Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck 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≥1. If a matrix in Mn(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 A 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) by writing that column as 0+0; cancellation in the additive group gives det⁡(A)=0.

L1algebra
2.1

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

step 1.1L1∎

Depends on

Used by

Dependency tree · two levels

6 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