However, the number of additions required to add those sets of matrices is much closer together. Generally, the number of additions is equal to the number of entries in the matrix, so four for the two-by-two matrices and 16 for the four-by-four matrices.
Maybe my linear algebra is REALLY rusty, but I that doesn't seem right to me.
For two 4x4 matrices, each element in the output matrix will be the sum of four products. So each of the 16 output elements will require 3 additions, and you'll need a total of 48 additions.
I'm not sure where they're getting "16 additions" from.
To multiply two nxn matrices according to the textbook method, I think you'll need n^3 multiplications and n^3-n^2 additions.
I don't doubt that the multiplication time dominates the algorithm. Maybe this is just a nitpick.