5. If A, B, and AB are symmetric n x n matrices and the characteristic roots of A are a1, a2,...,

a, and of B are b₁, b2,..., bn, show that

$$tr (AB) = \sum_{i=1}^{n} a_{j_i} b_i,$$

where ajı, aj..., a; is some ordering of a1, a2,..., a.