chap limits to computation at here stands for an matrix composed of zero valuesa is the original matrixand at stands for the transpose of matrix note that the resulting matrix is now symmetric we can convert matrix to symmetric matrix in similar manner if symmetric matrices could be multiplied "quickly(in particularif they could be multiplied together in th( time)then we could find the result of multiplying two arbitrary matrices in th( time by taking advantage of the following observationab bt at bt at in the above formulaab is the result of multiplying matrices and together hard problems there are several ways that problem could be considered hard for examplewe might have trouble understanding the definition of the problem itself at the beginning of large data collection and analysis projectdevelopers and their clients might have only hazy notion of what their goals actually areand need to work that out over time for other types of problemswe might have trouble finding or understanding an algorithm to solve the problem understanding spoken english and translating it to written text is an example of problem whose goals are easy to definebut whose solution is not easy to discover but even though natural language processing algorithm might be difficult to writethe program' running time might be fairly fast there are many practical systems today that solve aspects of this problem in reasonable time none of these is what is commonly meant when computer theoretician uses the word "hard throughout this section"hardmeans that the best-known algorithm for the problem is expensive in its running time one example of hard problem is towers of hanoi it is easy to understand this problem and its solution it is also easy to write program to solve this problem butit takes an extremely long time to run for any "reasonablylarge value of try running program to solve towers of hanoi for only disks the transpose operation takes position ij of the original matrix and places it in position ji of the transpose matrix this can easily be done in time for an matrix