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HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Objectives o Linear systems, matrices, and augmented matrices. o Gaussian elimination and row echelon form. o Gauss-Jordan elimination and reduced row echelon form.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Linear Systems and Matrices Matrices and Matrix Notation A matrix is a rectangular array of numbers, called elements or entries of the matrix. They naturally form rows and columns. We say that a matrix with m rows and n columns is an matrix (read “m by n”), or of order. By convention, the number of rows is always stated first. A is a 2x3 matrix.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Linear Systems and Matrices Matrices are often labeled with capital letters. The same letter in lower case, with a pair of subscripts attached, is usually used to refer to its individual elements. For instance, if A is a matrix, refers to the element in the row and the column of A.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 1: Linear Systems and Matrices Given the matrix below, determine the following: a.The order of. b.The value of. c.The value of. A has 4 rows and 2 columns, so A is a 4x2 matrix. The first subscript refers to the row and the second subscript refers to the column, so find the entry in the 3 rd row and 2 nd column. Similarly, find the entry in the 1 st row, 1 st column.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Linear Systems and Augmented Matrices Augmented Matrices The augmented matrix of a linear system of equations is a matrix consisting of the coefficients of the variables, with an adjoined column consisting of the constants from the right-hand side of the system. The matrix of coefficients and the column of constants are customarily separated by a vertical bar. For example, the augmented matrix for the system is.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 2: Linear Systems and Augmented Matrices Construct the augmented matrix for the linear system. Our first step is to write each equation in standard form. Now we can convert the coefficients and constants into an augmented matrix.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 3: Linear Systems and Augmented Matrices Construct the linear system for the augmented matrix. First, we need to assign each of the coefficient columns to a variable. Now we can create the system of equations.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Gaussian Elimination and Row Echelon Form Consider the following augmented matrix. If we translate this back into system form we obtain and can easily solve for the variables by back substitution.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Gaussian Elimination and Row Echelon Form The point of Gaussian elimination is that it transforms an arbitrary augmented matrix into a form (called row echelon form) like the one on the previous slide. Row Echelon Form A matrix is in row echelon form if: 1.The first non-zero entry in each row is 1. 2.Every entry below each 1 (called a leading 1) is 0, and each leading 1 appears one digit farther to the right than the leading 1 in the previous row. 3.All rows consisting entirely of 0’s appear at the bottom.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Row Echelon Form The matrix below is in row echelon form. However, the matrix below is not in row echelon form because the first non-zero entries in the second and third rows are not 1.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Gaussian Elimination and Row Echelon Form Elementary Row Operations Assume is an augmented matrix corresponding to a given system of equations. Each of the following operations on results in the augmented matrix of an equivalent system. In the notation, refers to row of the matrix. 1.Rows and can be interchanged. (Denoted ) 2.Each entry in row can be multiplied by a non-zero constant. (Denoted ) 3.Row can be replaced with the sum of itself and a constant multiple of row. (Denoted )

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 4: Gaussian Elimination and Row Echelon Form Use Gaussian Elimination to solve the system. Augmented matrix form Continued on the next slide…

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 4: Gaussian Elimination and Row Echelon Form (Cont.) The final matrix is in row echelon form. Continued on the next slide…

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 4: Gaussian Elimination and Row Echelon Form (Cont.) Now we can solve for x, y and z. Given by the last row of the matrix. Plug the value found for z into the equation given by the 2 nd row of the matrix. Plug the values found for y and z into the 1 st row of the matrix. The solution set to this system.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 5: Gaussian Elimination and Row Echelon Form Use Gaussian Elimination to solve the system. Augmented matrix form We can stop here because is a false statement. Therefore,

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Gauss-Jordan Elimination and Reduced Row Echelon Form The goal of Gauss-Jordan elimination is to put a given matrix into reduced row echelon form. Reduced Row Echelon Form A matrix is said to be in reduced row echelon form if: 1.It is in row echelon form. 2.Each entry above a leading 1 is also 0. For example, the following matrix is in reduced row echelon form.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Gauss-Jordan Elimination and Reduced Row Echelon Form Consider the last matrix obtained in Example 4. Reduced row echelon form. Now we can write the system as which is equivalent to the original system, but in a form that tells us the solution to the system.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 6: Gauss-Jordan Elimination and Reduced Row Echelon Form Use Gauss-Jordan elimination to solve the system.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 6: Gauss-Jordan Elimination and Reduced Row Echelon Form (Cont.) Thus, we can write this in system form and the solution set for this system is the ordered triple