Determinants

The determinant is a number that embeds most information about a square matrix. Chiefly, for a matrix used to transform bases, the determinant is the scaling factor of space in the transformation.

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

Definition

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

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

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.

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

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: |A^T| = |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|           └    ┘
    └        ┘

CategoryRicottone