CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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The determinant is alternating and multilinear in the rows as well as in the columns

Statement

For $n\ge1$ , determinant on $M_n(R)$ over a commutative ring is multilinear and alternating in the rows. Interchanging two rows negates the determinant.

Facts & Assumptions

Given: A square matrix whose selected row is varied, or whose two selected rows are interchanged.

[L1]

Transpose leaves determinant unchanged (For every square matrix over a commutative ring, $\det(A^{\mathsf T})=\det(A)$ ).

[L2]

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

[L3]

Transpose is defined by $(A^{\mathsf T})_{ji}=a_{ij}$ , so it interchanges rows and columns (Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose).

Proof

technique · direct

1.1

Transposition sends a selected row, a row sum, a row scaling or a row swap to the corresponding operation on a selected column, without changing determinant.

L1L3

2.1

Apply the column multilinearity and alternation of [L2] to the transpose, then apply [L1] again. The resulting identities are exactly row multilinearity and row alternation for the original matrix.

step 1.1L1L2∎

Depends on

Used by

For every square matrix, including singular ones, a row swap negates the determinant, scaling a row by any scalar scales it, and row addition leaves it unchanged Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 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

S. New, MATH 146 Linear Algebra 1 Lecture Notes, Theorem 4.23 (standard reference, not scraped)