Chapter 2 Section 1 Solving a System of Linear Equations (using Matrices)

The Gaussian Elimination Method Steps for the Gauss -Jordan elimination method listed on pages 64 and 63. Example 5 on page 57 is an example of using the Gauss-Jordan elimination method on a system that has been converted into a matirx

Convert the system of equations into a matrix form It is important that we first arrange the variables, and their coefficients, in the same exact order in each equation on the left side of the equal sign and have the constants on the right hand side of the equal sign. This is done so that each column will correspond to a variable (or the constant after the equal sign). Example: See Examples: 5 (page 57), 2 (page 65), 3 (page 65), and 5 (page 67).

Objective of the Gaussian Elimination Method To get the matrix in the following form: using the Three Elementary Row Operations 10 0 * * *

The Pivot Number/Row The circled number is a pivot number and the row that the pivot number is in is called the pivot row. Objective is to change the pivot number to 1 and to change all the entries above and/or below the pivot number into a 0 by using the any of the three elementary row operations.

The First Elementary Row Operations 1.Rearrange the equations in any order. Notation:R 2 R 3 Notation indicate to interchange Row 2 and Row 3

The Second Elementary Row Operation 2.Multiply an equation by a nonzero number. Notation: ( – ¼) R 2 Notation indicates to multiply – ¼ to all the coefficient in Row 2. (This operation is used to change the pivot number to a 1)

The Third Elementary Row Operation 3.Change the equation by adding to it a multiple of another equation. Notation:R 2 + ( – 5) R 3 Explanation of the notation on the next slide! (This operation is used to change values above and below the pivot number to zero)

Explanation of the Notation for the Third Elementary Row Operation Notation: R 2 + ( – 5) R 3 On a scratch piece of paper: 1.Write down the numbers in Row 2 2.Write down the result of the ( – 5) times the numbers in Row 3 below the corresponding numbers in Row 2. 3.Add the corresponding numbers together. On the original paper 4. Replace Row 2 (since it is the first row listed in the notation) by the result of the sum in the previous step in the matrix

Converting a Pivot Number to a 1 When converting a pivot number (i.e. the circled number) to a 1, use the second elementary row operation. Multiply the pivot row by the reciprocal of the circled number.

Converting Any Number Above or Below the Pivot Number to a 0 When converting a number, above or below the pivot number, to a zero, use the third elementary row operation. The row in which the number that you want to change to a zero, is listed first in the notation of the third elementary row operation. Take the number that you want to change to a zero, change the sign, and the resulting number is the number that you want to multiply to the pivot row (the second part of the third elementary row operation).

Examples Example 5 on page 57 Example 4 on page 56 (but converted into matrix form) Other examples.

Matrices on the Calculator See Appendix B page A10 – A11 for entering, editing, and erasing matrices. We will use the rref function in the calculator to perform the Gaussian Elimination Method on a matrix. –Get into the matrix mode ( [2 nd ] [x – 1 /MATRIX] ) –Select MATH –Select B:rref( and hit ENTER –Get back into the matrix mode and select the matrix you want to perform the Gaussian elimination method on –Hit enter and the matrix that appears on the calculator is the matrix that is the result of performing the Gaussian elimination method.