Gaussian Elimination:

Introduction

For a complex system of equations where the number of unknowns is greater than two, (the number of both the equations and the unknowns is equal to n), a common method of solving the system is elimination. Elimination as its name implies involves eliminating unknowns and equations in the system. To accomplish this task multiples of the first equation are subtracted from the other equations to eliminate the first variable in each equation. As a result, the first equation contains the first unknown and the rest of the equations can be treated as a subsystem of n-1 equations with n-1 unknowns. When executed repeatedly for the subsystem with reduced size, this process leads to a system in which one of the variables becomes known. This variable can then be substituted back into the previously modified equations to solve for the remaining variables. This process is demonstrated by the following example.

Example

Given the following system of 3 equations with 3 unknowns,

Summary

The elimination process can be summarized as follows:

• Observe the column of the matrix farthest to the left (unless it consists entirely of zeros).
• Swap the rows of the matrix until a non-zero entry appears in the top row of the left most column not consisting entirely of zeros.
• If this leading term is any number other than one multiply the row by its reciprocal to obtain a one.
• Subtract multiples of the first row from all other rows until a zero appears below our leading one.
• Now ignore the top row and begin the process again using the row second from the top as the top row.
• Repeat this process until the last row becomes an equation of one variable and a solution b.
• After obtaining a solution for one variable, substitute this value into the reduced matrix to obtain values for all other variables.