Matrices and Systems of Equations 8.1 Copyright © Cengage Learning. All rights reserved.

Objectives Write matrices and identify their orders. Perform elementary row operations on matrices.

Matrices

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.

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.

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.

Example 1 – Order of Matrices Determine the order of each matrix. a. b. c. d.

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 = –3 2x – 4z = 6

Matrices Augmented Matrix: Coefficient Matrix:

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.

Elementary Row Operations

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.

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.

Example 3 – Elementary Row Operations a. Interchange the first and second rows of the original matrix. Original Matrix New Row-Equivalent Matrix

Example 3 – Elementary Row Operations cont’d b. Multiply the first row of the original matrix by Original Matrix New Row-Equivalent Matrix

Example 3 – Elementary Row Operations cont’d c. Add –2 times the first row of the original matrix to the third row. Original Matrix New Row-Equivalent Matrix Note that the elementary row operation is written beside the row that is changed.

8.1 Example – Worked Solutions

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 of the matrix is 1  4.

Example 1 – Solution cont’d c. This matrix has two rows and two columns. The order of the matrix is 2  2. d. This matrix has three rows and two columns. The order of the matrix is 3  2.