Associativity in Matrix Multiplication

Question

I read that while multiplying $3$ matrices, $$(AB)C = A(BC) $$

is this property true for $n$ matrices?

Eg : Can I write, $$A(BC)(DE)F= (AB)C(DE)F = (AB)(CD)(EF) $$

– Thanks

Answer 1

Can I write $\quad A(BC)(DE)F= (AB)C(DE)F = (AB)(CD)(EF) $

Yes, because associativity guarantees that the order in which the multiplications are performed does not matter, so all of them are equal, and the product is usually written simply as $ABCDEF$.

If the operation were not associative, then the products you wrote would not even be well defined, because you'd have to fully specify the order e.g. $\Big(\big(A(BC)\big)(DE)\Big)F$.

Answer 2

Yes. There's nothing specific about matrices here. What you wrote applies to any associative binary operation.