Gaussian elimination is a method to solve systems of linear equations by transforming the matrix of coefficients into upper triangular form using elementary row operations. It has two steps: forward elimination reduces the matrix to row echelon form, and back substitution solves the system. The method is equivalent to computing an LU matrix decomposition that writes the original matrix as the product of an invertible matrix and an upper triangular matrix. Examples demonstrate solving systems by Gaussian elimination.

1. Gauss Elimination Method<br />Gaussian elimination is a method of solving a linear system (consisting of equations in unknowns) by bringing the augmented matrix<br /> <br />to an upper triangular form<br />The process of Gaussian elimination has two parts. The first part (Forward Elimination) reduces a given system to either triangular or echelon form, or results in a degenerate equation with no solution, indicating the system has no solution. This is accomplished through the use of elementary row operations. The second step uses back substitution to find the solution of the system above.<br />Stated equivalently for matrices, the first part reduces a matrix to row echelon form using elementary row operations while the second reduces it to reduced row echelon form, or row canonical form.<br />Another point of view, which turns out to be very useful to analyze the algorithm, is that Gaussian elimination computes a matrix decomposition. The three elementary row operations used in the Gaussian elimination (multiplying rows, switching rows, and adding multiples of rows to other rows) amount to multiplying the original matrix with invertible matrices from the left. The first part of the algorithm computes an LU decomposition, while the second part writes the original matrix as the product of a uniquely determined invertible matrix and a uniquely determined reduced row-echelon matrix.<br />This elimination process is also called the forward elimination method.<br />The following examples illustrate the Gauss elimination procedure.<br />EXAMPLE 2.2.11 Solve the linear system by Gauss elimination method. <br /> <br />Solution: In this case, the augmented matrix is The method proceeds along the following steps.<br />Interchange and equation (or ).<br />Divide the equation by (or ).<br />Add times the equation to the equation (or ).<br />Add times the equation to the equation (or ).<br /> <br />Multiply the equation by (or ).<br />The last equation gives the second equation now gives Finally the first equation gives Hence the set of solutions is A UNIQUE SOLUTION.<br />EXAMPLE 2.2.12 Solve the linear system by Gauss elimination method. <br /> <br />Solution: In this case, the augmented matrix is and the method proceeds as follows:<br />Add times the first equation to the second equation.<br />Add times the first equation to the third equation.<br />Add times the second equation to the third equation<br />Thus, the set of solutions is with arbitrary. In other words, the system has INFINITE NUMBER OF SOLUTIONS.<br />Bibliograpgy<br />http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node18.html<br />http://en.wikipedia.org/wiki/Gaussian_elimination<br />Numerical Methods for engineers. Steven C. Chapra-<br />Raymond P. Canale. Five edition. Mc Graw Hill <br />