Kunze Linear Algebra · Problem 5.2.8

Determinants

Mathematics — Methods · determinant-columns

Problem

Let KK be a commutative ring with identity. Define a function DD on 3×33 \times 3 matrices over KK by the rule D(A)=A11det⁡[A22A23A32A33]−A12det⁡[A21A23A31A33]+A13det⁡[A21A22A31A32].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 DD is alternating and 3-linear as a function of the columns of AA.

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