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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 $n\ge1$ and let $A\in M_n(R)$ be a matrix over a commutative ring.

Interchanging two rows changes $\det(A)$ to $-\det(A)$ .
Multiplying one row by any $c\in R$ changes it to $c\det(A)$ .
Adding $c$ times one row to a distinct row leaves it equal to $\det(A)$ .

No invertibility assumption is made on $A$ or on $c$ .

Facts & Assumptions

Given: A square matrix $A$ over a commutative ring and a row transformation.

[L1]

Determinant is alternating and multilinear in its rows (The determinant is alternating and multilinear in the rows as well as in the columns).

[L2]

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).

[L3]

Determinant takes values in the same commutative ring (For $n\ge1$ , the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix).

Proof

technique · direct

1.1

Row multilinearity gives $\det(\ldots,cR_p,\ldots)=c\det(A)$ for every scalar $c$ , including $c=0$ and zero divisors.

L1L2L3

2.1

Row antisymmetry gives $\det(A')=-\det(A)$ when $A'$ is obtained by a row swap.

step 1.1L1L2

3.1

Replacing $R_p$ by $R_p+cR_q$ expands the new determinant as $\det(A)+cD$ , where $D$ has row $R_q$ in both positions $p$ and $q$ ; alternation gives $D=0$ , so the determinant is unchanged.

step 2.1L1L2algebra

4.1

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.

step 3.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 57 results over 12 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)
D. Margalit and J. Rabinoff, Interactive Linear Algebra, §4.1 (standard reference, not scraped)