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| A '''determinant''' is a number that embeds most information about a matrix. | The '''determinant''' is a number that embeds most information about a matrix. For a matrix effecting a [[LinearAlgebra/Basis#Change_of_Basis|change of basis]], the determinant is the scaling factor of space between bases. |
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| Given an [[LinearAlgebra/SpecialMatrices#Upper_Triangular_Matrices|upper triangular matrix]] '''''U''''', the determinant is the product of the diagonal. | Given an [[LinearAlgebra/SpecialMatrices#Upper_Triangular_Matrices|upper triangular matrix]], the determinant is the product of the diagonal. |
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| If a matrix '''''A''''' cannot be converted into an upper triangular matrix through [[LinearAlgebra/Elimination|elimination]], it must be [[LinearAlgebra/MatrixProperties#Invertability|singular and non-invertable]], and so the determinant must be 0. If any two rows are the same, or if there are any rows of zeros, the matrix is non-invertible and the determinant is 0. | If a matrix cannot be [[LinearAlgebra/Elimination|eliminated]] into an upper triangular matrix, it is '''degenerate''' and [[LinearAlgebra/MatrixProperties#Invertible|non-invertible]]. This directly means that the determinant is zero. This generally only happens if there is multicolinearity. As noted in the properties below, elimination does not change the determinant (although row exchanges flip the sign of it). Given a [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]], the determinant is the product of the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. If a matrix cannot be [[LinearAlgebra/Diagonalization|diagonalized]], it is '''defective'''. This simply means that one of the eigenvalues is zero; it may still be invertible and therefore have a non-zero determinent. ---- |
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| === Properties === | == Test for Invertibility == |
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| Determinants are the test for '''invertability'''. if ''|'''A'''| != 0'', then '''''A''''' is invertable and non-singular. Conversely, if ''|'''A'''| = 0'', then '''''A''''' is singular and non-invertable. | There is a direct connection between [[LinearAlgebra/MatrixProperties#Invertible|invertibility]] and having a non-zero determinant. As a result, calculation of the determinant is a simple test for invertibility. ---- == Special Matrices and their Determinants == |
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| ---- == Properties == |
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Exchanging rows flips the sign of the determinant. If '''''U''' = '''PA''''', then ''|'''U'''| = |'''P'''| |'''A'''|''. |
Exchanging rows flips the sign of the determinant. This is the intuitive explanation for the determinant of the [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrix]] as noted above. If '''''U''' = '''PA''''', then ''|'''U'''| = |'''P'''| |'''A'''|''. |
Determinants
The determinant is a number that embeds most information about a matrix.
For a matrix effecting a change of basis, the determinant is the scaling factor of space between bases.
The determinant of A is often notated as |A|.
Contents
Definition
Given an upper triangular matrix, the determinant is the product of the diagonal.
If a matrix cannot be eliminated into an upper triangular matrix, it is degenerate and non-invertible. This directly means that the determinant is zero. This generally only happens if there is multicolinearity.
As noted in the properties below, elimination does not change the determinant (although row exchanges flip the sign of it).
Given a diagonal matrix, the determinant is the product of the eigenvalues.
If a matrix cannot be diagonalized, it is defective. This simply means that one of the eigenvalues is zero; it may still be invertible and therefore have a non-zero determinent.
Test for Invertibility
There is a direct connection between invertibility and having a non-zero determinant. As a result, calculation of the determinant is a simple test for invertibility.
Special Matrices and their Determinants
For the identity matrix, the determinant is 1.
For a permutation matrix, the determinant is 1 if there are an even number of row exchanges in the matrix or -1 if there are an odd number of row exchanges.
For an orthogonal matrix, the determinant is 1 or -1.
Properties
Determinants can be factored: |AB| = |A| |B|.
The determinant of the inverse is the inverse of the determinant: |A-1| = 1/|A|.
Transposition does not change the determinent: |AT| = |A|.
Exchanging rows flips the sign of the determinant. This is the intuitive explanation for the determinant of the permutation matrix as noted above. If U = PA, then |U| = |P| |A|.
Multiplying a single row of a matrix by some factor simply means that the determinant was multiplied by the same factor.
┌ ┐ ┌ ┐
│ ta tb│ │ a b│
det │ c d│ = t * det │ c d│
└ ┘ └ ┘Multiplying every row of a matrix by some factor means that the determinant was multiplied by the same factor to the nth power.
┌ ┐ ┌ ┐ ┌ ┐
│ ta tb│ │ a b│ │ a b│
det │ tc td│ = t * det │ tc td│ = t * t * det | c d|
└ ┘ └ ┘ └ ┘Adding to or subtracting from a single row of a matrix means that the determinant is the sum of the determinants of the two factored-out matrices.
┌ ┐ ┌ ┐ ┌ ┐
│ a+x b+y│ │ a b│ │ x y│
det │ c d│ = det │ c d│ + det │ c d│
└ ┘ └ ┘ └ ┘Furthermore, elimination does not change the determinant at all.
┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐
│ a b│ │ a b│ │ a b│ │ a b│ │ a b│ │ a b│
det │ c-ma d-mb│ = det │ c d│ - det │ ma mb│ = det │ c d│ - m * det │ a b│ = det │ c d│ - m * 0
└ ┘ └ ┘ └ ┘ └ ┘ └ ┘ └ ┘Large matrices, especially with mostly zeros, can be broken up.
┌ ┐
| 2 0 0 0| ┌ ┐
| 0 a b 0| │ a b│
det | 0 c d 0| = 2 * det │ c d│ * 3
| 0 0 0 3| └ ┘
└ ┘