matakuliah aljabar

1. Determinan

2. 5.1 Introduction
Every square matrix has associated with it a scalar called its
determinant.
Given a matrix A, we use det(A) or |A| to designate its determinant.
We can also designate the determinant of matrix A by replacing the
brackets by vertical straight lines. For example,







3
0
1
2
A
3
0
1
2
)
det( 
A
Definition 1: The determinant of a 11 matrix [a] is the scalar a.
Definition 2: The determinant of a 22 matrix is the scalar ad-bc.
For higher order matrices, we will use a recursive procedure to compute
determinants.






d
c
b
a

3. 5.2 Expansion by Cofactors
Definition 1: Given a matrix A, a minor is the determinant of any
square submatrix of A.
Definition 2: Given a matrix A=[aij] , the cofactor of the element
aij is a scalar obtained by multiplying together the term (-1)i+j
and the minor obtained from A by removing the ith row and the
jth column.
In other words, the cofactor Cij is given by Cij = (1)i+j
Mij.
For example,











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 
33
31
13
11
22
a
a
a
a
M 
21
21
1
2
21 )
1
( M
M
C 



 
22
22
2
2
22 )
1
( M
M
C 


 

4. 5.2 Expansion by Cofactors
To find the determinant of a matrix A of arbitrary order,
Pick any one row or any one column of the matrix;
For each element in the row or column chosen, find its cofactor;
Multiply each element in the row or column chosen by its
cofactor and sum the results. This sum is the determinant of the
matrix.
In other words, the determinant of A is given by
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

5. Example 1:
We can compute the determinant
1 2 3
4 5 6
7 8 9

T
by expanding along the first row,
     
1 1 1 2 1 3
5 6 4 6 4 5
1 2 3
8 9 7 9 7 8
  
        
T 3 12 9 0
    
Or expand down the second column:
     
1 2 2 2 3 2
4 6 1 3 1 3
2 5 8
7 9 7 9 4 6
  
        
T 12 60 48 0
   
Example 2: (using a row or column with many zeroes)
 
2 3
1 5 0
1 5
2 1 1 1
3 1
3 1 0

  


16


6. 5.3 Properties of determinants
Property 1: If one row of a matrix consists entirely of zeros, then
the determinant is zero.
Property 2: If two rows of a matrix are interchanged, the
determinant changes sign.
Property 3: If two rows of a matrix are identical, the determinant
is zero.
Property 4: If the matrix B is obtained from the matrix A by
multiplying every element in one row of A by the scalar λ, then
|B|= λ|A|.
Property 5: For an n  n matrix A and any scalar λ, det(λA)=
λndet(A).

7. 5.3 Properties of determinants
Property 6: If a matrix B is obtained from a matrix A by adding to
one row of A, a scalar times another row of A, then |A|=|B|.
Property 7: det(A) = det(AT).
Property 8: The determinant of a triangular matrix, either upper or
lower, is the product of the elements on the main diagonal.
Property 9: If A and B are of the same order, then
det(AB)=det(A) det(B).

8. 5.4 Pivotal condensation
Properties 2, 4, 6 of the previous section describe the effects on
the determinant when applying row operations.
These properties comprise part of an efficient algorithm for
computing determinants, technique known as pivotal
condensation.
-A given matrix is transformed into row-reduced form using
elementary row operations
-A record is kept of the changes to the determinant as a result of
properties 2, 4, 6.
-Once the transformation is complete, the row-reduced matrix is in
upper triangular form, and its determinant is easily found by
property 8.
Example in the next slide

