Determinan Aljabar Linear Elementer Introduction Sub chapter Introduction
Determinan Aljabar Linear Elementer
Introduction Sub chapter : – Introduction – Find Determinant a matrix using OBE – Find Determinant a matrix Cofactor Expansion Beberapa Aplikasi Determinan Ø Solution of system of linear equation Ø Optimization Ø Economi Model Ø etc. 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 2
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 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 3
Find Determinant of a matrix using ERO a. That’s Easy… Because determinan will be equal to multiplication of every diagonal elements b. When matrix is not triangular matrix. How ? c. 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 4
We need ERO to get triangular matrix. Solution : Rectangular matrix ~ ERO ~ Triangular matrix Influent of ERO relatively to determinant of a matrix : 1. Let matrix B be obtained from matrix A with one exchange of row, then Det (B) = - Det (A) Example : such that 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 5
2. 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 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 6
3. Let Matrix B be obtained from matrix A with third ERO, then Det (B) = Det (A) Example 3 : Then ERO is – 2 b 1 + b 2 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 7
Contoh 3 : Find determinant of matrix : Answer : 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 8
= 4 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 9
Exercise Find determinant of matrix using ERO : 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 10
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 : 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 11
• Matrix Cij is called cofactor - ij (Cij) = (-1)i+j Mij Example : Then = (– 1)3. 2 =– 2 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 12
Generally, cofactor expansion : • Find det (A) using cofactor expansion along ith row det (A) = ai 1 Ci 1 + ai 2 Ci 2 +. . . + ain Cin • Find det (A) using cofactor expansion along jth column det (A) = a 1 j C 1 j + a 2 j C 2 j +. . . + anj Cjn Example 6 : Find Det(A) using cofactor expansion : 06/01/2022 10: 54 Aljabar Linear-UIN Sunan Gunung Djati 13
Answer : We will find det (A) using cofactor expansion along 3 rd row = a 31 C 31 + a 32 C 32 + a 33 C 33 =0– 2+6 =4 06/01/2022 10: 55 Aljabar Linear-UIN Sunan Gunung Djati 14
We will find det (A) using cofactor expansion along 3 rd column = a 13 C 13 + a 23 C 23 + a 33 C 33 =0– 2+6 =4 06/01/2022 10: 55 Aljabar Linear-UIN Sunan Gunung Djati 15
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). 06/01/2022 10: 55 Aljabar Linear-UIN Sunan Gunung Djati 16
Let A be square matrix, some properties of determinant as follow : 1. Let A be invertible then 2. 3. 4. 5. 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 06/01/2022 10: 55 Aljabar Linear-UIN Sunan Gunung Djati 17
Example : Let Find adjoin matrix of A, and the find A-1 Answer : By definition cofactor : 06/01/2022 10: 55 Aljabar Linear-UIN Sunan Gunung Djati 18
Cofactor matrix of A : Then, Adjoin matrix of A : So, invers of matrix A : 06/01/2022 10: 55 Aljabar Linear-UIN Sunan Gunung Djati 19
• Find determinant matrix below using expansion and find the invers! 06/01/2022 10: 55 Aljabar Linear-UIN Sunan Gunung Djati cofactor 20
Exercise of Chapter 2 1. Find determinant matrix below using cofactor expansion ERO and 2. Let : and Using calculation, show that det (A) det (B) = det (AB) 06/01/2022 10: 55 Aljabar Linear-UIN Sunan Gunung Djati 21
3. Let : Find k such that det (D) = 29 4. Let If B = A-1 and At be a transpose of A. Find 06/01/2022 10: 55 Aljabar Linear-UIN Sunan Gunung Djati 22
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