16.14.2 Tensor Determinant Alternating Argument Rule

Tensor Determinant Alternating Argument Rule is the condition stating that the determinant, viewed as a multilinear function of its column or row vectors, evaluates to zero whenever two of its arguments are equal, and correspondingly changes sign whenever two of its arguments are swapped. It is the second of the three defining properties, alongside multilinearity and normalization, that together uniquely characterize the determinant function.

Statement of the Rule

Vanishing on Repeated Arguments

For a determinant viewed as a function of n vector arguments v₁, v₂, ..., vₙ, the alternating argument rule states:

whenever any two of the argument positions hold the same vector. This directly mirrors the repeated factor rule of exterior powers, since the determinant is precisely the alternating form induced by the top exterior power.

Sign Reversal Under Transposition

Expanding the equality det(..., u + w, ..., u + w, ...) = 0, using multilinearity to distribute across both occurrences of the sum, and canceling the two terms that repeat u or repeat w, yields the sign-reversal consequence: swapping any two arguments negates the determinant.

Thus vanishing on repeated arguments and sign reversal under transposition are logically equivalent formulations of the same alternating rule, differing only in which is taken as primary.

Consequences for Matrix Operations

Row and Column Swap Rule

The classical rule that swapping two rows or two columns of a matrix negates its determinant is a direct restatement of the alternating argument rule applied to the column or row vectors of the matrix.

Effect of Row Reduction

The alternating rule justifies the standard elementary row operation used in Gaussian elimination for computing determinants: adding a multiple of one row to another leaves the determinant unchanged. This follows because det(..., vᵢ + c·vⱼ, ..., vⱼ, ...) expands by multilinearity into det(..., vᵢ, ..., vⱼ, ...) + c·det(..., vⱼ, ..., vⱼ, ...), and the second term vanishes by the alternating rule since it contains vⱼ twice.

Singular Matrices

If a matrix has two identical rows or columns, its column or row vectors are trivially linearly dependent, and the alternating rule immediately forces the determinant to zero. This links the alternating rule directly to the broader fact that a matrix is singular, meaning noninvertible, whenever its determinant vanishes, and that linear dependence among its rows or columns is a sufficient condition for this vanishing.

Relation to the Exterior Algebra

Determinant as an Alternating Tensor

The alternating argument rule is exactly the property that identifies the determinant as an element of the space of alternating n-linear forms, which is naturally isomorphic to the dual of the top exterior power Λⁿ(V). Every alternating n-linear form on an n-dimensional space satisfies this same vanishing-on-repetition rule, and the determinant is the specific instance normalized to equal 1 on the standard basis.

Antisymmetric Sign Pattern Under Permutation

More generally, permuting all n arguments of the determinant by any permutation σ multiplies the result by the sign of that permutation:

This is obtained by decomposing σ into a sequence of transpositions and applying the pairwise sign-reversal rule once for each transposition, which is exactly how the Leibniz formula for the determinant, expressed as a sum over all permutations with sign weights, arises.

Significance of the Rule

The alternating argument rule is what transforms a merely multilinear function into a genuinely determinant-like function: it forces vanishing on linearly dependent inputs, produces the sign behavior underlying permutation expansions, and justifies the elementary row and column operations used throughout practical determinant computation. Combined with multilinearity and normalization, it completes the axiomatic characterization of the determinant.