la_m2_handouts

la_m2_handouts - MAC 2103 Module 2 Systems of Linear...

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1 1 MAC 2103 Module 2 Systems of Linear Equations and Matrices II 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to : 1. Find the inverse of a square matrix. 2. Determine whether a matrix is invertible. 3. Construct and identify elementary matrices; represent A and A -1 as a product of elementary matrices. 4. Solve systems of linear equations by using the inverse matrix. 5. Identify diagonal, triangular and symmetric matrices. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.
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2 3 Rev.09 Systems of Linear Equations and Matrices II http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Inverses Inverses Elementary Matrices Elementary Matrices Systems of Equations and Invertibility Diagonal, Triangular, and Symmetric Matrices Diagonal, Triangular, and Symmetric Matrices There are four major topics in this module: 4 Rev.F09 Inverse of a Square Matrix http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Let A represent a square matrix as follows: The inverse of matrix A can be obtained as follows: One important condition: This will let us know whether the matrix is invertible or not. What happens if ad - bc = 0 ? The matrix is not invertible; it has no inverse. If matrix A has no inverse, then A is said to be singular . A = a b c d ! " # $ % ad ! bc " 0 A ! 1 = 1 ad ! bc d ! b ! c a " # $ %
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3 5 Rev.F09 Example: Finding an Inverse http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Let A be a square matrix as follows: The inverse of matrix A is: Note: The matrix is invertible because ad - bc produces a nonzero value. How do we know the resulting matrix is the inverse of A? A = 1 ! 1 ! 3 6 " # $ % A ! 1 = 1 (1)(6) ! ( ! 1)( ! 3) 6 1 3 1 " # $ % = 1 6 ! 3 6 1 3 1 " # $ % = 1 3 6 1 3 1 " # $ % = 2 1 3 1 1 3 " # $ $ % = A ! 1 = 1 ad ! bc d ! b ! c a " # $ % 6 Rev.F09 Example: Finding an Inverse (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. How do we know the resulting matrix is the inverse of A? Multiply the two matrices: The product of a matrix and its inverse matrix is the identity matrix I . Notice that the inverse matrix of an inverse matrix is the original matrix. A = 1 ! 1 ! 3 6 " # $ % AA ! 1 = 1 ! 1 ! 3 6 " # $ % 2 1 3 1 1 3 " # $ $ % = 1 0 0 1 " # $ % = I = ( A ! 1 ) ! 1 A ! 1 A ! 1 = 2 1 3 1 1 3 " # $ $ % A ! 1 A = 2 1 3 1 1 3 " # $ $ % 1 ! 1 ! 3 6 " # $ % = 1 0 0 1 " # $ % = I = A ! 1 ( A ! 1 ) ! 1
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4 7 Rev.F09 Can We Use the Gauss-Jordan Elimination Method to Find the Inverse? http://faculty.valenciacc.edu/ashaw/
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la_m2_handouts - MAC 2103 Module 2 Systems of Linear...

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