 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2016-resit/2/solution

Expanding the determinant along its first row gives

det A = a (a^{2} - b c) - b (a c - b^{2}) + c (c^{2} - ab) = a^{3} + b^{3} + c^{3} - 3 ab c .

The required factorization is

(a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a) .

\frac{1}{2} [(a - b)^{2} + (b - c)^{2} + (c - a)^{2}]

, making its permutation symmetry especially transparent. This factorization holds over both real and complex coefficients.

 New to topics? Read the docs here!