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5.2Matrix Solution of Linear Systems

The Gauss-Jordan Method Special Systems

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Matrix Solutions

Since systems of linear equations occur in so many practical situations, computer methods have been developed for efficiently solving linear systems. Computer solutions of linear systems depend on the idea of a matrix (plural matrices), a rectangular array of numbers enclosed in brackets. Each number is called an element of the matrix.

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Gauss-Jordan Method

Matrices in general are discussed in more detail later in this chapter. In this section, we develop a method for solving linear systems using matrices. As an example, start with a system and write the coefficients of the variables and the constants as a matrix, called the augmented matrix of the system.

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Gauss-Jordan Method

Linear system of equations

3 2 1x y z 2 2x y z

2x y z

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Gauss-Jordan Method

The vertical line, which is optional, separates the coefficients from the constants. Because this matrix has 3 rows (horizontal) and 4 columns (vertical), we say its size is 3 4 (read “three by four”). The number of rows is always given first. To refer to a number in the matrix, use its row and column numbers.

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Matrix Row Transformations

For any augmented matrix of a system of linear equations, the following row transformations will result in the matrix of an equivalent system.1. Interchange any two rows.2. Multiply or divide the elements of any row by a nonzero real number.3. Replace any row of the matrix by the sum of the elements of that row and a multiple of the elements of another row.

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Gauss-Jordan Method

The Gauss-Jordan method is a systematic technique for applying matrix row transformations in an attempt to reduce a matrix to diagonal form, with 1s along the diagonal, such as

from which the solutions are easily obtained. This form is also called reduced-row echelon form.

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Using the Gauss-Jordan Method to Put a Matrix Into diagonal Form

Step 1 Obtain 1 as the first element of the first column.

Step 2 Use the first row to transform the remaining entries in the first column to 0.

Step 3 Obtain 1 as the second entry in the second column.

Step 4 Use the second row to transform the remaining entries in the second column to 0.

Step 5 Continue in this manner as far as possible.

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Example 1 USING THE GAUSS-JORDAN METHOD

Solve the system.

Solution

3 4 1x y 5 2 19x y

Both equations are in the same form, with variable terms in the same order on the left, and constant terms on the right.

Write the augmented matrix.

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Example 1 USING THE GAUSS-JORDAN METHOD

The goal is to transform the augmented matrix into one in which the value of the variables will be easy to see. That is, since each column in the matrix represents the coefficients of one variable, the augmented matrix should be transformed so that it is of the form for real numbers k and j. Once the augmented matrix is in this form, the matrix can be rewritten as a linear system to get

This form is our goal.

x k

.y j

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Example 2 USING THE GAUSS-JORDAN METHOD

Solve the system.5 6x y z

3 3 10x y z 3 2 5x y z

Solution

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Summary of Possible Cases

When matrix methods are used to solve a system of linear equations and theresulting matrix is written in diagonal form:1. If the number of rows with nonzero elements to the left of the vertical line is equal to the number of variables in the system, then the system has a single solution. See Examples 1 and 2.

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Summary of Possible Cases

2. If one of the rows has the form [0 0 0 a] with a ≠ 0, then the system has no solution. See Example 3.

3. If there are fewer rows in the matrix containing nonzero elements than the number of variables, then the system has either no solution or infinitely many solutions. If there are infinitely many solutions, give the solutions in terms of one or more arbitrary variables. See Example 4.