Determinants
A determinant is a number that embeds most information about a matrix.
The determinant of A is often notated as |A|.
Contents
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: |AT| = |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
└ ┘ └ ┘ └ ┘ └ ┘ └ ┘ └ ┘