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Is there a formula to calculate the number of multiplications that take place when multiplying 2 matrices? For example

$$\begin{pmatrix}1&2\\3&4\end{pmatrix} \times \begin{pmatrix}5&6\\7&8\end{pmatrix} = \text{8 multiplications and 4 additions} $$

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migrated from mathoverflow.net Sep 5 '13 at 6:51

This question came from our site for professional mathematicians.

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There is actually a way to multiply a pair of $2\times2$ matrices doing only 7 multiplications, and finding the minimum number of multiplications for $n\times n is ongoing research. But the number of multiplications required by the straightforward method is not a research question, and research is what this website is about. – Gerry Myerson Sep 5 '13 at 6:34

Doing a $k\times l$ times $l\times m$ matrix multiplication in the straightforward way, every entry of the result is a scalar product of of two $l$-vectors, which requires $l$ multiplications and $l-1$ additions. Multiply that by the number $km$ of entries of the result (or don't multiply if you have sufficiently many processors to do everything in parallel).

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