mae 101 fpt fall 2019 nguyenvantien0405

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Chapter 3

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OUR GOAL

o How to find the determinant of a square

matrix?

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Determinant of a square matrix

• Determinant of an nxn matrix A are

denoted by det(A) or |A|.

• For 2 x 2 matrices:

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• 3 x 3matrices:

det(A) =

+a.det - b.det + c.det

= aei – afh – (bdi – bgf) + cdh – cge

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Example

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The

determinant

of 3x3 matrix (only)

+

+ +

-3
2
2

-2
0
1

2 2

1 col col3 1 co

a b c a b

d e f d

col col l

e

g h i g h

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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

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The determinant of triangular

matrices

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Examples

Find det(A), det(B), det(AB), det(A+B)

•

det(A.B) = det(A).det(B)

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Examples

• Find det(A), det(3A), det(A2) if

•

o det(cA) = cndet(A)

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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)

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The determinant of triangular

matrices

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Examples

o Find the determinants

// from A, interchange row 1 and row 2

// from A, -2.(row 1)

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Examples

o Find the determinants
And

The second matrix is obtained from the first matrix
by (2*row1 + row3), they have the same

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Determinants and elementary

operators

1. Đổi chỗ 2 dòng (hoặc 2 cột) cho nhau,

matrix thu được và matrix ban đầu có
định thức trái dấu nhau. // ri  rj

2. Nhân một dòng (hoặc cột) với một số k 
matrix thu được có định thức gấp k lần
det của matrix cũ. //kri

3. Nếu nhân c vào dòng ri rồi cộng vào dòng
rj (hoặc thực hiện trên cột)  định thức

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Examples

1 2 1 5
0 6 4 3
Do yourself: Find

1 3 4 6
1 2 4 5

- 
2 3
2
1 3

1 2 1 4

4
2 3

2
7
3
3

0 2 1 9 1 1 2 3

2 2 4 6 0 2 1 9 0 2 1 9 0 2 1 9

3 2 2 1 3 2 2 1 3 2 2 1 0 1 4 8

3 4 2 0 3 4 2 0 3 4 2 0 0 7 4 9

1 1 2 3 1 1 2 3 1 1 2 3

0 1 4 8 0 1 4 8 0 1

2 2 4 6

4 8

2 2.7

0 2 1 9 0 0 7 7 0 0 1 1

0 7 4 9 0

1 1 2 3

0 24 4

2 2

7 0 0 24 4

2
 
 



 
   
  
    
   
  
     
 
 
  

  
 
  
r r

r r r r

r r r rr r

   

3 4

1 1 2 3

0 1 4 8

2.7 2.7.1. 1 .1. 23

0 0 1 1

0 0 0 23

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Next

• det(A) and existence of A-1

o A is invertible  det(A)  0

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(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) = 1

•

row 2
column 3

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How to find A

-1

?

• An nxn matrix A is invertible if and only if

det(A)  0

Furthermore,
A-1

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Adjugate matrix

• The adjugate matrix of A is the matrix

• For example,

21
11
12 2
1
2
2
2
1
...
...
... ... ... ...
...
n n
n
n
nn
c
c
c
c
adjA
c
c
c c
c
 
 
 


 
 
 

3 2 2

3 1 1

6 4 5

adjA
 
 
  
 
  
 

 1 1  1 2

11 12

11 12 13
21 22 23

1 2 0

3 1 1 1

1 3 1 . We have c 1 3, c 1 3,

0 1 2 1

2 0 1

c 3, c 3, c 6

c 2, c 1, c 4,

A  


 
 
 
       
 
 
 
  
  

31 32 33

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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 1

det A

A adjA



1 1 2 2 1 3

0 2 1 det 2 and adjA= 0 1 1

0 0 1 0 0 2

2 1 3 1 1 / 2 3 / 2

0 1 1 0 1 / 2 1 / 2

0 0 2 0 0 1

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Diagonal matrices

• An nxn matrix is called diagonal matrix if all its

entries off the main diagonal are zeros

• For example





0 0 0
0 0 0

2
3, 2,1,4

1
4

0 0 0
0 0 0

diag

 
 
 




 








2
1 2

1 0 ... 0

0 ... 0
... ... ... ...

0 0 ...

