Kunze Linear Algebra · Problem 5.2.1

Determinants

Mathematics — Methods · n-linear-functions

Problem

Each of the following expressions defines a function

D

on the set of

3 \times 3

matrices over the field of real numbers. In which of these cases is

D

a 3-linear function? ((a))

D(A) = A_{11} + A_{22} + A_{33}

; ((b))

D(A) = (A_{11})^2 + 3A_{11}A_{22}

; ((c))

D(A) = A_{11}A_{12}A_{33}

; ((d))

D(A) = A_{13}A_{22}A_{32} + 5A_{12}A_{22}A_{32}

; ((e))

D(A) = 0

; ((f))

D(A) = 1