For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix

Depends on

• Finite rectangular matrices over a commutative ring, their entries, rows and columns
• Inversions, inversion number, the sign $\operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}$, and even and odd permutations
• A finite sum in a commutative monoid indexed by an arbitrary finite set
• The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n = 0$) equal to the identity
• A finite set $A$ with $\lvert A\rvert = n$ has exactly $n!$ bijections onto itself, and $n!$ bijections onto any set of the same cardinality
• Absolute value in an ordered field
• The reals form a field
• The reals form a totally ordered field
• Every field is a commutative ring with $1 \ne 0$; it is an integral domain, and it is a commutative division ring

Used by

• A triangle has zero Jordan content if and only if its vertices are collinear Corollary
• For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries Corollary
• Lebesgue measure on ℝⁿ is invariant under every orthogonal linear map Corollary
• The complex Jacobian determinant of a composite of equidimensional holomorphic maps is the product Corollary
• A real invertible matrix with no real logarithm Counterexample
• Congruence need not preserve trace or determinant: the real 1×1 matrices [1] and [4] are congruent Counterexample
• Positive determinant does not imply positive definiteness Counterexample
• Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring Definition
• Determinantal divisors from the minors of a matrix over a PID Definition
• For A∈ Mₙ(F), the characteristic polynomial is χ_A(x)=det(xIₙ-A) when n≥1, with χ_A(x)=1 for the unique 0×0 matrix Definition
• For A∈ Mₙ(R), the coordinate endomorphism T_A:Rⁿ→ Rⁿ, with det(T_A):=det(A) and adj(T_A):=T_adj(A) Definition
• Submatrices and minors of a rectangular matrix Definition
• The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and 1 on the zero space Definition
• The Gram inner product on ΛᵏV Definition
• The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1 Definition
• The Jacobian determinant of a square-dimensional C¹ map is the determinant of its Jacobian matrix Definition
• A full 3×3 Leibniz expansion lists all six permutations and their signs Example
• Cramer's rule solves 2x+y=5 and x-y=1 as (x,y)=(2,1) Example
• Determinant is additive in one selected column but not under simultaneous whole-matrix addition Example
• For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
• General and special linear Lie groups Example
• One operator has matrices diag(2,3) and beginpmatrix2&01&3 endpmatrix in two bases, both with determinant 6 Example
• Polar cartan decomposition of sl n r Example
• SL(n) as a closed Lie subgroup of GL(n) Example
• Spectrum in a finite-dimensional matrix algebra Example
• SU(2) to SO(3) as a covering homomorphism Example
• The complex Jacobian and its determinant for (z₀z₁, z₀+z₁) Example
• The content of a concrete three-dimensional parallelepiped computed from its spanning matrix Example
• The hyperspherical-coordinate Jacobian is the standard product of a radial power and sine powers Example
• The kernel and image of the determinant homomorphism Example
• The Lebesgue measure of the image of the unit cube under an explicit linear map of the plane and of three-space Example
• The Leibniz formula gives detbeginpmatrixa&bc&d endpmatrix=ad-bc Example
• The special linear group is a codimension-one embedded submanifold Example
• Unitary and special unitary Lie groups Example
• FALSE: a square matrix over a commutative ring is invertible if and only if its determinant is nonzero False statement
• The exponential map is surjective on every connected Lie group False statement
• The plus exponential convention is not a homomorphism for left actions False statement
• ‖ v‖₂ d(w,ℝv)=|det[v w]| for v≠0 in ℝ² Lemma
• 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 cyclic permutation of the coordinates of ℝ³ preserves Jordan measurability and integrals Lemma

…and 26 more results.

Dependency tree · two levels

45 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, Definition 4.21 (standard reference, not scraped)
• P. Massot, Structures algébriques fondamentales, Definition 6.4.1 (standard reference, not scraped)
• D. Margalit and J. Rabinoff, Interactive Linear Algebra, §4.1 (standard reference, not scraped)