A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit

Depends on

• For every positive-sized square matrix over a commutative ring, $A\operatorname{adj}(A)=\operatorname{adj}(A)A=\det(A)I$
• An invertible square matrix over a commutative ring has unit determinant
• Invertible square matrices and similarity over a commutative ring
• The units of a ring are the invertible elements of its multiplicative monoid, and $R^{\times}$ is a group under multiplication; $0 \in R^{\times}$ only in the zero ring

Used by

• If det(A) is a unit, then A⁻¹=det(A)⁻¹adj(A) Corollary
• Over a field, for each right-hand side b, Ax=b has a unique solution exactly when det(A) is nonzero, and then Cramer's quotient formula holds Corollary
• The index of a full-rank subgroup of ℤⁿ is the absolute determinant of a generating matrix Corollary
• The unreduced Burau matrices Definition
• Multiplication by 2 on ℤ is injective but not surjective: its determinant is the non-unit 2, its adjugate is integral, and its inverse exists after extending scalars to ℚ Example
• The exceptional spectra for E₆ and H₃ computed exactly: characteristic polynomials, cyclotomic factorisations and the resulting degree tables Example
• The rank-one 2 by 2 determinantal cone 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
• Algebraicity of the coordinates over the invariant field and non-vanishing of the invariant Jacobian Lemma
• Finite matrix and module preliminaries: right inverses, rank invariance, finite length and nilpotent trace Lemma
• Finite matrix rank bounds under regular specialization Lemma
• For a finite Galois extension, (αⱼ) is a base-field basis exactly when the matrix (σᵢαⱼ) is invertible Lemma
• Rational representations of an affine group scheme are comodules of its coordinate Hopf algebra Lemma
• The Coxeter elements of the classical types Aₙ, Bₙ, Dₙ and I₂(m): characteristic polynomials, orders and spectral exponents from their reflection models Lemma
• The general linear group scheme and its coordinate ring Lemma
• The six exceptional Coxeter spectra: characteristic polynomials, orders and spectral exponents from exact matrices Lemma
• Triviality of finite free deformations of semisimple algebras over the power series ring Lemma
• A finite-dimensional linear operator over a field is invertible if and only if its determinant is nonzero Theorem
• A finitely generated affine group scheme has a faithful finite-dimensional representation Theorem
• A regular Coxeter eigenvector determines the basic degrees: the exponent-residue identification and the complete degree tables for all finite Coxeter types Theorem
• Cramer's rule over a commutative ring: every solution satisfies det(A)xⱼ=det(Aⱼ(b)), and a unit determinant gives the unique quotient formula Theorem
• Every finite Galois extension of an infinite field has a normal basis Theorem
• Fixed-tabloid reconstruction of the character table (Fox–Mullineux) Theorem
• On the invertible locus, Dinv(A)[H]=-A⁻¹HA⁻¹ Theorem
• Polynomial algebras over fields have finite integral closures Theorem
• Ramification is detected by the number-field discriminant Theorem
• The binary Sylvester resultant detects a common geometric projective root Theorem
• The formal quantum Serre half embeds in the shuffle algebra and is degreewise free Theorem
• The generator presentation is the planar diagram algebra over every commutative ring Theorem

Dependency tree · two levels

18 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

• András Pál, Introduction to Commutative Algebra, Proposition 7.2 (standard reference, not scraped)