, ,..., n

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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 3 0

0 2

P AP  

  

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Nếu t≠0 thì X=(t,t)
được gọi là véc tơ
riêng ứng với x=-2

Nếu t≠0 thì X=(-4t,t) được gọi
là véc tơ riêng (eigenvectors)
ứng với giá trị riêng x=3

Các giá trị riêng (eigenvalues) của 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
2 4

Find c =det : 2 1 4 6

1 1

c 0 3 2

4 0

x=3: solve the system 3 0

4 0

4 1

4 4 0

x=-2: solve the system 2 0

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
1
1

4 1 3 0

Choose P=

1 1 0 2

x

X t

t y

P AP

    
  
    

    

   

 
    
   
1

4 1 4 2 1 4 1 4 1 4 2 0

, , , ,...are allowed. In case P= ,

1 1 1 2 1 1 1 1 1 1 0 3

P   P    P   P      P AP  

    

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Find the eigenvalues ang eigenvectors and
then diagonalize the matrix

The characteristic polynomial of A is

Example

1
1 1
1 2
0 0
0 3

 
 
 

 
 
 
P
P AP
 
 

0 : Solve the system 0I-A X=0

1 1 0 1 1 0 1

2 2 0 0 0 0 1

3: Solve the system 3I-A X=0

2 1 0 2 1 0 1

2 1 0 0 0 0 2


         
  
     
   
   

       
  
     

  
   
x
X t
x
X t
1 1
2 2

A  

 

1 1

( ) ( 1)( 2) 2 ( 3) 0 0 3

2 2

0,3 are eigenvalues

 
           
 

A
x

c x x x x x x x

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Use the fact: if x1, x2,…, xm are eigenvalues of an nxn matrix , then
det(A) = x1.x2…xm

First, det(A) = 4

We know that det(A) = the product of eigenvalues

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Use the fact: if X is an eigenvector of a matrix A
corresponding an eigenvalue k, then

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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
2

0 1

is not diagonalizable.
1 2

In fact, c det 2 1 2 1 1 0 1

1 2

1 1 0 1 1 0

1 (multiplicity 2): solve the system 1 0

1 1 0 0 0 0

one parameter not diag

A

A
x

x xI A x x x x x x

x

x I A X

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When is A diagonalizable?

       

 

2
3

0 1 1

1 0 1 is not diagonalizable.
2 0 0

1 1

In fact, c det 1 1 3 2 1 2 0 1 2

2 0

1 1 1 0 1 1 1 0

1 (multiplicity 2): solve the system 1 0 1 1 1 0 0 0 0 0

2 0 1 0 0 2 1 0

A

A
x

x xI A x x x x x x x

x

x I A X

 
 
 
 
 
 
               

     
 
        
  
  


one parameter not diagonalizable



 

 

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When is A diagonalizable?

       

 

2
3

0 1 1

1 0 1 is not diagonalizable.
2 0 0

1 1

In fact, c det 1 1 3 2 1 2 0 1 2

2 0

1 1 1 0 1 1 1 0

1 (multiplicity 2): solve the system 1 0 1 1 1 0 0 0 0 0

2 0 1 0 0 2 1 0

A

A
x

x xI A x x x x x x x

x

x I A X

one parameter not diagonalizable



 

 

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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)  0

o A-1 = adj(A)/detA

o Diagonalization

o Characteristic polynomial

o Eigenvalues

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mae 101 fpt fall 2019 nguyenvantien0405

<b>Chapter 3</b>

<b>OUR GOAL</b>

<b>Determinant of a square matrix</b>

<b>Example </b>

<b>The </b>

<b>determinant</b>

<b> of 3x3 matrix (only)</b>

+

+ +

<b></b>

<b></b>

<b></b>

<b>Definition</b>

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











 

 

 

<b>The determinant of triangular </b>

<b>matrices</b>

<b>Examples </b>

<b>Examples</b>

<b>Properties </b>

<b>The determinant of triangular </b>

<b>matrices</b>

<b>Examples </b>

<b>Examples </b>

<b>Determinants and elementary </b>

<b>operators</b>

<b>Examples</b>

<b>Next </b>

<b>(i,j)-cofactor </b>

<b>How to find A</b>

<b>? </b>

<b>Adjugate matrix</b>

<b>Theorem of Adjugate Formula</b>

<b>Note that […]</b>

<b>Diagonal matrices</b>









<b>Diagonalization </b>

How to find P?

Example

When is A diagonalizable?

When is A diagonalizable?

When is A diagonalizable?

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