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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
Statement
Let and let be a matrix over a commutative ring.
- Interchanging two rows changes to .
- Multiplying one row by any changes it to .
- Adding times one row to a distinct row leaves it equal to .
No invertibility assumption is made on or on .
Facts & Assumptions
Given: A square matrix over a commutative ring and a row transformation.
Determinant is alternating and multilinear in its rows (The determinant is alternating and multilinear in the rows as well as in the columns).
Ring row transformations include swaps, scaling by any scalar, and addition of a multiple of a distinct row (Row swaps, arbitrary row scalings and row additions over a commutative ring, with reversible elementary cases distinguished).
Determinant takes values in the same commutative ring (For , the determinant over a commutative ring by the Leibniz formula, and for a real matrix).
Proof
Row multilinearity gives for every scalar , including and zero divisors.
Row antisymmetry gives when is obtained by a row swap.
Replacing by expands the new determinant as , where has row in both positions and ; alternation gives , so the determinant is unchanged.
The derivations used only row multilinearity and alternation, not cancellation, matrix invertibility or scalar invertibility. Hence all three formulas include singular matrices and arbitrary scalars.
Depends on
- The determinant is alternating and multilinear in the rows as well as in the columns
- Row swaps, arbitrary row scalings and row additions over a commutative ring, with reversible elementary cases distinguished
- For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix
Used by
- In positive dimension the determinant of an operator is computed from a representing matrix by row reduction, tracking swaps and row scalings Corollary
- A sparse 4×4 determinant is 30 by both cofactor expansion and row reduction Example
- Row operations track determinant correctly for a singular triangular matrix over ℤ/6 Example
- A coordinate scaling and a coordinate transposition send the unit cube to a set of measure equal to the absolute value of the determinant Lemma
- A shear sends the unit cube to a set of Lebesgue measure one Lemma
- Smooth orientation sign is the local integral homology multiplier Lemma
- A finite square real matrix is invertible if and only if its determinant is nonzero Theorem
- A linear endomorphism of ℝⁿ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant Theorem
Dependency tree · two levels
17 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
- S. New, MATH 146 Linear Algebra 1 Lecture Notes, Theorem 4.23 (standard reference, not scraped)
- D. Margalit and J. Rabinoff, Interactive Linear Algebra, §4.1 (standard reference, not scraped)