= Determinants =

The '''determinant''' is a number that embeds most information about a square matrix. Chiefly, for a matrix used to transform [[LinearAlgebra/Basis#Change_of_Basis|bases]], the determinant is the scaling factor of space in the transformation.

The determinant of '''''A''''' is often notated as ''|'''A'''|''.

<<TableOfContents>>

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== Definition ==

Given a matrix of shape 2 by 2, the determinant is calculated like:

{{{
    | a b |
det | c d | = ad - bc
}}}

Given an [[LinearAlgebra/SpecialMatrices#Upper_Triangular_Matrices|upper triangular matrix]], the determinant is the product of the diagonal.

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.

Only square matrices can be invertible, so non-square matrices have a determinant of zero.

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== Test for Invertibility ==

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.

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== Special Matrices and their Determinants ==

For the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]], the determinant is 1.

For a [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|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 [[LinearAlgebra/Orthogonality#Matrices|orthogonal matrix]], the determinant is 1 or -1.

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== Properties ==

Determinants can be factored: ''|'''AB'''| = |'''A'''| |'''B'''|''.

The determinant of the [[LinearAlgebra/MatrixInversion|inverse]] is the inverse of the determinant: ''|'''A'''^-1^| = 1/|'''A'''|''.

[[LinearAlgebra/MatrixTransposition|Transposition]] does not change the determinent: ''|'''A'''^T^| = |'''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'''|''.

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 ''n''th 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|           └    ┘
    └        ┘
}}}



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