On Minimizing the Number of Multiplications Necessary for Matrix Multiplication

This paper develops an algorithm to multiply a p × 2 matrix by a 2 × n matrix in (3pn + max (n,p)) 2 multiplications without use of commutativity of matrix elements. The algorithm minimizes the number of multiplications for matrix multiplication without commutativity for the special cases $p = 1$ or 2,n = 1,2, ⋯ and $p = 3,n = 3$. It is shown that commutativity actually reduces the number of multiplications required.