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 33 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
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