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Matrices and Systemsof Equations8.1

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< Write matrices and identify their orders.

< Perform elementary row operations on matrices.

< Use matrices and Gaussian elimination to solve systems of linear equations.

< Use matrices and Gauss-Jordan elimination to solve systems of linear equations.

Objectives

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Matrices

In this section, you will study a streamlined technique for solving systems of linear equations. This technique involves the use of a rectangular array of real numbers called a matrix. The plural of matrix is matrices.

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Matrices

The entry in the i th row and j th column is denoted by the double subscript notation aij.

For instance, a23 refers to the entry in the second row, third column.

A matrix that has only one row is called a row matrix, and a matrix that has only one column is called a column matrix.

A matrix having m rows and n columns is said to be of order m ´ n.

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Matrices

If m = n, then the matrix is square of order m ´ m(or n ´ n). For a square matrix, the entries a11, a22, a33, . . . are the main diagonal entries.

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Example 1 – Order of Matrices

Determine the order of each matrix.a. b.

c. d.

Solution:a. This matrix has one row and one column. The order of

the matrix is 1 ´ 1.

b. This matrix has one row and four columns. The order ofthe matrix is 1 ´ 4.

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Example 1 – Solution

c. This matrix has two rows and two columns. The order ofthe matrix is 2 ´ 2.

d. This matrix has three rows and two columns. The orderof the matrix is 3 ´ 2.

cont’d

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Matrices

A matrix derived from a system of linear equations (each written in standard form with the constant term on the right) is the augmented matrix of the system.

Moreover, the matrix derived from the coefficients of the system (but not including the constant terms) is the coefficient matrix of the system.

System:x – 4y + 3z = 5

–x + 3y – z = –32x – 4z = 6

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Matrices

Augmented Matrix:

Coefficient Matrix:

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Matrices

Note the use of 0 for the coefficient of the missing y-variable in the third equation, and also note the fourth column of constant terms in the augmented matrix.

When forming either the coefficient matrix or the augmented matrix of a system, you should begin by vertically aligning the variables in the equations and using zeros for the coefficients of the missing variables.

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Elementary Row Operations

You have studied three operations that can be used on a system of linear equations to produce an equivalent system.

1. Interchange two equations.

2. Multiply an equation by a nonzero constant.

3. Add a multiple of an equation to another equation.

In matrix terminology, these three operations correspond to elementary row operations.

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Elementary Row Operations

An elementary row operation on an augmented matrix of a given system of linear equations produces a new augmented matrix corresponding to a new (but equivalent) system of linear equations.

Two matrices are row-equivalent when one can be obtained from the other by a sequence of elementary row operations.

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Example 3 – Elementary Row Operations

a. Interchange the first and second rows of the originalmatrix.

Original Matrix

New Row-Equivalent Matrix

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Example 3 – Elementary Row Operations

b. Multiply the first row of the original matrix by

Original Matrix

New Row-Equivalent Matrix

cont’d

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Example 3 – Elementary Row Operations

c. Add –2 times the first row of the original matrix to thethird row.

Original Matrix

New Row-Equivalent Matrix

Note that the elementary row operation is written beside the row that is changed.

cont’d

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Gaussian Elimination with Back-Substitution

The term echelon refers to the stair-step pattern formed by the nonzero entries of the matrix. To be in this form, a matrix must have the following properties.

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Gaussian Elimination with Back-Substitution

It is worth noting that the row-echelon form of a matrix is not unique.

That is, two different sequences of elementary row operations may yield different row-echelon forms.

However, the reduced row-echelon form of a given matrix is unique.

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Gaussian Elimination with Back-Substitution

Gaussian elimination with back-substitution works well for solving systems of linear equations by hand or with a computer.

For this algorithm, the order in which the elementary row operations are performed is important.

You should operate from left to right by columns, using elementary row operations to obtain zeros in all entries directly below the leading 1’s.

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Example 6 – Gaussian Elimination with Back-Substitution

Solve the systemy + z – 2w = –3

x + 2y – z = 22x + 4y + z – 3w = –2x – 4y – 7z – w = –19

Solution:

Write augmented matrix.

.

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Example 6 – Solution cont’d

Interchange R1 and R2 so first column has leading 1 in upper left corner.

Perform operations on R3 and R4 so first column has zeros below its leading 1.

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Example 6 – Solution cont’d

Perform operations on R4so second column haszeros below its leading 1.

Perform operations on R3and R4 so third and fourthcolumns have leading 1’s.

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Example 6 – Solution

The matrix is now in row-echelon form, and the corresponding system is

x + 2y – z = 2y + z – 2w = –3

z – w = –2w = 3

Using back-substitution, you can determine that the solution is x = –1, y = 2, z = 1, and w = 3.

cont’d

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Gaussian Elimination with Back-Substitution

The procedure for using Gaussian elimination with back-substitution is summarized below.

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Gaussian Elimination with Back-Substitution

When solving a system of linear equations, remember that it is possible for the system to have no solution.

If, in the elimination process, you obtain a row of all zeros except for the last entry, then the system has no solution, or is inconsistent.

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Gauss-Jordan Elimination

With Gaussian elimination, elementary row operations are applied to a matrix to obtain a (row-equivalent) row-echelon form of the matrix.

A second method of elimination, called Gauss-Jordan elimination, after Carl Friedrich Gauss and Wilhelm Jordan (1842–1899), continues the reduction process until a reduced row-echelon form is obtained.

This procedure is demonstrated in Example 8.

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Example 8 – Gauss-Jordan Elimination

Use Gauss-Jordan elimination to solve the systemx – 2y + 3z = 9

–x + 3y = –4.2x – 5y + 5z = 17

Solution: The row-echelon form of the linear system can be obtained using the Gaussian elimination.

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Example 8 – Solution

Now, apply elementary row operations until you obtain zeros above each of the leading 1’s, as follows.

Perform operations on R1so second column has azero above its leading 1.

Perform operations on R1and R2 so third column haszeros above its leading 1.

cont’d

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Example 8 – Solution

The matrix is now in reduced row-echelon form. Converting back to a system of linear equations, you have

x = 1y = –1.z = 2

Now you can simply read the solution, x = 1, y = –1, and z = 2 which can be written as the ordered triple (1, –1, 2).

cont’d