### CHAPTER 3

```Chapter 3
Determinants
3.1 The Determinant of a Matrix


Note:
 a 11

 a 21
a 12 
a 11
  a
a 22 
21
a 12
a 22
The determinant of a matrix can be positive, zero, or negative.
3-1

Notes: Sign pattern for cofactors








 

































3-2
3-3
3-4
• The determinant of a matrix of order 3:
Subtract these three products.
 a 11

A  a 21

 a 31
a 12
a 22
a 32
a 13 

a 23

a 33 
a 11
a 12
a 13
a 11
a 12
a 21
a 22
a 23
a 21
a 22
a 31
a 32
a 33
a 31
a 32
 det( A )  | A | a 11 a 22 a 33  a 12 a 23 a 31  a 13 a 21 a 32  a 31 a 22 a 13
 a 32 a 23 a 11  a 33 a 21 a 12
3-5
• Upper triangular matrix:
All the entries below the main diagonal are zeros.

Lower triangular matrix:
All the entries above the main diagonal are zeros.

Diagonal matrix:
All the entries above and below the main diagonal
are zeros.

Ex:
 a 11 a 12 a 13 
 0 a 22 a 23 
 0

0
a
33 

0 
 a 11 0
 a 21 a 22 0 
a

a
a
32
33 
 31
upper triangular lower triangular
0 
 a 11 0
 0 a 22 0 
 0

0
a
33 

diagonal
3-6
• A row-echelon form of a square matrix is always upper triangular.
3-7
3.2 Evaluation of a Determinant Using Elementary
Operations
3-8
3-9
• Determinants and Elementary Column Operations: The
elementary row operations can be replaced by the column
operations and two matrices are called column-equivalent if one
can be obtained form the other by elementary column operations.
3-10
3-11
3.3 Properties of Determinants
• Notes:
(1) det( A  B )  det( A )  det( B )
(2)
a 11
a 12
a 13
a 22  b 22
a 22  b 22
a 23  b 23
a 31
a 32
a 33
a 11
a 12
a 13
a 11
a 12
a 13
 a 21
a 22
a 23  b 21
b 22
b 23
a 31
a 32
a 33
a 32
a 33
a 31
3-12
3-13
3-14
3-15
3-16
3-17
3.4 Applications of Determinants

Matrix of cofactors of A:
 C 11

C 21

C ij  
 

 C n1

C 12

C 22


Cn2

C 1n 

C 2n

 

C nn 
C ij  (  1)
i j
M ij
adj ( A )  C ij 
T
 C 11

C 12


 

C 1n
C 21

C 22


C 2n

C n1 

Cn2

 

C nn 
3-18
3-19
3-20
3-21
3-22
3-23
3-24
```