General Equation for the Determinant of a 2x2 Matrix
ad - bc
When is a 2x2 matrix invertible?
when ad - bc ≠ 0
Inductive Definition of the Determinant
(j=1->n) Σ (-1)^(1+j) * a1j * det(A1j)
where Aij is the (i,j)th minor of A, obtained by deleting the ith row and jth column of A
Advantages of the Inductive Definition of the Determinant
- easy to remember
- easy to programme in recursive computer languages
- easy to compute if n is small (<4)
Disadvantages of the Inductive Definition of the Determinant
-major disadvantage is that it is extremely inefficient and involves potentially n! multiplications
Main Diagonal
Definition
- let A be an nxn matrix with (i,j)th entry aij
- the diagonal with entries aii is the main diagonal of A
Upper Triangular
Definition
- let A be an nxn matrix with (i,j)th entry aij
- A is upper triangular if all the entries below the main diagonal are zero, i.e. aij=0 for i>j
Lower Triangular
Definition
- let A be an nxn matrix with (i,j)th entry aij
- A is lower triangular if all the entries above the main diagonal are zero, i.e. if aij=0 for i
Diagonal
Definition
- let A be an nxn matrix with (i,j)th entry aij
- A is diagonal if all the entries not on the main diagonal are zero i.e. aij=0 for i≠j
Properties of Determinants
i) if B is obtained from A by swapping two rows in A then det(B) = -det(A)
ii) if B is obtained from A by adding a multiple of a row to another in A then
det(B) = det(A)
iii) if B is obtained from A by multiplying a row from A by λ then det(B) = λdet(A)
Determinant of a Triangular Matrix
- if A=(aij) is a triangular matrix then det(A) = a11a22a33…annn
- i.e. the determinant of a triangular matrix is the product of the entries on its main diagonal
Determinants
Scalar Multiplication
if A is an nxn matrix and λ is a real number
then:
det(λA) = λ^n det(A)
Determinants and Invertibility
if det(A) ≠ 0, then A is invertible
if A is invertible then
det(A) ≠ 0
Determinants of Elementary Matrices
det(EA) = det(E) det(A)
det(AB)
det(AB) = det(A) det(B)
What is the detmerminant of the inverse?
det(A^-1) = 1/det(A)
What is the determinant of the transpose?
det(A^T) = det(A)