Determinants are scalar values computed from square matrices that encode important properties. The determinant measures whether a matrix is invertible ($\det \neq 0$), gives the volume scaling factor of the linear transformation, and appears in eigenvalue theory, Cramer rule, and change of variables.
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- Particular types of matricesParticular types of matrices are special matrices with specific structural properties or algebraic characteristics...
- Definition of trace of a matrixDefinition: The trace of a square matrix $A$ is the sum of its diagonal entries: $\text{tr}(A) = \sum i a {ii}$....
- CofactorsCofactors: The cofactor $C {ij}$ of entry $a {ij}$ is $C {ij} = (-1)^{i+j} M {ij}$, where $M {ij}$ is the minor...
- Determinants and operations on matricesDeterminants and matrix operations: - $\det(AB) = \det(A)\det(B)$ - $\det(A^T) = \det(A)$ - $\det(cA) = c^n \det(A)$...
- Determinants axiomaticallyThe determinant can be defined axiomatically as the unique function $\det: M n(\mathbb{F}) \to \mathbb{F}$...
- The determinant of a matrix measures the area/volume of the parallelogram/parallelipiped determined by its columns.Theorem: The absolute value of the determinant of a matrix measures the area (2D), volume (3D), or hypervolume (nD)...
- The determinant of the matrix of a linear transformation is the factor by which the area/volume changes.Theorem: The determinant of the matrix of a linear transformation $T$ is the factor by which area/volume changes...
- Definition of adjugate/classical adjoint of a matrixDefinition: The adjugate (or classical adjoint ) of a matrix $A$, denoted $\text{adj}(A)$, is the transpose of the...
- Ill-conditioning is measured by relative distance to singularityAn invertible matrix is ill-conditioned when its condition number is large in the chosen norm. A determinant close to...
- Condition number measures relative sensitivityFor an invertible square matrix and an induced norm, $\kappa(A)=\ A\ \ A^{-1}\ $. In the 2-norm, $\kappa 2(A)=\sigma...
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