Explanation Augmented matrix: The system of equation can be written in the form of matrix called as augmented matrix. In augmented matrix, the elements are the coefficients of a set of linear equation with the constant terms of the equations at the right column of the matrix. The constant term in the matrix is separated by a dashed line. Example: [ A] = [ 1 1 7 1 − 1 − 1 ] Pivoting: The element in the diagonal of a matrix by which other elements are divided in an algorithm such as Gauss-Jordan elimination is called as pivot element. Divide all the entries in the row by the pivot appear in that row so that the pivot entry becomes 1. Obtaining zeros above and below the pivot element by using elementary row operations is called as pivoting. Gauss-Jordan elimination: Gauss-Jordan elimination is an algorithm that is used to solve systems of linear equation. It efficiently uses the elementary row operations to form a diagonal matrix. First pivot is the element in first row and first column; second pivot is the element in second row and second column; third pivot is the element in third row and third column. Pivot element is used to eliminate elements above and below it in a column by using elementary row operations. Steps followed in Gauss-Jordan elimination: Form an augmented matrix from the set of linear system equations. Select the first pivot and perform pivoting. Select the second pivot and pivot. Repeat the process until at the last row or until the pivot element is zero. If the pivot element is zero, interchange the row. Elementary row operation *Row+ : The third elementary row operation is combination of both row multiplication and row addition. It performs multiplication of a row element by nonzero number and adds the result with another row. It stores the result in the target row and the pivot row should not be changed. It is represented as, *Row+ ( 2 , [ A], 3, 1 ) Where [A] represents the matrix. 2 is the scalar, 3 is the pivot row and 1 is the target row. It multiplies the scalar with the elements in pivot row 3 and add the result to the first row. Elementary row operation *Row : The third elementary row operation is row multiplication. It performs multiplication or division of a row element by nonzero number. In a matrix, the solution will not be changed if a row is multiplied or divided by a nonzero constant. It stores the result in the target row. It is represented as, *Row ( 2 , [ A], 1 ) Where [A] represents the matrix. 2 is the scalar and 1 is the target row. It multiplies the scalar with the elements in row 1 and store the result in row 1. Conversion of augmented matrix into a linear system equation: Multiply the coefficient of matrix with the variables of the system. Insert answers on the other side of equal sign of the equation. Calculation: The system equation given is, { 4 x + y = − 2 3 x + 2 z = − 9 2 y + 3 z = − 5 The system has three equation and three variables x , y , z . Form an augmented matrix by placing the coefficients and constant in a matrix form and name it as A. [ A] = [ 4 1 0 − 2 3 0 2 − 9 0 2 3 − 5 ] Select the element in first row and first column as first pivot and perform pivoting. Perform pivoting by multiplying the row 2 of a matrix previous matrix by the scalar − 1 and add the corresponding entries to row 1. → [ 1 1 − 2 7 3 0 2 − 9 0 2 3 − 5 ] *Row+ ( − 1 , [ A], 2, 1 ) Multiply row 1 of the previous matrix by − 3 and add the corresponding entries to row 2. → [ 1 1 − 2 7 0 − 3 8 − 30 0 2 3 − 5 ] *Row+ ( − 3 , [ Ans], 1, 2 ) Select the second pivot and pivot. Multiply the row 3 of the previous matrix by 1and add the corresponding entries to row 2. → [ 1 1 − 2 7 0 − 1 11 − 35 0 2 3 − 5 ] *Row+ ( 1 , [ Ans], 3, 2 ) Multiply row 2 of previous matrix by - 1

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ISBN: 9781285697734

Author: karl J. smith

Publisher: Cengage Learning US

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Chapter 16.3, Problem 52PS

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The system of equation {4x+y=−23x+2z=−92y+3z=−5 by Gauss-Jordan method.

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Chapter 16.3, Problem 51PS

Chapter 16.3, Problem 53PS

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The system of equation { 4 x + y = − 2 3 x + 2 z = − 9 2 y + 3 z = − 5 by Gauss-Jordan method.

Solution Summary: The author explains how Gauss-Jordan elimination solves linear equations in the form of an augmented matrix.

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To solve:

The system of equation {4x+y=−23x+2z=−92y+3z=−5 by Gauss-Jordan method.

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Chapter 16 Solutions

The system of equation { 4 x + y = − 2 3 x + 2 z = − 9 2 y + 3 z = − 5 by Gauss-Jordan method.

Solution Summary: The author explains how Gauss-Jordan elimination solves linear equations in the form of an augmented matrix.

Videos

To solve: The system of equation {4x+y=−23x+2z=−92y+3z=−5 by Gauss-Jordan method.

Want to see the full answer?

Chapter 16 Solutions

To solve:

The system of equation {4x+y=−23x+2z=−92y+3z=−5 by Gauss-Jordan method.