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Nhân một dòng (hoặc cột) với một số k matrix thu được có định thức gấp k lần det của matrix cũ.. Use the fact: if X is an eigenvector of a matrix A corresponding an eigenvalue k, th[r]
(1)Chapter 3
(2)OUR GOAL
o How to find the determinant of a square
matrix?
(3)Determinant of a square matrix
• Determinant of an nxn matrix A are
denoted by det(A) or |A|
• For x matrices:
Or
(4)• 3 x 3matrices:
det(A) =
+a.det - b.det + c.det
= aei – afh – (bdi – bgf) + cdh – cge
(5)Example
(6)The determinant of 3x3 matrix (only)
+ + +
-
-3 2
-2
2
1 col col3 co
a b c a b
d e f d
col col l
e
g h i g h
(7)Definition
If A is an mxm matrix then the determinant of A is defined by
• detA=ai1ci1(A)+ai2ci2(A)+…+aimcim(A)
• or detA= a1jc1j(A)+a2jc2j(A)+…+amjcmj(A)
1 2 1 5
6 4 0
0 6 4 0
1 7 1 0 68
0 7 1 0
1 8 2
0 1 8 2
11 12 13
(1,1) (1,2) (1,3)
det
cofactor c
a a a
cofactor
ofactor
e f d
a b c
a f d e
A d e f
h i g i g h
g h
c
i
(8)The determinant of triangular matrices
(9)Examples
Find det(A), det(B), det(AB), det(A+B)
•
det(A.B) = det(A).det(B)
(10)Examples
• Find det(A), det(3A), det(A2) if
•
o det(cA) = cndet(A)
(11)Properties
For all nxn matrices A, B:
o det(A.B) = det(A).det(B) o det(kA) = kndet(A)
o det(AT) = det(A)
o det(A-1) = 1/det(A)
(12)The determinant of triangular matrices
(13)Examples
o Find the determinants
// from A, interchange row and row
// from A, -2.(row 1)
(14)Examples
o Find the determinants And
The second matrix is obtained from the first matrix by (2*row1 + row3), they have the same
(15)Determinants and elementary operators
1 Đổi chỗ dòng (hoặc cột) cho nhau, matrix thu matrix ban đầu có định thức trái dấu // ri rj
2 Nhân dòng (hoặc cột) với số k matrix thu có định thức gấp k lần det matrix cũ //kri
3 Nếu nhân c vào dòng ri cộng vào dòng rj (hoặc thực cột) định thức
(16)Examples
1 Do yourself: Find
1
- 3
1
4 3
0 1
2 9
3 2 2 2 1
3 4 0
1 1 1
0 8
2
4
2 2.7
0 0 7 0 1
0
1
0 24
2
7 0 24
2 r r
r r r r
r r r rr r
3
24
7
1
0
2.7 2.7.1 1 23
0 1
0 0 23
(17)Next
• det(A) and existence of A-1
o A is invertible det(A)
(18)(i,j)-cofactor
• (i,j)-cofactor of a matrix [aij]
is defined by
cij = (-1)i+jdet(Aij),
where Aij is the matrix obtained from A by
deleting row ith and column jth
For example, given A = Then, c23 = (-1)2+3det
= -1.(-1) =
•
row column
(19)How to find A-1?
• An nxn matrix A is invertible if and only if
det(A) 0
Furthermore, A-1
(20)Adjugate matrix
• The adjugate matrix of A is the matrix
• For example,
21 11 12 2 2 n n n n nn c c c c adjA c c c c c
3 2
3 1
6
adjA
1 1
11 12
11 12 13 21 22 23
1
3 1
1 We have c 3, c 3,
0
2
c 3, c 3, c
c 2, c 1, c 4,
A
31 32 33
(21)Theorem of Adjugate Formula
If A is any square matrix, then
• A(adjA)=(detA)I
• In particular, if detA≠0 then A is invertible and
• For example,
Note that […]
1
det A
A adjA
1
1 2
0 det and adjA= 1
0 0
2 1 / /
1
0 1 / /
2
0 0
(22)Diagonal matrices
• An nxn matrix is called diagonal matrix if all its
entries off the main diagonal are zeros
• For example
0 0 0
3
2 3, 2,1,4
1
0 0 0
diag 2
1
0
0
, , , n
(23)Diagonalization
• Diagonalizing a matrix A is to find an invertible matrix P such
that P-1AP is a diagonal matrix P-1AP=diag(
1, 2,…, n)
For example,
o Given a matrix A,
o Find a matrix P,
o Compute P-1AP, •
1
0
P AP
(24)Nếu t≠0 X=(t,t) gọi véc tơ riêng ứng với x=-2
Nếu t≠0 X=(-4t,t) gọi véc tơ riêng (eigenvectors) ứng với giá trị riêng x=3
Các giá trị riêng (eigenvalues) A
Đa thức đặc trưng (characteristic polynomial)
How to find P?
Relationship between eigenvalues and eigenvectors
: eigenvalue (a number)
X: -eigenvector (remember: vector X≠0) (I-A)X=0 AX=X
2
Find c =det :
1
c
4
x=3: solve the system
4
4
4
4
x=-2: solve the system
0
A
A
x
x xI A x x x x
x
x x x
x y
I A X
x y
y t x
X t
x t y
x y
I A X
x y y t x 1
4
Choose P=
1
x
X t
t y
P AP
4 4 4
, , , , are allowed In case P= ,
1 1 1 1 1
P P P P P AP
(25)Find the eigenvalues ang eigenvectors and then diagonalize the matrix
The characteristic polynomial of A is
Example 1 1 0 P P AP
0 : Solve the system 0I-A X=0
1 1 1
2 0 0
3: Solve the system 3I-A X=0
2 1
2 0 0
x X t x X t 1 2
A
1
( ) ( 1)( 2) ( 3) 0
2
0,3 are eigenvalues
A x
c x x x x x x x
(26)Use the fact: if x1, x2,…, xm are eigenvalues of an nxn matrix , then det(A) = x1.x2…xm
First, det(A) =
We know that det(A) = the product of eigenvalues
(27)Use the fact: if X is an eigenvector of a matrix A corresponding an eigenvalue k, then
(28)Theorem
A is diagonalizable iff every eigenvalue of multiplicity m yields
exactly m basic eigenvectors, that is, iff the general solution of the
system (I-A)X=0 has exactly m parameters
For example,
When is A diagonalizable?
2
0
is not diagonalizable
1
In fact, c det 2 1
1
1 1
1 (multiplicity 2): solve the system
1 0 0
1
one parameter not diag
1
A
A x
x xI A x x x x x x
x
x I A X
(29)When is A diagonalizable?
2
0 1
1 is not diagonalizable 0
1
In fact, c det 1 2
2
1 1 1
1 (multiplicity 2): solve the system 1 1 0 0
2 0
A
A x
x xI A x x x x x x x
x
x I A X
one parameter not diagonalizable
(30)When is A diagonalizable?
2
0 1
1 is not diagonalizable 0
1
In fact, c det 1 2
2
1 1 1
1 (multiplicity 2): solve the system 1 1 0 0
2 0
A
A x
x xI A x x x x x x x
x
x I A X
one parameter not diagonalizable
(31)SUMMARY
o Determinants of nxn matrices
o Properties:
o det(AB) = det(A)det(B)
o det(cA) = cndet(A)
o det(AT) = det(A)
o det(A-1) = 1/det(A)
o Determinants and elementary operators
o Determinants and inverse of a matrix
o An nxn matrix has an inverse if and only if det(A)
o A-1 = adj(A)/detA
o Diagonalization
o Characteristic polynomial
o Eigenvalues
(32)