In this paper we give a new algorithm for matrix multiplication which for n large uses $n^2 + o(n^2 )$ multiplications to multiply $n \times p$ matrices by $p \times n$ matrices provided $p \leqq \log _2 n$. Multiplication and division by 2 is necessary in this algorithm. This is to be compared with $pn^2 $ for the standard algorithm and $ \simeq p^{.58} n^2 + o(n^2 )$ for an algorithm of Hopcroft and Kerr [1] which, however, requires no multiplication and division by 2.