sec reductions sortinginteger array transform integers to integer array pairing array of pairs transform sorted integer array figure the reduction of sorting to pairing shown as "blackboxdiagram our next example of reduction concerns the multiplication of two matrices for this problemwe will assume that the values stored in the matrices are simple integers and that multiplying two simple integers takes constant time (because multiplication of two int variables takes fixed number of machine instructionsthe standard algorithm for multiplying two matrices is to multiply each element of the first matrix' first row by the corresponding element of the second matrix' first columnthen adding the numbers this takes th(ntime each of the elements of the solution are computed in similar fashionrequiring total of th( time faster algorithms are known (see the discussion of strassen' algorithm in section )but none are so fast as to be in ( nowconsider the case of multiplying two symmetric matrices symmetric matrix is one in which entry ij is equal to entry jithat isthe upper-right triangle of the matrix is mirror image of the lower-left triangle is there something about this restricted case that allows us to multiply two symmetric matrices faster than in the general casethe answer is noas can be seen by the following reduction assume that we have been given two matrices and we can construct symmetric matrix from an arbitrary matrix as follows