Determinant untuk kuliahteknik sipil atau umum

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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 Aby replacing the
brackets by vertical straight lines. For example,






30
12
A 30
12
)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.





dc
ba

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=[a
ij] , the cofactorof the element
a
ijis a scalar obtained by multiplying together the term (-1)
i+j
and the minor obtained from Aby removing the ith row and the
jth column.
In other words, the cofactor C
ijis given by C
ij= (1)
i+j
M
ij.
For example,










333231
232221
131211
aaa
aaa
aaa
A 3332
1312
21
aa
aa
M 3331
1311
22
aa
aa
M 2121
12
21 )1( MMC 
 2222
22
22 )1( MMC 


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 Ais given byininiiii
n
j
ijij CaCaCaCaAA  


2211
1
)det( njnjjjjj
n
i
ijij CaCaCaCaAA  


2211
1
)det(
ith row
expansion
jth column
expansion

5. Example 1:
We can compute the determinant1 2 3
4 5 6
7 8 9
T
by expanding along the first row,  
1 1 1 2 1 35 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 24 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)
23
1 5 0
15
2 1 1 1
31
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 Bis obtained from the matrix Aby
multiplying every element in one row of Aby the scalar λ, then
|B|=λ|A|.
Property 5:For an n n matrix Aand any scalar λ, det(λA)=
λ
n
det(A).

7. 5.3 Properties of determinants
Property 6:If a matrix Bis obtained from a matrix Aby adding to
one row of A, a scalar times another row of A, then |A|=|B|.
Property 7:det(A) = det(A
T
).
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.
Examplein the next slide

9. •Find the determinant of310
1032
221



 













310
221
1032
A 310
221
1032


 310
1470
221




(2)
Factor 7out of the 2nd row310
210
221
7




(1
)100
210
221
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
Ais an n n matrix A
c
obtained from Aby replacing each
element of Aby its cofactor.
Definition 2:The adjugateof an n n matrix Ais the transpose
of the cofactor matrix of A: A
a
= (A
c
)
T

11. •Find the adjugate of
Solution:
The cofactor matrix of A:













201
120
231
A 
































20
31
10
21
12
23
01
31
21
21
20
23
01
20
21
10
20
12 










217
306
214 










232
101
764
a
A
Example of finding adjugate

12. Inversion using determinants
Theorem 2:A A
a
= A
a
A = |A| I .
If |A| ≠ 0 then from Theorem 2, 0
1
1




















AifA
A
A
IA
A
A
A
A
A
a
aa
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 ,







dc
ba
A 










ac
bd
bcad
A
A
A
a 11
1

13. Use the adjugate of to find A
-1













201
120
231
A 










232
101
764
A
a 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
232
101
764
3
1
A
1
a
AA
Inversion using determinants: example

14. •If a system of nlinear equations in nvariables Ax=bhas
a coefficient matrix with a nonzerodeterminant |A|,
then the solution of the system is given by
where A
iis a matrix obtained from Aby replacing the ith
column of Aby 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
  







3333232131
2323222121
1313212111
bxaxaxa
bxaxaxa
bxaxaxa 333231
232221
131211
33231
22221
11211
3
3
aaa
aaa
aaa
baa
baa
baa
A
A
x 
5.6 Cramer’s rule

15. •Use Cramer’s Rule to solve the system of linear equation.2443
02
132



zyx
zx
zyx 10
443
102
321



A 10
442
100
321
1 


A
A
x 5
4
10
8
10
)1)(1)(4()2)(1)(2(


 5
8
,
2
3
 zy
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