BME 130-Numerical Methods in Biomedical Engineering Lab 5 - Linear Systems and Gaussian Elimination 9/19/2018 Fall 2018 DUE: 9/26/2018 Row Operations & Gaussian Elimination Gaussian Elimination is based on the use of Elementary Row Operations, which are a set of allowable computations that can be performed on a system of equations without changing its solution for Typically, they are implemented when the system is expressed in matrix form. These operations are as follows: 1. Switch the order of any two rows (equations) in a matrix. 2. Multiply any row by a nonzero constant. 3. Add any row to another row and replacing one with the sum of the two. Switching Rows Conceptualy, this operation is very easy to understand. Let's look at an example system: 2x2 4x3 16 3x1 +2x2 +x,-10 x1 +3x2 +3x 16 Our solution for variables x1,x2 and x3 does not depend on the order of how the three equations are written on the ince the mathematical relationships between the variables remains 2x1 + x2 +4x3 16 x1 + 3x2 + 3x3 = 16 3x1 2x2 x3 10 x +3x2 3x3 16 2x++4x16 x1 3x2 3x3 16 2x1 +x2 4x3 16 All produce the same solution: x 1x2x33. Therefore, since a matrix is just another way to represent the system of equations, we can apply the same reasoning to its matrix form. NOTE, however that we must apply this operation to both sides of the equation (not just the left side)! To help keep track of the entire equation in matrix form when performing row operations, we can use an augmented matrix which combines both left and right sides into a single matrix, [Alb]: 2 1 416 3 2 110 1 3 3 16 Therefore 1 41161 [2 1 4161 [3 2 110 3 2 1 101 3 3 161 3 3 16 1 3 3l 16 Activity 5 Let's test our first elementary row operation by performing some row-switching, and testing equivalency A=[2 1 4; 321; 133]; b [16:10:16]: aug [A bl switchl-aug: switchl (1,aug (3,) switchl (3,aug (,) % switch 1: swap first and third rows operations/a/matrix-row-operations BME 130 Page 6 of 10