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Flashcards in Chapter 3 Deck (17)
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1
Q

General Equation for the Determinant of a 2x2 Matrix

A

ad - bc

2
Q

When is a 2x2 matrix invertible?

A

when ad - bc ≠ 0

3
Q

Inductive Definition of the Determinant

A

(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

4
Q

Advantages of the Inductive Definition of the Determinant

A
  • easy to remember
  • easy to programme in recursive computer languages
  • easy to compute if n is small (<4)
5
Q

Disadvantages of the Inductive Definition of the Determinant

A

-major disadvantage is that it is extremely inefficient and involves potentially n! multiplications

6
Q

Main Diagonal

Definition

A
  • let A be an nxn matrix with (i,j)th entry aij

- the diagonal with entries aii is the main diagonal of A

7
Q

Upper Triangular

Definition

A
  • 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

8
Q

Lower Triangular

Definition

A
  • 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

9
Q

Diagonal

Definition

A
  • 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

10
Q

Properties of Determinants

A

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)

11
Q

Determinant of a Triangular Matrix

A
  • if A=(aij) is a triangular matrix then det(A) = a11a22a33annn
  • i.e. the determinant of a triangular matrix is the product of the entries on its main diagonal
12
Q

Determinants

Scalar Multiplication

A

if A is an nxn matrix and λ is a real number
then:
det(λA) = λ^n det(A)

13
Q

Determinants and Invertibility

A

if det(A) ≠ 0, then A is invertible
if A is invertible then
det(A) ≠ 0

14
Q

Determinants of Elementary Matrices

A

det(EA) = det(E) det(A)

15
Q

det(AB)

A

det(AB) = det(A) det(B)

16
Q

What is the detmerminant of the inverse?

A

det(A^-1) = 1/det(A)

17
Q

What is the determinant of the transpose?

A

det(A^T) = det(A)