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HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Hawkes Learning Systems College Algebra Section 8.2: Matrix Notation and Gaussian Elimination HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists 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 Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists 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 m n matrix (read “m by n”), or of order m n . By convention, the number of rows is always stated first. 5 9 7 A 2 6 0 A is a 2x3 matrix. HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists 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, aij refers to the element in the i th row and the j th column of A. 5 9 7 A 2 6 0 a12 9 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: 5 9 2 6 A 7 0 5 3 a. The order of A . A has 4 rows and 2 columns, so A is a 4x2 matrix. 4 2 b. The value of a32. The first subscript refers to the row and the second subscript refers to the column, so find the 0 entry in the 3rd row and 2nd column. c. The value of a11 . Similarly, find the entry in the 1st row, 1st column. 5 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 3 x 7 y 4 3 7 4 is . 9 x 4 y 9 1 4 HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 2: Linear Systems and Augmented Matrices Construct the augmented matrix for the linear system. 2 x 5 y 1 6 z 6 x 12 z 2 y 3 z 3 x 7 y 9 2x 5 y 6z 1 2x y 4z 2 3x 7 y z 9 2 5 6 1 2 1 4 2 3 7 1 9 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. 0 8 3 5 12 First, we need to assign each of the 9 1 0 0 7 coefficient columns to a variable. 5 Now we can create the system of equations. 2 0 0 2 a b c d 0a 8b 3c 5d 12 9a b 0c 0d 7 2a 0b 0c 2d 5 8b 3c 5d 12 9a b 7 2a 2d 5 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. 1 2 3 10 0 1 4 7 0 0 1 1 If we translate this back into system form we obtain x 2 y 3z 10 y 4 z 7 z 1 and can easily solve for the variables by back substitution. z 1 y 4 1 7 x 2 3 3 1 10 x7 y 3 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. 1 9 1 1 A 0 1 0 2 0 0 1 3 However, the matrix below is not in row echelon form 1 2 3 4 B 0 4 7 2 0 6 0 0 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 A is an augmented matrix corresponding to a given system of equations. Each of the following operations on A results in the augmented matrix of an equivalent system. In the notation, Ri refers to row i of the matrix A. 1. Rows i and j can be interchanged. (Denoted Ri R j ) 2. Each entry in row i can be multiplied by a non-zero constant c . (Denoted cRi ) 3. Row j can be replaced with the sum of itself and a constant multiple of row i . (Denoted cRi R j ) HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 4: Gaussian Elimination and Row Echelon Form Use Gaussian Elimination to solve the system. 7 x y 4 z 11 x 3y 2 z 13 6 x 2 y 3z 22 R1 R2 Augmented matrix form 7 1 4 11 1 3 2 13 6 2 3 22 1 3 2 13 7R R 7 1 4 11 1 2 6R1 R3 6 2 3 22 3 2 13 1 0 20 10 80 0 20 15 100 Continued on the next slide… HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 4: Gaussian Elimination and Row Echelon Form (Cont.) 3 2 13 1 0 20 10 80 0 20 15 100 20R2 R3 1 3 2 13 1 0 1 4 2 0 0 5 20 1 R2 20 1 R3 5 1 0 0 1 0 0 3 2 13 1 1 4 2 20 15 100 3 2 13 1 1 4 2 0 1 4 The final matrix is in row echelon form. Continued on the next slide… HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 4: Gaussian Elimination and Row Echelon Form (Cont.) 1 3 2 13 1 0 1 4 2 0 0 1 4 Now we can solve for x, y and z. z 4 Given by the last row of the matrix. Plug the value found for z into the equation 1 y 4 4 given by the 2nd row of the matrix. 2 y2 x 3 2 2 4 13 x 1 1,2, 4 Plug the values found for y and z into the 1st row of the matrix. The solution set to this system. HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 5: Gaussian Elimination and Row Echelon Form Use Gaussian Elimination to solve the system. x y 9 z 16 x 3y 4 z 6 2 x 6 y 38z 10 R1 R2 2R1 R3 1 1 9 16 0 2 5 10 0 8 20 22 1 1 9 16 Augmented 1 3 4 6 matrix form 2 6 38 10 4R2 R3 1 1 9 16 0 2 5 10 0 0 0 62 We can stop here because 0 62 is a false statement. Therefore, the solution set 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. 1 0 0 2 0 1 0 17 0 0 1 3 HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Gauss-Jordan Elimination and Reduced Row Echelon Form Consider the last matrix obtained in Example 4. 1 3 2 13 1 5 R3 R2 1 3 0 2 1 0 1 0 2 0 1 4 2 2R3 R1 0 0 1 4 0 0 1 4 3R2 R1 1 0 0 1 0 1 0 2 0 0 1 4 Reduced row echelon form. Now we can write the system as x 1 y 2 z 4 which is equivalent to the original system, but in a form that tells us the solution to the system. HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 6: Gauss-Jordan Elimination and Reduced Row Echelon Form Use Gauss-Jordan elimination to solve the system. x 2 y 3 z 7 3x y 5 z 7 8 x 10 y 19 z 1 3R1 R2 8R1 R3 2R2 R1 6R2 R3 1 2 3 7 3 1 5 7 8 10 19 1 1 2 3 7 0 7 14 28 0 6 43 55 1 R2 7 1 2 3 7 0 1 2 4 0 6 43 55 1 0 1 1 0 1 2 4 0 0 31 31 1 R3 31 1 0 1 1 0 1 2 4 0 0 1 1 HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 6: Gauss-Jordan Elimination and Reduced Row Echelon Form (Cont.) 1 0 1 1 0 1 2 4 0 0 1 1 2R3 R2 1R3 R1 1 0 0 0 0 1 0 2 0 0 1 1 Thus, we can write this in system form x 0 y 2 z 1 and the solution set for this system is the ordered triple 0, 2,1 .