Determinants

A determinant is a number that embeds most information about a matrix.

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

Definition

Given an upper triangular matrix U, the determinant is the product of the diagonal.

If a matrix A cannot be converted into an upper triangular matrix through elimination, it must be 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.

Properties

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.

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.

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: |A^T| = |A|.

Exchanging rows flips the sign of the determinant. 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|           └    ┘
    └        ┘

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