Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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A square matrix with a zero column or two equal columns has determinant zero

Statement

Let n≥1n\ge1. If a matrix in Mn(R)M_n(R) over a commutative ring has a zero column or has two equal columns, then its determinant is zero.

Facts & Assumptions

Given: A square matrix AA over a commutative ring.

[L1]

Determinant is column-multilinear and alternating (The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring).

Proof

technique · direct
1.1

If one column is zero, multilinearity gives det⁡(A)=det⁡(A)+det⁡(A)\det(A)=\det(A)+\det(A) by writing that column as 0+00+0; cancellation in the additive group gives det⁡(A)=0\det(A)=0.

L1algebra
2.1

If two columns are equal, alternation in [L1] gives det⁡(A)=0\det(A)=0 directly, regardless of whether the columns are adjacent.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources