16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.2.8

Determinants

Mathematics — Linear Algebra · alternating-multilinear-columns

Problem

Let

K

be a commutative ring with identity. Define a function

D

3 \times 3

matrices over

K

by the rule

D(A) = A_{11} \det \begin{bmatrix} A_{22} & A_{23} \\ A_{32} & A_{33} \end{bmatrix} - A_{12} \det \begin{bmatrix} A_{21} & A_{23} \\ A_{31} & A_{33} \end{bmatrix} + A_{13} \det \begin{bmatrix} A_{21} & A_{22} \\ A_{31} & A_{32} \end{bmatrix}.

Show that

D

is alternating and 3-linear as a function of the columns of

A