On Minimizing the Number of Multiplications Necessary for Matrix Multiplication

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This paper develops an algorithm to multiply a $p \times 2$ matrix by a $2 \times n$ matrix in $\lceil (3pn + \max (n,p)) 2\rceil $ 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, \cdots $ and $p = 3,n = 3$. It is shown that commutativity actually reduces the number of multiplications required.