Copyright © Cengage Learning. All rights reserved. 2 SYSTEMS OF LINEAR EQUATIONS AND MATRICES Solve p. 87 #45 by elimination

Copyright © Cengage Learning. All rights reserved. 2.2 Systems of Linear Equations: Unique Solutions

3 Augmented Matrices

4 Observe from the preceding example that the variables x, y, and z play no significant role in each step of the reduction process, except as a reminder of the position of each coefficient in the system. With the aid of matrices, which are rectangular arrays of numbers, we can eliminate writing the variables at each step of the reduction and thus save ourselves a great deal of work.

5 Augmented Matrices For example, the system 2x + 4y + 6z = 22 3x + 8y + 5z = 27 –x + y + 2z = 2 may be represented by the matrix The augmented matrix representing System (5) (5) (6)

6 Augmented Matrices The submatrix consisting of the first three columns of Matrix (6) is called the coefficient matrix of System (5). The matrix itself, Matrix (6), is referred to as the augmented matrix of System (5) since it is obtained by joining the matrix of coefficients to the column (matrix) of constants. The vertical line separates the column of constants from the matrix of coefficients.

7 Example Write the augmented matrix corresponding to each equivalent system in the given systems. Solution: The required sequence of augmented matrices follows. Equivalent System Augmented Matrix a. 2x + 4y + 6z = 22 3x + 8y + 5z = 27 –x + y + 2z = 2

8 Example – Solution b. x + 2y + 3z = 11 3x + 8y + 5z = 27 –x + y + 2z = 2 c. x + 2y + 3z = 11 2y – 4z = –6 –x + y + 2z = 2 cont’d

9 Example 2 – Solution d. x = 3 y = 1 z = 2 cont’d

10 Augmented Matrices The augmented matrix on the previous slide is an example of a matrix in row-reduced form. In general, an augmented matrix with m rows and n columns (called an m  n matrix) is in row-reduced form if it satisfies the following conditions.

11 Example Determine which of the following matrices are in row-reduced form. If a matrix is not in row-reduced form, state the condition that is violated.

12 Example – Solution The matrices in parts (a)–(c) are in row-reduced form. d. This matrix is not in row-reduced form. Conditions 3 and 4 are violated: The leading 1 in row 2 lies to the left of the leading 1 in row 1. Also, column 3 contains a leading 1 in row 3 and a nonzero element above it. e. This matrix is not in row-reduced form. Conditions 2 and 4 are violated: The first nonzero entry in row 3 is a 2, not a 1. Also, column 3 contains a leading 1 and has a nonzero entry below it.

13 Example – Solution f. This matrix is not in row-reduced form. Condition 2 is violated: The first nonzero entry in row 2 is not a leading 1. g. This matrix is not in row-reduced form. Condition 1 is violated: Row 1 consists of all zeros and does not lie below the nonzero rows. cont’d

14 Practice p. 86 Exercises #1 & 2