This document discusses determinants and their properties. It begins by defining the determinant of a matrix as a real number associated with the matrix. It then covers evaluating determinants using cofactor expansion and elementary row/column operations. The document provides examples of calculating determinants for various matrices through expanding along different rows and columns based on which entries are zeros. It also states that the determinant of an upper or lower triangular matrix is the product of its diagonal entries.

1. Chap. 2
Determinants
Dr. Litan Kumar Saha
Associate Professor
Department of Applied Mathematics
University of Dhaka

2. Chap. 2
Determinants
2.1 The Determinants of a Matrix by cofactor expansion
2.2 Evaluation of a Determinant Using Elementary Operations
2.3 Properties of Determinants
Dr. Litan Kumar Saha, MAT125

3. 3-3
 Every square matrix can be associated with a real number
called its determinant.
 Definition: The determinant of the matrix
is given by
 Example 1:
2.1 The Determinant of a Matrix
+

?
4
2
3
0
?
2
4
1
2
?
2
1
3
2




?
]
2
[ 


 A
A
12
21
22
11
22
21
12
11
)
det( a
a
a
a
a
a
a
a
A
A 










22
21
12
11
a
a
a
a
A
Dr. Litan Kumar Saha, MAT125

4. 3-5
Minors and Cofactors of a Matrix
 If A is a square matrix, then the minor Mij of the element aij
is the determinant of the matrix obtained by deleting the ith
row and jth column of A.
The cofactor Cij is given by Cij = (1)i+j
Mij.

 Sign pattern for cofactors:











33
32
31
23
22
21
13
12
11
a
a
a
a
a
a
a
a
a
A 33
32
13
12
21
a
a
a
a
M  21
21
1
2
21 )
1
( M
M
C 



 
33
31
13
11
22
a
a
a
a
M  22
22
2
2
22 )
1
( M
M
C 


 
4
4
3
3

 














































Dr. Litan Kumar Saha, MAT125

5. 3-6
Theorem 2.1
Expansion by Cofactors
 Let A be a square matrix of order n. Then the determinant of
A is given by
 For any 33 matrix:
in
in
i
i
i
i
n
j
ij
ij C
a
C
a
C
a
C
a
A
A 




 


2
2
1
1
1
)
det(
nj
nj
j
j
j
j
n
i
ij
ij C
a
C
a
C
a
C
a
A
A 




 


2
2
1
1
1
)
det(
ith row
expansion
jth column
expansion











33
32
31
23
22
21
13
12
11
a
a
a
a
a
a
a
a
a
A
12
21
33
11
23
32
13
22
31
32
21
13
31
23
12
33
22
11 a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
A 





33
32
31
23
22
21
13
12
11
a
a
a
a
a
a
a
a
a
32
31
22
21
12
11
a
a
a
a
a
a
+

+ +
 
Dr. Litan Kumar Saha, MAT125

6. 3-7
Examples 2 & 3
 Find all the minors and cofactors of A, and then find the
determinant of A.












1
0
4
2
1
3
1
2
0
A
Dr. Litan Kumar Saha, MAT125

7. 3-9
Example 4
 Find the determinant of
Sol: Expansion by which row
or which column?
 the 3rd column: three of the entires are zeros
















2
0
4
3
3
0
2
0
2
0
1
1
0
3
2
1
A
2
4
3
3
2
0
2
1
1
2
4
3
3
2
0
2
1
1
)
1
( 3
1
13






 
C
13
)
7
)(
3
(
)
4
)(
2
(
0
4
3
1
1
)
1
)(
3
(
2
3
2
1
)
1
)(
2
(
2
4
2
1
)
1
)(
0
( 3
2
2
2
1
2















 


13
12
12
9
4
)
2
)(
1
(
0
)
1
)(
3
(
4
)
2
)(
2
(
3
)
3
)(
3
(
1
)
2
)(
4
(
0
)
2
)(
2
)(
1
(















39
)
13
(
3
13
13 

 C
a
A
Dr. Litan Kumar Saha, MAT125

8. 3-10
Example 5
 Find the determinant of
Sol:













1
4
4
2
1
3
1
2
0
A
1
4
4
2
1
3
1
2
0


4
4
1
3
2
0


+(0) +(16)
(4)
+(12)
(6)
(0)
2
6
0
)
4
(
)
12
(
16
0 








A
1
4
4
2
1
3
1
2
0


Dr. Litan Kumar Saha, MAT125

9. 3-11
Triangular Matrices
Upper triangular Matrix Lower triangular Matrix
 Theorem 3.2: If A is a triangular matrix of order n, then its
determinant is the product of the entires on the main
diagonal. That is,
















nn
n
n
n
a
a
a
a
a
a
a
a
a
a









0
0
0
0
0
0
3
33
2
23
22
1
13
12
11
















nn
n
n
n a
a
a
a
a
a
a
a
a
a









3
2
1
33
32
31
22
21
11
0
0
0
0
0
0
nn
a
a
a
a
A
A 
33
22
11
)
det( 

Dr. Litan Kumar Saha, MAT125

10. 3-12
Example
?
3
0
0
2
1
0
1
3
2



3
0
2
1
)
1
(
2 1
1 
 
6
)]
2
(
0
)
3
)(
1
[(
2 




?
2
0
0
0
0
0
4
0
0
0
0
0
2
0
0
0
0
0
3
0
0
0
0
0
1
?
3
3
5
1
0
1
6
5
0
0
2
4
0
0
0
2






12
)
3
)(
1
)(
2
(
2




48
)
2
)(
4
)(
2
)(
3
)(
1
(




Dr. Litan Kumar Saha, MAT125