d i E Determinant of a matrix a l l u d Dr. Abdulla Eid b - PowerPoint PPT Presentation

Section 2.1 d i E Determinant of a matrix a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 211: Linear Algebra Dr. Abdulla Eid (University of Bahrain) Inverse of a matrix 1 / 5

Goal: 1 To define the determinant of a matrix. d i 2 To find the determinant of a matrix using cofactor expansion (Section E 2.1). a l l 3 To find the determinant of a matrix using row reduction (Section 2.2). u d 4 Explore the properties of the determinant and its relation to the b A inverse. (Section 2.3) 5 To solve linear system using the Cramer’s rule. (Section 2.3) . r D . Dr. Abdulla Eid (University of Bahrain) Inverse of a matrix 2 / 5

Theorem 1 If A is an n × n triangular matrix (upper triangular, lower triangular, or diagonal), then det ( A ) is the product of the entries on the main diagonal of the matrix, that is det ( A ) = a 11 a 22 . . . a nn . d Theorem 2 i E (Row operations and determinant) If A is an n × n matrix. a l 1 If B ∼ A by multiplying a row of A by k, then l u d b det ( B ) = k det ( A ) A . r D 2 If B ∼ A by exchanging two rows of A, then det ( B ) = − det ( A ) 3 If B ∼ A by adding a multiple of one row to another row of A, then Dr. Abdulla Eid (University of Bahrain) Inverse of a matrix 3 / 5 det ( B ) = det ( A )

Example 3 Find det ( A ) for  1 2 4  A = − 3 3 5   7 0 6 d i E Solution: a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Inverse of a matrix 4 / 5

Example 4 Find det ( A ) for  5 2 − 2 0  3 2 − 2 0   A =   1 0 − 1 1   d 0 − 1 5 7 i E a Solution: l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Inverse of a matrix 5 / 5