gistfile1.txt

Do we have to correctly compute the determinant of matrices whose determinant contains components that don't fit in a long, like this 4x4 matrix:

M=

0000001 4999999 4999999 4999999
0000001 5000000 9999998 9999998
0000001 5000000 9999998 9999999
0000000 0000000 0000001 4999999

This matrix's determinant is -1 (this can be discovered by noticing that it is the product of the 3 matrices

U=
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0

S=

1 0 0 0
1 1 0 0
0 0 1 0
1 1 0 1

A=
0000001 4999999 4999999 4999999

0000000 0000001 4999999 4999999
0000000 0000000 0000001 4999999
0000000 0000000 0000000 0000001


whose can be easily shown to be -1, 1, 1 respectively).

However, the computation of the determinant of M requires calculating 1*5000000*9999998*4999999, which is ~2^67.76 , and does not fit in a long.

you can Get Reduced Row Echelon form By simple steps and then it will be converted into the forms given below. Any Arithmetic operation on rows/columns does not change Determinant of matrix. you can subtract one row from another and can obtain the Reduced Row Echelon Form (Make sure that you are Applying functions on rows only)