, , , , ' , , , , , , , , figure decomposing ab into four quadrants for long time it was assumed that ( was required for matrix multiplication howeverin the late sixtiesstrassen showed how to break the ( barrier the basic idea of strassen' algorithm is to divide each matrix into four quadrantsas shown in figure then it is easy to show that , , , , , , , , , , , , , , , , , , , , as an exampleto perform the multiplication ab ab we define the following eight / -by- / matricesa , , , , , , , , we could then perform eight / -by- / matrix multiplications and four / -by- / matrix additions the matrix additions take ( time if the matrix multiplications are done recursivelythen the running time satisfies ( ( / ( from theorem we see that (no( )so we do not have an improvement as we saw with integer multiplicationwe must reduce the number of subproblems below strassen used strategy similar to the integer multiplication divide-and-conquer algorithm and showed how to use only seven recursive calls by carefully arranging the computations the seven multiplications are