Determinants Flashcards
Determinant gives what
volume of cuboid
Determinants applied on ___________
Determinants applied on square matrix
Definition of determinant
A factor which determines whether the linear system of equations have unique solution or not
Determinant Formula
|Aₙₓₙ|= Σ ⁿⱼ₌₁ (-1)¹⁺ʲ a₁ⱼ δ₁ⱼ
(fix i = 1 (understood as first row is fixed))
fix any row or column
where δᵢⱼ= minor of an element aᵢⱼ
= it is determinant of sub-matrix obtained by eliminating iᵗʰ row and jᵗʰ column of element aᵢⱼ
δᵢⱼ what it is
δᵢⱼ= minor of an element aᵢⱼ
= it is determinant of sub-matrix obtained by eliminating iᵗʰ row and jᵗʰ column of element aᵢⱼ
Minor of an element
δᵢⱼ= minor of an element aᵢⱼ
= it is determinant of sub-matrix obtained by eliminating iᵗʰ row and jᵗʰ column of element aᵢⱼ
Number of terms in determinant expansion of A
Number of terms in determinant expansion of Aₙₓₙ = n!
|A| in terms of transpose
|A| = |Aᵀ|
|AB|
|AB| = |A||B| = |B||A|=|BA|
|Aⁿ|
|A|ⁿ
|A⁻¹|
|A⁻¹| = |A|⁻¹ = 1/|A|
What happens to the determinant of a 2x2 matrix when a scalar multiplies one row or one column?
What is the determinant of a scalar multiplied by a marix?
|k Aₙₓₙ|
kⁿ |A|
What is the result of multiplying the determinant of a matrix A by a scalar k? State the formula.
|k Aₙₓₙ|= kⁿ |A|
|adj Aₙ|
|A|ⁿ⁻¹
adj(adj Aₙ)
|A|ⁿ⁻² A
|adj(adj Aₙ)|
Determinant of lower triangular matrix or upper triangular matrix
Product of its principal diagonal elements
Determinant of diagonal matrix
Product of its principal diagonal elements
Determinant of skew symmteric matrix
The determinant of skew matrix of odd order is zero and of even order is a perfect square
Determinant of idempotent matrix
0 or 1
Determinant of involuntary matrix
± 1
Determinant of orthogonal matrix
± 1
Determinant of unitary matrix
± 1
Relationship between linear dependency/linear independency and determinant
If any two row or column are linear dependent then the determinant of matrix is zero
Determinant in Rᵢ <—> Rⱼ elementary transformation
|B| = - |A|
Determinant in Rᵢ —> kRⱼ elementary transformation
|B| = k|A|
Determinant in Rᵢ —> Rᵢ + cRⱼ elementary transformation
|B| = |A|
Determinant in Cᵢ <—> Cⱼ elementary transformation
|B| = - |A|
Determinant in Cᵢ —> CRⱼ elementary transformation
|B| = k|A|
Determinant in Cᵢ —> Cᵢ + CRⱼ elementary transformation
|B| = |A|
