Multiplying two n by n matrices the schoolbook way costs n cubed multiplications: every entry of the answer is a row dotted with a column. For a long time that was assumed to be the end of the story, on the reasonable grounds that you have to combine every row with every column somehow.
In 1969 Volker Strassen showed it was not. Two 2 by 2 matrices can be multiplied with seven scalar multiplications instead of eight, using a set of combinations that looks like a conjuring trick the first time you see it. The saving of one looks trivial until you recurse: apply it to block matrices and the cost of an n by n product falls from n^3 to about n^2.807.
Why the exponent is the whole game
A constant-factor saving is worth having once. An exponent saving compounds. At 4 by 4 the schoolbook method needs 64 multiplications and Strassen needs 49, a saving of about a quarter. At 4096 by 4096 the schoolbook method needs about 68.7 billion and Strassen about 13.8 billion, a saving of four fifths. The gap widens with every doubling, which is what a smaller exponent means.
So the race has always been to push the exponent down. And at the 4 by 4 case, Strassen's own method had held the record since 1969: recursing 2 by 2 twice gives 7 times 7, which is 49.
Forty-eight
In May 2025 Google DeepMind reported that AlphaEvolve, an evolutionary coding agent, had found a way to multiply two 4 by 4 complex-valued matrices using 48 scalar multiplications. One fewer than 49, and the first improvement on that case in 56 years. Recursing on a 4 by 4 base case of 48 gives an exponent of log base 4 of 48, which is about 2.793, against Strassen's 2.807.
Two qualifications belong with that, and they are usually left out. The first is that the algorithm is for complex-valued entries. For ordinary real matrices, 49 remains the best known at 4 by 4, so the result answers a slightly different question than the one Strassen answered. The second is that this is not the lowest exponent known in theory: methods in the Coppersmith-Winograd family get below 2.372, but with constant factors so large that they are never used on a real matrix. The interest in 48 is that it is small, explicit and could plausibly be implemented.
The matrix multiplication cost calculator compares the three methods at any size. It also shows the unglamorous side: below the first block size the fast methods lose badly, because a 2 by 2 matrix has to be padded up to 4 by 4 and charged 48 multiplications where doing nothing clever costs 8.
The wider run
The matrix result was not isolated. AlphaEvolve improved the state of the art on fourteen other matrix multiplication cases, and when it was later pointed at 67 problems across analysis, combinatorics, geometry and number theory it matched the best known construction on most and improved it on roughly a fifth. One of those improvements was a kissing-number configuration in eleven dimensions, covered in the piece on the kissing number.
None of this has made matrix multiplication faster in practice yet. Turning a smaller base case into faster library code means dealing with extra additions, memory traffic and numerical stability, which is why libraries often only switch to Strassen above a few hundred. The significance is that a bound which had not moved since 1969 moved, and a machine moved it.