Lect7 - Chapter 1. Matrices and Systems of Equations...

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Unformatted text preview: Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion (Cont’d) Theorem Let A , B be nonsingular n × n matrices. Then AB is nonsingular and ( AB )- 1 = B- 1 A- 1 . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion (Cont’d) Theorem Let A , B be nonsingular n × n matrices. Then AB is nonsingular and ( AB )- 1 = B- 1 A- 1 . Proof. Want to show (i) AB is a square matrix, (ii) AB ( B- 1 A- 1 ) = I = ( B- 1 A- 1 ) AB . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion (Cont’d) Theorem Let A , B be nonsingular n × n matrices. Then AB is nonsingular and ( AB )- 1 = B- 1 A- 1 . Proof. Want to show (i) AB is a square matrix, (ii) AB ( B- 1 A- 1 ) = I = ( B- 1 A- 1 ) AB . Recall Given a square matrix M . If M N = I = M N , then M is nonsingular and M- 1 = N . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion (Cont’d) Theorem Let A , B be nonsingular n × n matrices. Then AB is nonsingular and ( AB )- 1 = B- 1 A- 1 . Proof. Want to show (i) AB is a square matrix, (ii) AB ( B- 1 A- 1 ) = I = ( B- 1 A- 1 ) AB . Recall Given a square matrix M . If M N = I = M N , then M is nonsingular and M- 1 = N . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Transpose Definition Let A = ( a ij ) m × n be an m × n matrix. The transpose of A is the matrix ( a ji ) n × m and denoted by A T . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Transpose Definition Let A = ( a ij ) m × n be an m × n matrix. The transpose of A is the matrix ( a ji ) n × m and denoted by A T . Example . Let A = 1 2 3 , B = 1 2 3 4 5 6 . Find A T and B T . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Transpose Definition Let A = ( a ij ) m × n be an m × n matrix. The transpose of A is the matrix ( a ji ) n × m and denoted by A T . Example . Let A = 1 2 3 , B = 1 2 3 4 5 6 . Find A T and B T . Theorem Let A , B be matrices and α be a scalar. 1. ( A T ) T = A 2. ( α A ) T = α A T 3. ( A + B ) T = A T + B T 4. ( AB ) T = B T A T Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Transpose Definition Let A = ( a ij ) m × n be an m × n matrix. The transpose of A is the matrix ( a ji ) n × m and denoted by A T . Example . Let A = 1 2 3 , B = 1 2 3 4 5 6 . Find A T and B T . Theorem Let A , B be matrices and α be a scalar. 1. ( A T ) T = A 2. ( α A ) T = α A T 3. ( A + B ) T = A T + B T 4. ( AB ) T = B T A T Definition A matrix A is symmetric if A T = A . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Transpose (Cont’d) Proof....
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Lect7 - Chapter 1. Matrices and Systems of Equations...

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