ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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A concrete second exterior-power matrix matches the signed $2 \times 2$ minors

Example

Let $T : R^{3} \to R^{3}$ have matrix

$A = (\begin{array}{ccc} 1 & 2 & 0 \\ 0 & 1 & 3 \\ 2 & 0 & 1 \end{array}) .$

In the wedge basis $(e_{1} \land e_{2}, e_{1} \land e_{3}, e_{2} \land e_{3})$ of $Λ^{2} R^{3}$ , the matrix of $Λ^{2} T$ is

$[Λ^{2} T] = (\begin{array}{ccc} 1 & 3 & 6 \\ - 4 & 1 & 6 \\ - 2 & - 6 & 1 \end{array}),$

whose entry in row $J$ and column $I$ is exactly the signed $2 \times 2$ minor of $A$ with rows $J$ and columns $I$ .

Facts & Assumptions

Given: The operator $T$ with the displayed matrix $A$ and the wedge basis of $Λ^{2} R^{3}$ .

[L1]

In the wedge bases, the matrix of $Λ^{k} T$ has entries $\det A_{J, I}$ , where $A_{J, I}$ is the $k \times k$ submatrix with rows $J$ and columns $I$ (In basis-wedge coordinates, the matrix of $Λ^{k} T$ is the signed matrix of $k$ -minors).

Verification

technique · direct

1.1

Direct computation of the images of the three basis wedges gives $Λ^{2} T (e_{1} \land e_{2}) = (e_{1} + 2 e_{3}) \land (2 e_{1} + e_{2}) = e_{1} \land e_{2} - 4 e_{1} \land e_{3} - 2 e_{2} \land e_{3}$ , $Λ^{2} T (e_{1} \land e_{3}) = (e_{1} + 2 e_{3}) \land (3 e_{2} + e_{3}) = 3 e_{1} \land e_{2} + e_{1} \land e_{3} - 6 e_{2} \land e_{3}$ , and $Λ^{2} T (e_{2} \land e_{3}) = (2 e_{1} + e_{2}) \land (3 e_{2} + e_{3}) = 6 e_{1} \land e_{2} + 2 e_{1} \land e_{3} + e_{2} \land e_{3}$ , so the columns of the matrix are $(1, - 4, - 2)$ , $(3, 1, - 6)$ , $(6, 2, 1)$ .

algebra

1.2

By [L1], the expected entries are the nine $2 \times 2$ minors: $\det A_{12, 12} = 1$ , $\det A_{13, 12} = - 4$ , $\det A_{23, 12} = - 2$ , $\det A_{12, 13} = 3$ , $\det A_{13, 13} = 1$ , $\det A_{23, 13} = - 6$ , $\det A_{12, 23} = 6$ , $\det A_{13, 23} = 2$ , $\det A_{23, 23} = 1$ .

L1algebra

2.1

The nine coordinates of step 1.1 coincide entry by entry with the nine minors of step 1.2, so the displayed matrix equals the signed minor matrix.

step 1.1step 1.2∎

Depends on

In basis-wedge coordinates, the matrix of $\Lambda^kT$ is the signed matrix of $k$-minors

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Keith Conrad, Exterior Powers (standard reference, not scraped)

A concrete second exterior-power matrix matches the signed 2×22×2 minors