16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.2.12

Determinants

Mathematics — Linear Algebra · multiplicative-determinant-function

Problem

Let FF be a field and DD a function on n×nn \times n matrices over FF (with values in FF). Suppose D(AB)=D(A)D(B)D(AB) = D(A)D(B) for all AA, BB. Show that either D(A)=0D(A) = 0 for all AA, or D(I)=1D(I) = 1. In the latter case show that D(A)≠0D(A) \neq 0 whenever AA is invertible.

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