We give a short proof of the optimality of Strassen's classical algorithm for 2 x 2 matrix multiplication, and more generally give a 2n^2 - n + 1 lower bound for n x n matrix multiplcation, in terms of the border rank (and, hence, also rank) of the associated tensor, over an arbitrary field.

We call the tool we use “inner rank,” which seems to have gone unnoticed in the literature (at least in the way we use it). While our bounds are not competitive with the best bounds to date over the complex numubers, we argue that “inner rank” may lead to improved results and merits further study.