My point is only that the commutative property is not inherent to all multiplication operations, so there can be a distinction between the operands. It's not ne

My point is only that the commutative property is not inherent to all multiplication operations, so there can be a distinction between the operands. It's not necessarily a useful distinction, and in the usual use of multiplication it's utterly useless and only adds confusion. But matrix multiplication is taught in high school (and usually promptly forgotten), it's not particularly advanced math. Personally I'm of the opinion that the terminology is muddled. There's no need to distinguish the operands of a multiplication over any of the usual domains (reals, rationals, integers, etc). And when you reach the point where it does become important there's generally more than one product operation and we should stop calling it multiplication. "Matrix multiplication" is a bad term. You also typically wouldn't name the operands, since as you note there can be more than two!