For same-sized finite square matrices over a commutative ring, $\det (AB)=\det (A)\det (B)$

Depends on

• The determinant is the unique normalized alternating multilinear function on the columns
• Every alternating multilinear $F$ satisfies $F(A)=F(I)\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_i a_{\sigma(i),i}$
• Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products
• For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix

Used by

• An invertible square matrix over a commutative ring has unit determinant Corollary
• If A is invertible over a commutative ring, then det(A⁻¹)=det(A)⁻¹ Corollary
• Polar integration may discard the cut locus Corollary
• Similar matrices over a commutative ring have the same determinant Corollary
• The complex Jacobian determinant of a composite of equidimensional holomorphic maps is the product Corollary
• The index of a full-rank subgroup of ℤⁿ is the absolute determinant of a generating matrix Corollary
• Full-rank lattices, covolume, and the dual lattice Definition
• Orientation of a finite-dimensional real vector space Definition
• The modular group and its action on the upper half-plane Definition
• The real projective line and the action of SL2(R) Definition
• A unimodular change of generators preserves the regulator determinants Example
• Chebyshev extremals and the exact disk Fekete polynomial Example
• For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
• Regular and singular torus characters in GL₃(F_q) Example
• Spectrum in a finite-dimensional matrix algebra Example
• The Coxeter complex of I₂(5): the circle triangulated by the ten chambers Example
• The group schemes Ga, Gm, and GLn 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 special linear group is a codimension-one embedded submanifold Example
• A determinant and Frobenius norm give a certified lower bound for small-block separation Lemma
• A metabolic Seifert matrix has Fox–Milnor factorization Lemma
• A nonsingular principal minor of the symmetrized Cartan matrix of size the rank Lemma
• A uniform positive eigenvalue floor controls log-determinant error Lemma
• An explicit root-separation bound for squarefree integer polynomials Lemma
• Bfgs trace and determinant identities give a matrix potential inequality Lemma
• Bounded elementary generation of SLn(R) by transvections Lemma
• Cellular basis ambiguities vanish in the Whitehead group Lemma
• Changing an error-contrast basis changes the REML likelihood by a constant Lemma
• Chart and partition independence of surface measure Lemma
• Cyclotomic Schur weight Gram determinants are products of monic semistandard norms Lemma
• Determinant-one Clifford+T words are dense in SU(2) Lemma
• Determinant-one reduction for optimal Rz approximation Lemma
• Determinants of one-qubit Clifford+T words are eighth roots Lemma
• Finite matrix and module preliminaries: right inverses, rank invariance, finite length and nilpotent trace Lemma
• For symmetric M=beginpmatrixA&BB^T&C endpmatrix with A invertible, a block-unitriangular congruence gives A⊕(C-B^TA⁻¹B) and factors det M Lemma
• Gaussian integral for a positive-definite quadratic form Lemma
• Gaussian mixture lower bound for weakly sparse covariance Lemma
• Holomorphic Morse normalization and Gaussian contour Lemma
• Length-additive products of the standard intertwiners Lemma

…and 51 more results.

Dependency tree · two levels

22 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 4.24 (standard reference, not scraped)
• P. Massot, Structures algébriques fondamentales, Lemma 6.4.2 (standard reference, not scraped)