determinant of anti-diagonal matrix

Let A=adiag(a1,,an) be an anti-diagonal matrix. Using the sum over all permutationsMathworldPlanetmath formula for the determinantMathworldPlanetmath 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 n-1 permutations are needed so


Now bring the last element to the second position. To do this n-2 permutations are needed. Repeat this procedure n-1 times to get the permutation (1,,n) 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 n or n-1 is a multiple of 4 and negative otherwise.

Title determinant of anti-diagonal matrix
Canonical name DeterminantOfAntidiagonalMatrix
Date of creation 2013-03-22 15:50:25
Last modified on 2013-03-22 15:50:25
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 6
Author cvalente (11260)
Entry type Result
Classification msc 15-00