Alphabeta MathThe Determinant of an Operator: Examples and Counterexamples
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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
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
2 · Logical flowchart
3 · Items
Example
A nonidentity projection of the plane has determinant zero and is not invertible
Example
One operator has matrices diag(2,3) and beginpmatrix2&01&3 endpmatrix in two bases, both with determinant 6
Example
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
False statement
FALSE: a square matrix over a commutative ring is invertible if and only if its determinant is nonzero
Example
A sparse 4×4 determinant is 30 by both cofactor expansion and row reduction
Example
The adjugate gives the inverse of a 3×3 rational matrix with determinant 3
Example
Cramer's rule solves 2x+y=5 and x-y=1 as (x,y)=(2,1)
Example