Kunze Linear Algebra · Problem 5.2.6

Determinants

Mathematics — Methods · n-linear-functions

Problem

Let

K

be a subfield of the complex numbers and

n

a positive integer. Let

j_1, \ldots, j_n

and

k_1, \ldots, k_n

be positive integers not exceeding

n

. For an

n \times n

matrix

A

over

K

define

D(A) = A(j_1, k_1)A(j_2, k_2) \cdots A(j_n, k_n).

Prove that

D

n

-linear if and only if the integers

j_1, \ldots, j_n

are distinct.