honors Calc in prep school, and they didn’t come up there, either. Admittedly, it was not my best subject.
But then the English department would have to teach the definition of "commutative," which has too many syllables. Also, they would then have to teach "associative" and "distributive," too.
The last one would be particularly difficult after spending so much time explaining "redistribute."
-PJ