Universal Identities , Ii : ⊗ and ∧

Theorem 1.1. Let R be a commutative ring and n be a positive integer. For any A ∈ Mn(R), write its characteristic polynomial as det(TIn −A) = T + c1(A)T + · · ·+ cn−1(A)T + cn(A) ∈ R[T ]. Then ck(A) = (−1)k Tr(∧k(A)). Note the indexing: the coefficient of Tn−k is associated to the kth exterior power of A. In the special cases k = 1 and k = n this recovers the more familiar result that c1(A) = −Tr(A) and cn(A) = (−1)n det(A). Here are two more theorems about multilinear operations on matrices.