### 30th Putnam 1969

**Problem A2**

A is an n x n matrix with elements a_{ij} = |i - j|. Show that the determinant |A| = (-1)^{n-1} (n - 1) 2^{n-2}.

**Solution**

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) 2^{n-2}.

30th Putnam 1969

© John Scholes

jscholes@kalva.demon.co.uk

14 Jan 2002