A is an n x n matrix with elements aij = |i - j|. Show that the determinant |A| = (-1)n-1 (n - 1) 2n-2.
For i = 1, 2, 3, ... , n-2 subtract twice row i+1 from row i and add row i+2 to row i. For i < n-1, row i becomes all 0s except for a 2 in column i+1.
Now expand successively by the first, second, third ... rows to get (-2)n-2 times the 2 x 2 determinant with first row n-2, 1 and second row n-1, 0. This 2 x 2 determinant has value -(n-1), so the |A| = (-1)n-1 (n-1) 2n-2.
30th Putnam 1969
© John Scholes
14 Jan 2002