Prove that if a b c and d arc integers such that a b c d

Prove that if a, b, c and d arc integers such that a + b = c + d, then the matrix A = [a b c d] has integer eigenvalues.

Solution

A = ( a b , c d)

A- xI = ( a -x b , c d-x)

Charactereistice polynomial: (a-x)(d-x) - bc =0

ad -x(a+d) +x^2 -bc =0

x^2 -x (a+d) + ad -bc =0

As a + b = c + d then so a + d = (c + d - b) + d = (c + d) + (d - b) and ad - bc = (c + d - b) d - bc = cd + d^2 - bd - bc = d(c + d) - b (c + d) = (c + d)(d - b)

Quadratic function can be written as :

x^2 - [(c + d) + (d - b)] x + (c + d)(d - b) =0

factorised = (x - (c + d)) (x - (d - b))

Roots are c + d and d - b as given a + b = c + d, are the same things as a + b, and a - c

(subtract c + b from both sides of the equality a + b = c + d you find that a - c = d - b.)

As a, b, c, d are integers , so the eigen values are integers