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
• 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
• Orientation of a finite-dimensional real vector space Definition
• For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
• Spectrum in a finite-dimensional matrix algebra 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
• Chart and partition independence of surface measure 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
• Smooth orientation sign is the local integral homology multiplier Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• The real Jacobian determinant of a complex-linear automorphism is the squared modulus of its complex determinant Lemma
• A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent Theorem
• A linear map T of ℝⁿ sends Lebesgue measurable sets to Lebesgue measurable sets, with λₙ(T[E])=|det T| λₙ(E) when T is invertible and T[E] Lebesgue null when it is not Theorem
• A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique Theorem
• A square matrix has an unpivoted unit-lower LU factorisation exactly when all leading principal minors are nonzero Theorem
• Change of variables for an injective C¹ map on a compact Jordan set Theorem
• For A,B∈ Mₙ(F), the products AB and BA have the same characteristic polynomial Theorem
• Norm is multiplicative, trace is F-linear, and both are transitive in towers Theorem
• Similar matrices have the same characteristic polynomial Theorem
• Sylvester's criterion: a real symmetric n× n matrix with n≥1 is positive definite if and only if all leading principal minors are positive Theorem

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)