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Solving systems using matrices4.4 Solving systems using matrices
“A” Matrix “The Matrix” A matrix is a rectangular array of numbers.“The Matrix” is a movie with Keanu Reeves. “The Matrix”
Example of a matrix Columns Rows ElementNote: A Square matrix has the same # of rows and columns
Writing an Augmented MatrixLinear Equations 1: Linear Equations 2: Augmented Matrix Note these are Standard Form
Writing an Augmented MatrixLinear Equations 1: Linear Equations 2: Augmented Matrix Note these are Standard Form EX. 1
Writing an Augmented MatrixLinear Equations 1: Linear Equations 2: Augmented Matrix Write in Standard Form!!! EX. 2
Row Transformations All numbers in a row may be multiplied or divided by any nonzero real number. You can replace rows by adding them to other rows and placing the sum in the row.
Transformations Example 1All numbers in a row may be multiplied or divided by any nonzero real number. Multiply R1 by -2 =
Transformations Example 2All numbers in a row may be multiplied or divided by any nonzero real number. Divide R2 by 3 =
Example 2 ANSWER
Transformations Example 3All numbers in a row may be multiplied or divided by any nonzero real number. Multiply R1 by 2 and multiply R2 by -4 =
Example 3 ANSWER
Transformations Example 4You can replace rows by adding them to other rows and placing the sum in the row. Replace R1 with R1+R2 =
Transformations Example 5You can replace rows by adding them to other rows and placing the sum in the row. Replace R2 with R1-R2 =
Example 5 ANSWER
Transformations Example 6Replace R1 with : -2R1 + R2 =
Example 6 ANSWER Note: R2 does not change!!!!
Transformations Example 7Replace R2 with : -1/2R2 – R1 =
Example 7 ANSWER Note: R1 does not change!!!!
Triangular form The 1’s and the 0 in these locationsa, p, and q are just constants
Use row transformation to get a matrix in triangular form1.Work in column 1 to get the one. 2. Get the zero in column 1. 3. Get the 1 in column 2. 1st 2nd 3rd
Triangular form Example 1Write the matrix in Triangular form =
Example 1 Steps 1st : 1/6 R1 2nd : Replace R2 with 10R1 + R23rd : -1/28 R2 Let’s Look at it !
Example 1 ANSWER
Triangular form Example 2Write the Linear Equations in standard form. Write the Augmented Matrix. Get the matrix in Triangular Form. Write the matrix back into Standard form. Solve for x and y.
Put in Standard form. 2. Write the Augmented Matrix
3. Try for Triangular Form.4. Back to Standard Form.
5. Solve for x and y. Looking here. Therefore ( 7/2 , -1)y = -1, now substitute into equation 1. x = 7/2 Therefore ( 7/2 , -1) is where the lines cross.
Make sure to review these notes!
Gaussian Elimination Matrices Solutions By Dr. Julia Arnold.
§ 3.4 Matrix Solutions to Linear Systems.
4.3 Matrix Approach to Solving Linear Systems 1 Linear systems were solved using substitution and elimination in the two previous section. This section.
Chapter 4 Systems of Linear Equations; Matrices Section 2 Systems of Linear Equations and Augmented Matrics.
1 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-1 Systems of Equations and Inequalities Chapter 4.
Matrices. Special Matrices Matrix Addition and Subtraction Example.
10.1 Gaussian Elimination Method
Arithmetic Operations on Matrices. 1. Definition of Matrix 2. Column, Row and Square Matrix 3. Addition and Subtraction of Matrices 4. Multiplying Row.
Section 8.1 – Systems of Linear Equations
HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems College Algebra.
Multivariate Linear Systems and Row Operations.
Matrix Solution of Linear Systems The Gauss-Jordan Method Special Systems.
1 1.1 © 2012 Pearson Education, Inc. Linear Equations in Linear Algebra SYSTEMS OF LINEAR EQUATIONS.
Notes 7.3 – Multivariate Linear Systems and Row Operations.
Barnett/Ziegler/Byleen Finite Mathematics 11e1 Review for Chapter 4 Important Terms, Symbols, Concepts 4.1. Systems of Linear Equations in Two Variables.
AN INTRODUCTION TO ELEMENTARY ROW OPERATIONS Tools to Solve Matrices.
Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Solving Systems of Equations by matrices
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Three variables Systems of Equations and Inequalities.
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