9. • Find the determinant of
3
1
0
10
3
2
2
2
1



















3
1
0
2
2
1
10
3
2
A
3
1
0
2
2
1
10
3
2



3
1
0
14
7
0
2
2
1





(2)
Factor 7 out of the 2nd row
3
1
0
2
1
0
2
2
1
7




(1
) 1
0
0
2
1
0
2
2
1
7



 7
)
1
)(
1
)(
1
(
7 



5.4 Pivotal condensation

10. 5.5 Inversion
Theorem 1: A square matrix has an inverse if and only if its
determinant is not zero.
Below we develop a method to calculate the inverse of
nonsingular matrices using determinants.
Definition 1: The cofactor matrix associated with an n  n matrix
A is an n  n matrix Ac obtained from A by replacing each
element of A by its cofactor.
Definition 2: The adjugate of an n  n matrix A is the transpose
of the cofactor matrix of A: Aa = (Ac)T

11. • Find the adjugate of
Solution:
The cofactor matrix of A:














2
0
1
1
2
0
2
3
1
A


































2
0
3
1
1
0
2
1
1
2
2
3
0
1
3
1
2
1
2
1
2
0
2
3
0
1
2
0
2
1
1
0
2
0
1
2











2
1
7
3
0
6
2
1
4











2
3
2
1
0
1
7
6
4
a
A
Example of finding adjugate

12. Inversion using determinants
Theorem 2: A Aa = Aa A = |A| I .
If |A| ≠ 0 then from Theorem 2,
0
1
1





















A
if
A
A
A
I
A
A
A
A
A
A
a
a
a
That is, if |A| ≠ 0, then A-1 may be obtained by dividing the
adjugate of A by the determinant of A.
For example, if
then
,







d
c
b
a
A












a
c
b
d
bc
ad
A
A
A a 1
1
1

13. Use the adjugate of to find A-1














2
0
1
1
2
0
2
3
1
A











2
3
2
1
0
1
7
6
4
Aa
3
)
2
)(
2
)(
1
(
)
1
)(
1
)(
3
(
)
2
)(
2
)(
1
( 







A
























3
2
3
2
3
1
3
1
3
7
3
4
1
1
0
2
2
3
2
1
0
1
7
6
4
3
1
A
1 a
A
A
Inversion using determinants: example

14. • If a system of n linear equations in n variables Ax=b has
a coefficient matrix with a nonzero determinant |A|,
then the solution of the system is given by
where Ai is a matrix obtained from A by replacing the ith
column of A by the vector b.
• Example:
,
)
det(
)
det(
,
,
)
det(
)
det(
,
)
det(
)
det( 2
2
1
1
A
A
x
A
A
x
A
A
x n
n 

 














3
3
33
2
32
1
31
2
3
23
2
22
1
21
1
3
13
2
12
1
11
b
x
a
x
a
x
a
b
x
a
x
a
x
a
b
x
a
x
a
x
a
33
32
31
23
22
21
13
12
11
3
32
31
2
22
21
1
12
11
3
3
a
a
a
a
a
a
a
a
a
b
a
a
b
a
a
b
a
a
A
A
x 

5.6 Cramer’s rule

15. • Use Cramer’s Rule to solve the system of linear equation.
2
4
4
3
0
2
1
3
2









z
y
x
z
x
z
y
x
10
4
4
3
1
0
2
3
2
1





A
10
4
4
2
1
0
0
3
2
1
1 



A
A
x
5
4
10
8
10
)
1
)(
1
)(
4
(
)
2
)(
1
)(
2
(





5
8
,
2
3



 z
y
5.6 Cramer’s rule: example

16. Menghitung Luas Segitiga
Segitiga dengan A(x1,y1), B(x2,y2) dan C(x3,y3)
https://akar-
kuadrat.blogspot.com/2011/01/menghitung-luas-
segitiga-dengan-titik.html

17. 1. https://www.powershow.com/view4/6d6019-
ZmE1Z/Chap_3_Determinants_powerpoint_ppt_pr
esentation
2. http://aees.gov.in/htmldocs/downloads/e-
content_06_04_20/PPT%20of%20Module%204%
20(Class%2012%20Maths%20%20chapter%204
%20Determinants)%20%20by%20Mini%20Maria
%20Tomy.pdf
3. https://akar-
kuadrat.blogspot.com/2011/01/menghitung-luas-
segitiga-dengan-titik.html
Referensi