Aljabar Linear Elementer Determinan Aljabar Linear Elementer

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. 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

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 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

Find Determinant of a matrix using ERO b. c. That’s Easy… Because determinan will be equal to multiplication of every diagonal elements When matrix is not triangular matrix. How ? 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

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 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

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 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

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 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

Aljabar Linear-UIN Sunan Gunung Djati Contoh 3 : Find determinant of matrix : Answer : 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

Aljabar Linear-UIN Sunan Gunung Djati = 4 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

Aljabar Linear-UIN Sunan Gunung Djati Exercise Find determinant of matrix using ERO : 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

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 : 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

Aljabar Linear-UIN Sunan Gunung Djati Matrix Cij is called cofactor - ij (Cij) = (-1)i+j Mij Example : Then = (– 1)3 .2 = – 2 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

Aljabar Linear-UIN Sunan Gunung Djati Generally, cofactor expansion : Find det (A) using cofactor expansion along ith row det (A) = ai1 Ci1 + ai2 Ci2 + . . . + 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 : 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

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 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

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 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

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). 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

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 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

Aljabar Linear-UIN Sunan Gunung Djati Example : Let Find adjoin matrix of A, and the find A-1 Answer : By definition cofactor : 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

Aljabar Linear-UIN Sunan Gunung Djati Cofactor matrix of A : Then, Adjoin matrix of A : So, invers of matrix A : 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

Aljabar Linear-UIN Sunan Gunung Djati Find determinant matrix below using cofactor expansion and find the invers! 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

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) 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati

Aljabar Linear-UIN Sunan Gunung Djati 3. Let : Find k such that det (D) = 29 4. Let If B = A-1 and At be a transpose of A. Find 24/02/2019 5:43 Aljabar Linear-UIN Sunan Gunung Djati