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

Section 2.3 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) Determinant of a matrix 1 / 14

Goal: 1 To define the determinant of a matrix. d 2 To find the determinant of a matrix using cofactor expansion (Section i E 2.1). a 3 To find the determinant of a matrix using row reduction (Section 2.2). l l u 4 Explore the properties of the determinant and its relation to the d b inverse. (Section 2.3) A 5 To solve linear system using the Cramer’s rule. (Section 2.3) . r 6 The equation A x = b (Section 2.3) D . Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 2 / 14

Properties of the determinant 1 det ( kA ) = k n det ( A ) d i 2 E det ( A + B ) � = det ( A ) + det ( B ) a l l u d 3 b det ( AB ) = det ( A ) · det ( B ) A . r D 4 (Corollary) 1 det ( A − 1 ) = det ( A n ) = ( det ( A )) n , det ( A ) 5 If det ( A ) � = 0, then A has an inverse (invertible) Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 3 / 14

Example 1 Assume A is 5 × 5 matrix for which det ( A ) = − 3 Find the following: 1 det ( 3 A ) d i E 2 det ( A − 1 ) a l l u d 3 det ( A T ) b A . r D 4 det ( A 6 ) 5 det (( 2 A ) − 1 ) Solution: Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 4 / 14

Example 2 Use determinant to decide whether the given matrix is invertible or not   2 0 − 1 A = 0 2 3   d − 1 0 5 i E Solution: a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 5 / 14

Example 3 Find the value(s) of k for which A is invertible.   2 1 0 � 3 � k A = , A = 2 k k   3 k d 2 4 2 i E Solution: a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 6 / 14

Adjoint matrix Definition 4 Let A ∈ Mat ( n , n , R ) and C ij is the cofacotr of a ij , then the matrix with entries ( C ij ) is called the matrix of cofactors from A . The transpose of d this matrix is called the adjoint of A and is denoted by adj ( A ) . i E Example 5 a l l u Use the adjoint method to find the inverse (if exists) to the following d matrices: b   A − 2 4 3 A = 1 2 0 .   r 2 − 1 − 2 D Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 7 / 14

Theorem 6 1 A − 1 = det ( A ) adj ( A ) Example 7 d i Use the adjoint method to find the inverse (if exists) to the following E matrices: a  − 2 4 3  l l u A = 1 2 0 d   b 2 − 1 − 2 A Solution: . r D Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 8 / 14

Example 8 Use the adjoint method to find the inverse (if exists) to the following matrices:   3 0 0 A = − 2 1 0   d 4 3 2 i E Solution: a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 9 / 14

Example 9 Use the adjoint method to find the inverse (if exists) to the following matrices:   1 0 0 A = 0 cos θ − sin θ   d 0 sin θ cos θ i E Solution: a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 10 / 14

Cramer’s Rule Theorem 10 If A x = b is a system of n linear equations in n unknowns such that det ( A ) = � = 0 , then the system has a unique solution given by d i E x 1 = det ( A 1 ) det ( A ) , x 2 = det ( A 2 ) det ( A ) , . . . x n = det ( A n ) det ( A ) , a l l u where A j is the matrix obtained by replacing the entries in the jth column d b of A by the entries in the matrix A .   b 1 r D b 2     ·   b =   ·     ·   b n Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 11 / 14

Example 11 Solve using Cramer’s rule the following system of linear equations 3 x 1 + x 2 = 2 4 x 1 + x 2 = 3 d i E Solution: a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 12 / 14

Example 12 Solve using Cramer’s rule the following system of linear equations 3 x 1 + 5 x 2 = 7 6 x 1 + 2 x 2 + 4 x 3 = 10 d − x 1 + 4 x 2 − 3 x 3 = 0 i E a l l u Solution: d b A . r D Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 13 / 14

The equation A x = b d Theorem 13 i E The following are equivalent: a l 1 A is invertible. l u d 2 det ( A ) � = 0 . b 3 The reduced row echelon form is I n . A 4 A x = b is consistent for every n × 1 matrix b . . r D 5 A x = b has a unique solution for every n × 1 matrix b . Dr. Abdulla Eid (University of Bahrain) Determinant of a matrix 14 / 14