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
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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: Slide3Slide4
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.