A recent manuscript of OpenAI proves that the exponent of matrix multiplication satisfies $\omega\le9/4$. We give an elementary, self-contained proof of this bound. We follow the ideas of the original manuscript, but implement them differently to obtain explicit algorithms. Specifically, for every $\varepsilon>0$, we give a matrix multiplication algorithm with exponent below $9/4+\varepsilon$ and base matrix size at most $2^{(1/\varepsilon)^{O(1)}}$.

We also show that techniques similar to those used in the square case apply to rectangular matrix multiplication. Starting from Strassen's tensor, we give elementary proofs of the bounds $\omega\bigl(1,\frac{2}{3\log 3},1\bigr)=2$ and $\omega(1,0.539,2)=3$. Stronger rectangular bounds are known, but we include these proofs for their simplicity.

In an appendix we describe the original proof, which goes through Strassen's theory of the asymptotic spectrum.