12th Putnam 1952 Problem B3 Let A(x) = be the matrix 0 a-x b-x -a-x 0 c-x -b-x -c-x 0 For which (a, b, c) does det A(x) = 0 have a repeated root in x?
Multiplying out, we find that det A(x) = -2x 3 + 2(bc - ac + ab)x. This has roots 0 and ±√(ab + bc - ac). This has a repeated root iff ac = ab + bc. 12th Putnam 1952 © John Scholes jscholes@kalva.demon.co.uk 5 Mar 2002