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Determinants 林育崧 蘇育劭 Determinants 林育崧 蘇育劭

Determinants 林育崧 蘇育劭 - PowerPoint Presentation

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Uploaded On 2019-03-12

Determinants 林育崧 蘇育劭 - PPT Presentation

The Properties of the Determinant 1 The determinant of the n by n identity matrix is 1 2 The determinant changes sign when two rows are exchanged sign reversal 3 The determinant is a linear function of each row separately ID: 755452

determinant det determinants formula det determinant formula determinants properties rows row simple product matrix sign big rule pivot singular

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Presentation Transcript

Slide1

Determinants

林育崧

蘇育劭Slide2

The Properties of the Determinant

(1) The determinant of the n by n identity matrix is 1.

(2) The determinant changes sign when two rows are exchanged (sign reversal)

:(3) The determinant is a linear function of each row separately:  Slide3
Slide4

The Properties of the Determinant

(4) If two rows of A are equal , then

det

(A) = 0.(5) Subtracting a multiple of one row from another row leaves det(A) unchanged.Slide5

The Properties of the Determinant

(6) A matrix with a row of zeros has

det

(A) = 0. By (4)(5)(7) If A is triangular then det A = product of diagonal entries.Slide6

The Properties of the Determinant

(8) If A is singular then

det

A = 0.If A is invertible then det A != 0. If PA = LU then det P

det A = det L det USlide7

The Properties of the Determinant

(9) |AB| =|A||B|

(10) |A| = |A

T| L,U,P has the same determinant asLT UT PT

Important comment on columns : Every rule for rows can apply to the columns(Since(10) )Slide8

Three ways to compute determinants

(1) Pivot Formula (Multiply the n pivots)

(2) Big Formula (Add up n! terms)

(3) Cofactor Formula (Combine n smaller determinants)Slide9

Pivot Formula

PA = LU

det P det A = det L det U det A = Slide10

Big Formula

There are possibly (3*2*1)=3! terms that may be non-zero.Slide11

Big Formula

Matrices of n by n

n!

simple determinants that need to be summed up.Each simple determinant chooses one entry from every row and columnThe value of a simple determinant is the product times +1 or -1The complete determinant of A is the sum of these n! simple determinants.Slide12

Cofactor FormulaSlide13

Cramer’s Rule

Solving Ax = b

n x n system , need

to evaluate (n + 1) determinants.Slide14

Formula for

A

-1

(using Cramer’s rule)Slide15

Direct proof of

A

-1

= cT /det AA A

-1 = I → A cT

= det (A) I

How to explain the zores off the diagonal? *two equal rows

Slide16

Applications of

determiants

Area is the absolute value of the determinant

Area has the same properties as the determinantSlide17

(1) When , A = 1.

(2) When rows are exchanged , the determinant reverses sign.(The absolute value stays the same)

(3)

LinearitySlide18

Cross Product

Scalar

Triple Product (b×c) .a = = The volume of the

parallelpiped If (b×c

) .a = 0 (1) a,b,c

lie in the same plain (2) a,b,c are dependent. The matrix is singular.

(3) zero volume.