1. Construct 6x6, 5x5, 4x4 and 3x3 matrices that has the largest determinant possible using only 1 and -1 I have attempted to reduce this problem by applying determinant properties, here is an example of my work for a 3x3 matrix. If we have 3x3 and fill it with 1's and -1's the total possible matrices are 512 However, property two says and row exchange or column exchange does not change the absolute values Thus, there are 8*7*6 = 336 Property three says if we multiply a matrix its determinant stays the same (in this case by -1) Thus there are 336/2 = 168 Property 5 says that two rows or columns that are equal (linear dep) Since we already applied that no rows are the same with P2. Not sure.. Property 10 says the det of the transpose is equal to the original matrix Thus 168/2 = 84 Where my true problem lies is not in only in simplifying the problem, but what to do afterwards? Once I have a reasonable number, how do I construct those matrices? Will I have to construct all 512 and then cut down...there must be a simple way to do this. Thanks for the help.