Alphabeta MathThe Determinant of a Linear Operator, Cofactors and Cramer's Rule
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1 · Prerequisites
Abstract Algebra
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
2 · Logical flowchart
3 · Items
Lemma
An alternating top-degree form is determined by its value on one ordered basis
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
Theorem
On a positive-dimensional space, det(T) is the unique scalar by which T scales every alternating top-degree form
Theorem
The determinant of a linear operator is independent of the chosen ordered basis
Theorem
For endomorphisms S and T of one finite-dimensional vector space, det(ST)=det(S)det(T)
Corollary
In positive dimension the determinant of an operator is computed from a representing matrix by row reduction, tracking swaps and row scalings
Definition
Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring
Theorem
Laplace expansion computes the determinant along every row and every column over a commutative ring
Theorem
For every positive-sized square matrix over a commutative ring, Aadj(A)=adj(A)A=det(A)I
Corollary
A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit
Corollary
If det(A) is a unit, then A⁻¹=det(A)⁻¹adj(A)
Lemma
For A∈ Mₙ(R) and columns u,v over a commutative ring, det(A+uv^T)=det(A)+v^Tadj(A)u
Theorem
Over a commutative ring, adj(P⁻¹AP)=P⁻¹adj(A)P for every invertible P
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)
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
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
Theorem
A finite-dimensional linear operator over a field is invertible if and only if its determinant is nonzero
Corollary
Determinant is a group homomorphism GL(V)→ F^×, and det(T⁻¹)=det(T)⁻¹
Definition
The adjugate of an operator on a positive-dimensional finite-dimensional vector space, defined by the adjugate matrix in any basis
Theorem