> By definition, a matrix multiplication takes rows and multiplies them by columns.
> you (by definition) need to multiply them by each other at least once
Why does multiplying a row with a column require at least one multiplication? "By definition" it takes n multiplications (!), but in the context of matrix multiplication we can do it with less - possibly with O(1) multiplications. And it seems neither you nor me can come up with a good argument why it can't take less, despite discussing it for a while.
Which is why "you need to write n² entries" is used as an argument for the O(n²) bound, even if it has nothing to do with the number of multiplications.
> you (by definition) need to multiply them by each other at least once
Why does multiplying a row with a column require at least one multiplication? "By definition" it takes n multiplications (!), but in the context of matrix multiplication we can do it with less - possibly with O(1) multiplications. And it seems neither you nor me can come up with a good argument why it can't take less, despite discussing it for a while.
Which is why "you need to write n² entries" is used as an argument for the O(n²) bound, even if it has nothing to do with the number of multiplications.