determinant of anti-diagonal matrix
Let be an anti-diagonal matrix. Using the sum over all permutations formula for the determinant of a matrix and since all but possibly the anti-diagonal elements are null we get directly at the result
so all that remains is to calculate the sign of the permutation. This can be done directly.
To bring the last element to the beginning permutations are needed so
Now bring the last element to the second position. To do this permutations are needed. Repeat this procedure times to get the permutation which has positive sign.
Summing every permutation, it takes
permutations to get to the desired permutation.
So we get the final result that
Notice that the sign is positive if either or is a multiple of and negative otherwise.
|Title||determinant of anti-diagonal matrix|
|Date of creation||2013-03-22 15:50:25|
|Last modified on||2013-03-22 15:50:25|
|Last modified by||cvalente (11260)|