Systems of Linear Equations: Matrices

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Systems of Linear Equations: Matrices Section 11.2 Systems of Linear Equations: Matrices Copyright © 2013 Pearson Education, Inc. All rights reserved

Write the augmented matrix of a system of linear equations. Objectives Write the augmented matrix of a system of linear equations. Perform row operations on a matrix. Solve a system of linear equations using matrices. Copyright © 2013 Pearson Education, Inc. All rights reserved

You already know two methods for solving a system of equations. Substitution and elimination Another approach to the elimination method is to use matrices. Copyright © 2013 Pearson Education, Inc. All rights reserved

The matrix used to represent a system of linear equations is call an augmented matrix. Copyright © 2013 Pearson Education, Inc. All rights reserved

Write the augmented matrix of each system of equations. Copyright © 2013 Pearson Education, Inc. All rights reserved

If the constants are not included, the resulting matrix is called the coefficient matrix. Copyright © 2013 Pearson Education, Inc. All rights reserved

Row operations can be used to solve a system of equations written as an augmented matrix. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

The advantages of solving a system of equations this way are To solve a system of linear equations using matrices, we use row operation son the augmented matrix of the system to obtain a matrix that is in row echelon form. The advantages of solving a system of equations this way are The process is algorithmic meaning that a computer can perform the repetitive steps. The process works for any number of variables or equations. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

The good news is that the calculator can do this for you! To enter the matrix into the calculator: MATRIX (2nd x‒1) EDIT: enter # by # and entries To get row echelon form: MATRIX MATH A: ref( ) (You can display the answers and fractions using ►FRAC if you prefer.) Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Reduced Row Echelon Form Reduced row echelon form is even better because the solution is given without having to substitute. The method to get a matrix in this form is called Gauss-Jordan elimination. Copyright © 2013 Pearson Education, Inc. All rights reserved

The calculator can do this too! MATRIX MATH B:rref( ) Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

The system is inconsistent. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Homework 11.2 #5, 7, 13, 15, 17-29 odd #37, 47, 51, 59 (with calculator) Copyright © 2013 Pearson Education, Inc. All rights reserved