SECTION 2.2 Solving Systems of Linear

Equations Part 2

2.2 Solving Systems of Linear Equations, II

1. Pivot a Matrix2. Gaussian Elimination Method3. Infinitely Many Solutions4. Inconsistent System5. Geometric Representation of System

Pivot a Matrix

Method To pivot a matrix about a given nonzero entry:

1. Transform the given entry into a one;2. Transform all other entries in the same

column into zeros.

3

Example Pivot a Matrix

Pivot the matrix about the circled element.

4

18 6 15

5 2 4

1 [1]653 118 6 15 2

5 2 4 5 2 4

[2] 2[1]53 1 2

1 0 1

Gaussian Elimination Method

Gaussian Elimination Method to Transform a System of Linear Equations into Diagonal Form

1. Write down the matrix corresponding to the linear system.

2. Make sure that the first entry in the first column is nonzero. Do this by interchanging the first row with one of the rows below it, if necessary.

5

Gaussian Elimination Method (2)

Gaussian Elimination Method to Transform a System of Linear Equations into Diagonal Form

3. Pivot the matrix about the first entry in the first column.

4. Make sure that the second entry in the second column is nonzero. Do this by interchanging the second row with one of the rows below it, if necessary.

6

Gaussian Elimination Method (3)

Gaussian Elimination Method to Transform a System of Linear Equations into Diagonal Form

5. Pivot the matrix about the second entry in the second column.

6. Continue in this manner.

7

Infinitely Many Solutions

When a linear system cannot be completely diagonalized,1. Apply the Gaussian elimination method to as many columns as possible. Proceed from left to right, but do not disturb columns that have already been put into proper form.2. Variables corresponding to columns not in proper form can assume any value.

8

Infinitely Many Solutions (2)

3. The other variables can be expressed in terms of the variables of step 2.4. This will give the general form of the solution.

9

Example Infinitely Many Solutions

Find all solutions of

2 2 4 8

2 2

5 2 2.

x y z

x y z

x y z

10

General Solutionz = any real numberx = 3 - 2zy = 1

1 [1]2[2] ( 1)[1][3] 1 [1]

2 2 4 8 1 1 2 4

1 1 2 2 0 2 0 2

1 5 2 2 0 6 0 6

1 [2]2[1] ( 1)[2][3] 6 [2]

1 0 2 3

0 1 0 1

0 0 0 0

Inconsistent System

When using the Gaussian elimination method, if a row of zeros occurs to the left of the vertical line and a nonzero number is to the right of the vertical line in the same row, then the system has no solution and is said to be inconsistent.

11

Example Inconsistent System

Find all solutions of

3

5

2 4 4 1.

x y z

x y z

x y z

12

Because of the last row, the system is inconsistent.

[2] ( 1)[1][3] 2 [1]

1 1 1 3 1 1 1 3

1 1 1 5 0 2 2 2

2 4 4 1 0 2 2 7

1 [2]2[1] (1)[2][3] 2 [2]

1 0 0 4

0 1 1 1

0 0 0 5

Summary Section 2.2 - Part 1

The process of pivoting on a specific entry of a matrix is to apply a sequence of elementary row operations so that the specific entry becomes 1 and the other entries in its column become 0.

To apply the Gaussian elimination method, proceed from left to right and perform pivots on as many columns to the left of the vertical line as possible, with the specific entries for the pivots coming from different rows.

13

Summary Section 2.2 - Part 2

After an augmented matrix has been completely reduced with the Gaussian elimination method, all the solutions to the corresponding system of linear equations can be obtained.

If the reduced augmented matrix has a 1 in every column to the left of the vertical line, then there is a unique solution.

14

Summary Section 2.2 - Part 3

If one row of the reduced augmented matrix has the form 0 0 0 … 0 | a where a ≠ 0, then there is no solution.

Otherwise, there are infinitely many solutions. In this case, variables corresponding to columns that have not been pivoted can assume any values, and the values of the other variables can be expressed in terms of those variables.

15