1. Aljabar Linear Elementer
Determinan
Aljabar Linear Elementer

2. Aljabar Linear-UIN Sunan Gunung Djati
Introduction
Sub chapter :
Find Determinant a matrix using OBE
Find Determinant a matrix Cofactor Expansion
Beberapa Aplikasi Determinan
Solution of system of linear equation
Optimization
Economi Model
etc.
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Aljabar Linear-UIN Sunan Gunung Djati

3. Aljabar Linear-UIN Sunan Gunung Djati
Introduction
Let A be a square matrix nxn,
Determinant of A can be found by
OBE
Cofactor Expansion
Notation of determinant of A :
Det(A) or |A|
Det(A) is not equal to zero if and only if matrix A is invertible
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Aljabar Linear-UIN Sunan Gunung Djati

4. Find Determinant of a matrix using EROb. c.
That’s Easy…
Because determinan will be equal to multiplication of every diagonal elements
When matrix is not triangular matrix.
How ?
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Aljabar Linear-UIN Sunan Gunung Djati

5. Aljabar Linear-UIN Sunan Gunung Djati
We need ERO to get triangular matrix.
Solution :
Rectangular matrix ~ ERO ~ Triangular matrix
Influent of ERO relatively to determinant of a matrix :
Let matrix B be obtained from matrix A with one exchange of row, then Det (B) = - Det (A)
Example :
such that
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Aljabar Linear-UIN Sunan Gunung Djati

6. Aljabar Linear-UIN Sunan Gunung Djati
Let Matrix B be obtained from matrix A which
elements in a row are multiplied with k  0.
Then Det (B) = k Det (A)
Example :
and
then
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Aljabar Linear-UIN Sunan Gunung Djati

7. Aljabar Linear-UIN Sunan Gunung Djati
Let Matrix B be obtained from matrix A with
third ERO, then Det (B) = Det (A)
Example 3 :
Then
ERO is –2b1 + b2
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Aljabar Linear-UIN Sunan Gunung Djati

8. Aljabar Linear-UIN Sunan Gunung Djati
Contoh 3 :
Find determinant of matrix :
Answer :
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Aljabar Linear-UIN Sunan Gunung Djati

9. Aljabar Linear-UIN Sunan Gunung Djati
= 4
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Aljabar Linear-UIN Sunan Gunung Djati

10. Aljabar Linear-UIN Sunan Gunung Djati
Exercise
Find determinant of matrix using ERO :
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Aljabar Linear-UIN Sunan Gunung Djati

11. Aljabar Linear-UIN Sunan Gunung Djati
Find Determinant using Cofactor ekspansion
Let
Preliminary concepts needed :
Mij (Minor- ij ) is determinant of a matrix (n-1 x n-1)
which the elements are obtained from matrix A
without ith row and jth column of matrix matriks A.
e.g :
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Aljabar Linear-UIN Sunan Gunung Djati

12. Aljabar Linear-UIN Sunan Gunung Djati
Matrix Cij is called cofactor - ij (Cij) = (-1)i+j Mij
Example :
Then
= (– 1)3 .2
= – 2
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Aljabar Linear-UIN Sunan Gunung Djati

13. Aljabar Linear-UIN Sunan Gunung Djati
Generally, cofactor expansion :
Find det (A) using cofactor expansion along ith row
det (A) = ai1 Ci1 + ai2 Ci ain Cin
Find det (A) using cofactor expansion along jth column det (A) = a1j C1j + a2j C2j anj Cjn
Example 6 :
Find Det(A) using cofactor expansion :
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Aljabar Linear-UIN Sunan Gunung Djati

14. Aljabar Linear-UIN Sunan Gunung Djati
Answer :
We will find det (A) using cofactor expansion
along 3rd row
= a31 C31 + a32 C32 + a33 C33
= 0 – 2 + 6
= 4
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Aljabar Linear-UIN Sunan Gunung Djati

15. Aljabar Linear-UIN Sunan Gunung Djati
We will find det (A) using cofactor expansion
along 3rd column
= a13 C13 + a23 C23 + a33 C33
= 0 – 2 + 6
= 4
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Aljabar Linear-UIN Sunan Gunung Djati

16. Aljabar Linear-UIN Sunan Gunung Djati
Given An x n. Let Cij be cofactor aij, then Is called cofactor matrix A. Transpose matrix of cofactor matrx is called adjoin A, Denoted by adj(A).
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Aljabar Linear-UIN Sunan Gunung Djati

17. Aljabar Linear-UIN Sunan Gunung Djati
Let A be square matrix, some properties of
determinant as follow :
1. Let A be invertible then
A is invertible if and only if det (A)  0.
det (A) = det (At)
det (A) det (B) = det (AB)
Let A be invertible then
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Aljabar Linear-UIN Sunan Gunung Djati

18. Aljabar Linear-UIN Sunan Gunung Djati
Example : Let Find adjoin matrix of A, and the find A-1 Answer : By definition cofactor :
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Aljabar Linear-UIN Sunan Gunung Djati

19. Aljabar Linear-UIN Sunan Gunung Djati
Cofactor matrix of A : Then, Adjoin matrix of A : So, invers of matrix A :
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Aljabar Linear-UIN Sunan Gunung Djati

20. Aljabar Linear-UIN Sunan Gunung Djati
Find determinant matrix below using cofactor expansion and find the invers!
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Aljabar Linear-UIN Sunan Gunung Djati

21. Aljabar Linear-UIN Sunan Gunung Djati
Exercise of Chapter 2
Find determinant matrix below using ERO and cofactor expansion
2. Let :
and
Using calculation, show that det (A) det (B) = det (AB)
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Aljabar Linear-UIN Sunan Gunung Djati

22. Aljabar Linear-UIN Sunan Gunung Djati
3. Let : Find k such that det (D) = Let If B = A-1 and At be a transpose of A. Find
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Aljabar Linear-UIN Sunan Gunung Djati