Gaussian Elimination
About Gaussian Elimination (Row Reduction)
Gaussian elimination is a method for solving a system of linear equations. The procedure can be used for any number of equations in any number of variables. Given a system of \(n\) equations in \(m\) variables \[ \begin{align*} a_{11}x_1 + a_{12}x_2 + &\cdots + a_{1m}x_m = y_1 \\ a_{21}x_1 + a_{22}x_2 + &\cdots + a_{2m}x_m = y_2 \\ &\vdots \\ a_{n1}x_1 + a_{n2}x_2 + &\cdots + a_{nm}x_m = y_n \end{align*} \] we can create the augmented matrix \[ \left[\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1m} & y_1 \\ a_{21} & a_{22} & \cdots & a_{2m} & y_2 \\ \vdots&\vdots& \ddots & \vdots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nm} & y_n \\ \end{array}\right] \] Elementary row operations are performed on this system to reduce it to row echelon form. If the matrix with entries \((a_{ij})\) is an invertible, square matrix, then the left side of the augmented matrix can be reduced to the identity and the system has a unique solution. Otherwise, the system may have multiple or no solutions. The reduced form of the matrix is independent of the operations chosen.Using the Applet
This applet generates a system of 3 linear equations in 3 variables, which will have a unique solution (indeterminate systems will not appear in this applet). Row operations can be performed on the system using the controls under the augmented matrix. When the matrix on the left is reduced to the identity, the process is complete. At this point, the right column of the augmented matrix displays the solution to the system.On large screens, the elementary matrix of the operation will be displayed, and the operations performed will be recorded next to the augmented matrix.