# Fast Commutative Matrix Algorithm

We show that the product of an nx3 matrix and a 3x3 matrix over a commutative ring can be computed using 6n+3 multiplications. For two 3x3 matrices this gives us an algorithm using 21 multiplications. This is an improvement with respect to Makarov algorithm using 22 multiplications[10]. We generalize our result for nx3 and 3x3 matrices and present an algorithm for computing the product of an lxn matrix and an nxm matrix over a commutative ring for odd n using n(lm+l+m-1)/2 multiplications if m is odd and using (n(lm+l+m-1)+l-1)/2 multiplications if m is even. Waksman algorithm for odd n needs (n-1)(lm+l+m-1)/2+lm multiplications[16], thus in both cases less multiplications are required by our algorithm.

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